MATH AND MEASUREMENT
1. Given 4 x  4   1 , calculate x.
x
2. Given
1
1  (v / c )
= 10, calculate v.
2
[0.995c]
1
= 8, calculate  .
1 2
4. Given x = 2, calculate s.
3. Given
5. Given
2
= 15, calculate x.
1 x
6. Given 4-4/x = -1/x , calculate x.
8
7. Light travels at 310 m/s. Express its speed in units of mm/nsec.
[3105]
8. How many nsec are there in a century (100yr)?
9. How many kg are there in a ng?
10. A fortnight is 14 days, a furlong is 220 yards. 22 m/s is how many furlongs/fortnight?
11. If light travels 3 10 m/s express its speed in units of mm/nsec.
12. How many m are there in an inch?
13. How many nsec are there in a year?
14. Express ‘g’ in units of Mm/(ksec) .
15. Dave Barry has written about the Hubble Space Telescope that “it would have been cheaper
to take a regular telescope and put it on top of an 87-mile-high pile of $50 bills.” Estimate how
much money would be in such a pile. (This is an order-of-magnitude estimate: be aware of how
many digits are significant.)
ONE-DIMENSIONAL KINEMATICS
1 A ball travels in a vertical direction and lands with a speed of 6m/s.
a) What might the initial velocity have been, if the ball was thrown from a height of 1m?
b) How long would it have taken the ball to fall to the ground if it had started from rest and ended with a
speed of 6m/s?
2. One car travelling on a test track, accelerating at –3m/s2, can stop in a distance of 400m. A second car
travelling at 50m/s tries to stop in 400m, but still has a speed of 5m/s at the end of 400m.
a) What is the initial velocity of the first car?
b) What is the acceleration of the second car?
[49m/s, -3.1m/s2]
3. To avoid concussion in a a car accident, you need to accelerate less than about l0g or 98m/s2. To keep
from being crashed, you need the car to travel less than the length of the hood, about 1.5m.
a) How fast can a car be travelling before crashing into a brick wall, in order to avoid exceeding either
limit?
b) How long does such a collision last?
4. You throw a ball straight up in the air from ground level in such a manner that it takes 2s for it to reach
its maximum height.
a) What was its initial velocity?
b) How high did it travel?
c) How long does it take until it hits the ground again?
[20m/s, 4s]
5. Carl Lewis, the Olympic sprinter, was timed as taking 1.88s to sprint 10m, starting from rest. Assuming a
constant acceleration,
a) What was his average acceleration?
b) What was his final speed?
6. During a car crash, a car starting at 30mph slows to 0mph in a distance of 0.6m.
a) What is the average acceleration of the car in m/s2? (One mile is 1609m.)
b) How many seconds does the collision last?
[150m/s2, 0.089s]
7. A driver begins at rest at t=0s. She accelerates uniformly to 25m/s in 5s, then decelerates to 0m/s in the
next 10s, reverses gear and accelerates uniformly to a backwards 25m/s in 7s, travels 3s at this spced and
then brakes uniformly to 0m/s in 5s.
a) What is her average acceleration over the first 22s?
b)What is her instantaneous acceleration at 11s?
c) What is her net displacement after all this driving?
[-1.1m/s2, -2.5m/s2, -37.5m]
8. The area under a velocity vs time curve is equal to ...
a) maximum velocity
b) maximum acceleration
c) minimum acceleration
d) displacement
e) It has no physical significance.
9. A car goes from 0 to 80 km/hr in 6 seconds.
a) What is its acceleration (in MKS units)?
b) How are does it travel in 6 seconds?
c) How far would it take for the car to go from 0 to 80 km/hr in free fall?
10. A car travels with a speed v. With its wheels locked and skidding, its minimum stopping
distance is d. Assuming that all other factors remain constant, tripling the initial speed will
increase this stopping distance to ...
11. A ball rolls down an inclined plane with constant acceleration, starting from rest. Distances
are marked every 3s, and the second mark after the starting point is 2.3m from the starting point.
a) What is the acceleration?
b) Where is the first mark?
c) Where is the third mark?
d) What is its speed after it travels 2m?
12. Fill in the following blanks with a verbal description of what each of the items in Fig. E
represents:
a) The slope of line A represents ________________
b) The slope of line B represents ________________
c) The slope of line C represents ________________
d) The shaded area D represents ________________
Note: by ‘slope’ I mean a number that is positive or negative. I do not mean the magnitude of the
slope.
TWO-DIMENSIONAL KINEMATICS
Vectors
1. Add these two vectors analytically. Find the magnitude and direction of the sum.
Vector A: magnitude =10m/s, at 45°
Bx=2m/s; By =5m/s
2. Subtract these two vectors analytically. Find the magnitude and direction of the difference.
Px=3m/s2; Py= -2m/s2
Vector Q: magnitude = 5m/s2 at 1800
[8.2m/s2 at -14o]
3. A hiker travels 13 mi in a direction 30 east of due north, then 5 mi in a direction 60 north of due west.
a) What is the hiker’s net displacement?
b) Give directions to the hiker to get to a point 20 mi due north of the starting point. (For example, “Go 20
miles North, then 5 miles East.”)
4. Thelma and Louise drive their car off a 75m high cliff. The car starts with a horizontal velocity of 80
km/hr.
a) How many seconds later does the car hit the ground?
b) What is the diagonal distance between the edge of the cliff and the impact site?
c) What is the car’s final speed in m/s?
[3.91s, 115m, 44m/s]
5. A polar bear, seeking a yummy harp seal meal, travels first 50 miles due East, then 75 miles at 230°. How
far and in what direction must he travel in order to return to his starting point7
[57mi at 92o]
6. A runner travels for 60s at 4 m/s due north, for 100s at 3.5 m/s at 30  north of due east, and for
120s at 4 m/s due west. Calculate
a) the runner’s net displacement,
b) the runner’s average velocity,
c) the runner’s average speed,
d) the runner’s average (nonzero) acceleration.
7. A hiker travels 13 mi in a direction 30  east of due north, then 5 mi in a direction 60  north of
due west.
a) What is the hiker’s net displacement?
b) Give directions to the hiker to get to a point 20 mi due north of the starting point.
8. If I begin a journey 11 mi north of Potsdam and end, 30 minutes later, 15 mi at 15  south of
west of Potsdam, what is the magnitude and direction of my average velocity?
9. When a vector is resolved into components, they must be
a) perpendicular to each other
b) in the vertical and horizontal directions
c) of unequal lengths
d) All of the above must be true
e) None of the above is necessarily true.
10. Given the two vectors,


A , with a magnitude of 2m and a direction of 225  from the positive x-
axis, and B , with a magnitude of 3m and a direction of -30, sketch the following (use a separate
sketch for each) and solve for magnitude and direction of:
 
a) A  B.
 
b) A  B
Two-dimensional kinematics
11. A car travels due East from Canton to Potsdam (We'll call this the positive x-direction.), covering 11
miles in 15 minutes. Once there, it travels 15 miles due North (toward Massena) in 18 minutes. Assuming
both parts of the trip to be constant velocity, and assuming that the car is in motion at the start and end of
the trip,
a) What is the magnitude of the average velocity (in m/s) of the entire trip?
b) What is the magnitude of the average acceleration (in m/s2)?
12. A runner travels for 60s at 4 m/s due north, for 100s at 3.5 m/s due east, and for 120s at 4 m/s due south.
Calculate
a) the runner’s net displacement,
b) the runner’s average velocity,
c) the runner’s average speed,
d) the runner’s average (nonzero) acceleration.
[420m at -34o, 1.5m/s at -34o, 3.8m/s, -0.029m/s2 due S]
13. A runner travels around a circular track of radius 64m at a uniform speed of 6.7m/s. It takes her 30s to
travel halfway around the track. If her starting point is at the Easternmost point on this circle, and if she
maintains a constant speed throughout, and we use the Eastern direction as the positive x-axis, and North is
the positive y-direction, then calculate -- for the first 30s of her journey -a) the magnitude and direction of her average velocity.
b) the magnitude and direction of her average acceleration.
[4.3m/s West, 0.44m/s2 South]
14. A cannon is fired horizontally from the edge of a cliff, 150m above sea level. The cannonball reaches
the ocean 150m away from the base of the cliff. What is the initial velocity of the cannonball?
15. You pitch a ball at 28m/s, 45° above the horizon from a second story window 5m above the ground.
a) What are the components of the intial velocity?
b) How long does it take for the ball to hit the ground?
c) How far from the base of the building does the ball land?
16. A tennis player hits a ball 100 below the horizon at 30m/s, from a height of 2.5m.
a) How long does it take the ball to hit the ground?
b) What are the components of the velocity the instant before the ball hits the ground?
[0.36s, 30m/s and -9m/s]
17. A tennis player serves a tennis ball so that its velocity as it leaves the racket is totally horizontal. The
ball is 2.5m above the ground when served. The ball must land within about 15m from the server for the
serve to be inbounds.
a) How fast may the tennis player hit the ball (maximum speed)?
b) How long does it take the ball to reach the ground?
18. A diver jumps from a diving platform into a pool.
a) If the diver's initial velocity was 4m/s at an angle of 30° above the horizontal, what are the
components of the intial velocity?
Assume now that the platform was 4m above the level of the pool and that the diver jumps at 4m/s in a
horizontal direction.
b) How many seconds does the dive last?
c) How far from the base of the platform does the diver land?
[3.5m/s, 2m/s, 0.90s, 3.6m]
19. Adiver dives from a platform 10m above water. She leaves the diving board with a velocity such that
vox=2m/s and voy= 10m/s.
a) What are the components of the diver's final velocity as she enters the water?
b) How long is she in the air?
c) How far has she traveled in the horizontal direction when she enters the water?
[2m/s, -17m/s, 2.8s, 5.6m]
20. For the ‘monkey-and-the-blowdart’ demonstration, a gun is set at 1.5m above the floor. A ball
is fired that must travel 8m horizontally to hit a can. The ball is fired with an initial velocity v at a
45 angle. Calculate the minimum value for v such that the ball hits the can before it hits the
ground.
21. A cannon is fired horizontally from the edge of a cliff, 150m above sea level. The cannonball
reaches the ocean 150m away from the base of the cliff. W hat is the initial velocity of the
cannonball?
22. Show that, for a projectile fired from the ground with an initial speed, v , fired at an angle ,
the range is given by
2v 2 sin  cos 
R
g
23. Thelma and Louise drive their car off a 75m high cliff. The car starts with a horizontal speed
of 80 km/hr.
a) How many seconds later does the car hit the ground?
b) At what angle does the car hit the ground?
c) What is the car’s final speed in km/hr?
FORCES AND FREE-BODY DIAGRAMS
1. Acceleration is produced by
a) inertia
b) velocity
c) mass
d) force
e) any of the above
2. It was once thought that space travel would be achieved by firing astronauts out of a gigantic cannon.
(They would, of course, be inside some sort of space vessel when fired.) To achieve the necessary 11km/s
escape speed, what average force would each 70kg astronaut need to experience inside a hypothetical 2.2km
long gun?
[1.9MN or 200tons]
3. A 600kg sailboat experiences two forces, a northerly one from the wind and an easterly one due to the
water. The net acceleration is 0.5m/s2 in a direction 60o North of East.
a) What force is exerted by the wind? by the water?
b) If the boat starts with a speed of 3m/s due West, what is its displacement vector after three seconds of
uniform acceleration?
[150N, 260N, 8m at 104o]
4. Two 5kg bags of sugar are hanging from the ceiling of an elevator car. A massless rope attached to the
ceiling holds the top bag, and a massless chain hanging from the top bag supports the lower bag. What are
the tensions in the rope (T1) and the chain (T2) if the elevator is travelling upward but slowing down at a rate
of 2m/s per second?
[39N, 78N]
5. Draw a free-body diagram for each of the following. Write Newton's Second Law in component form for
each, plug in components for each, but do not solve.
a) A lawnmower is being pushed at a constant velocity. The handle is at an angle  above the
horizontal.
b) Two masses are connected by a string passing over a pulley. The mass on the left, m1, is on a
frictionless inclined plane that slopes down from the pulley, the other mass, m2, hangs freely.
c) An empty roller coaster car is at the top of a loop-the-loop, and is travelling just fast enough to
stay on the tracks.
6. Draw a free-body diagram for each of the following. Write Newton’s Second Law in component form for
each, plug in components for each, but do not solve:
a) Two masses are connected by a string passing over a pulley. The mass on the right, which is lighter, is
initially moving downward.
b) A block on an incline is accelerating downward despite its friction with the surface.
c) A car traveling at a speed v is rounding the top of a hill and the passengers are briefly weightless.
7. Draw a free-body diagram for the following and write the appropriate equation[s] for Newton’s Second
Law. Plug in the appropriate quantities, but do not solve.
A gardener is pushing a lawnmower of mass m, with a force of magnitude F, along its handle at an angle
30° below the horizon. The mower is traveling at a constant speed. The magnitude, f, of the frictional force
between the mower and ground is unknown.
8. Draw a free body diagram for the following and write the appropriate equation[s] for Newton’s Second
Law. Plug in the appropriate quantities, but do not solve.
a) Consider a passenger of weight W in an elevator that is rising but slowing down.
b) Consider that same elevator car, of mass M, which is supported by a steel cable.
9. Draw a free-body diagram for each of the following 3 figures, making sure to indicate the
direction of acceleration, if any. Make clear if any of the forces are obviously larger than others:
a) A skier slows down as she skis down a ski slope.
b) Mass 2 is sliding down the slope and picking up speed. There is friction on both.
c) The painting m is motionless.
10. In 1834, people and goods were moved across the rugged Allegheny mountains on the
‘Allegheny Portage Railroad’, which consisted on ten inclines planes, situated between
Hollidaysburg, PA and Johnstown. The inclines ranged in slope between 6% and 10% (0.06 < tan
 < 0.10). On each incline, a hemp rope passed from one railroad car on one set of tracks to the
top ov the incline, and down another set of tracks on the same incline to another car. As one car
fell down the slope, the other rose up the very same slope. If two such cars having mass 1600 kg
each are attached via rope on an incline of 7% slope,
a) What is the angle of elevation  of the incline?
b) What is the tension in the rope? Assume zero friction.
c) If the rope snaps, what is the acceleration of either cart down the slope? Again, assume zero
friction.
11. A 3-kg mass on an inclined plane is on the verge of slipping when the angle of the incline is
35 .
a) What is the normal force?
b) What is the frictional force?
c) How large a mass must be put on the other side of the pulley shown in the diagram in order to
budge the 3-kg mass.?
Torque/equilibrium
12. Consider the safety certificate of an elevator, which has mass M, hangs from two wires at
angle  and  . If the elevator is accelerating downward, show how you would solve this problem
for the tensions in the two wires.
13. SET UP, but do not solve the following problem. Write down all of the appropriate equations
you need to start this problem, WITHOUT any actual numbers in these equations. Draw a box
around each equation you want me to consider, and below each box list the quantities that you
would plug into the boxed equations to solve them: A 2.5 m, 0.75 kg ladder leans against a slick,
frictionless wall at an angle of 60  above the horizontal. SET UP, but do not solve, this problem to
determine all the forces acting on the lader.
14. A 124-kg museum display case is support by two wires: one at 30 above the positive x-axis,
the other at 60 above the negative x-axis.
a) Draw the appropriate free-body diagram for the problem and write the appropriate force
equations.
b) Solve for the unknown forces.
c) Which wire would break first if they are identical?
15. A 5m long board is balanced on a point pivot to make a seesaw. A 30kg child sits on one end of the
seesaw, 3m from the pivot. A 47kg child sits on the opposite end.
a) If the board is equilibrium with both kids off the ground, what is the board’s mass?
b) What force is needed at the pivot to support the seesaw?
[8kg, 833N]
16. A 2kg painting is suspended by two wires, one of which makes an angle of 30º with the vertical, and the
other of which makes an angle of 40º with the vertical. Find the tensions, T1 in the first wire, and T2 in the
second wire.
[10.4N, 13.4N]
17. A crowbar of 24 inch length is being used to extract a nail from a board. Assume that the nail is at one
end of the bar and one inch from the contact point of the crowbar with the board and a carpenter is exerting
a ten pound force at the other end,
a) With what force is the nail lifted?
b) What force must the board exert on the crowbar?
c) What is the mechanical advantage of the crowbar in this case?
[230lb, 240lb, 23]
18. A 5kg, 3m long ladder leans against a wall, making a 60 deg angle with the floor. There is no friction
between the ladder and the wall, but a normal force of magnitude N1. Any other normal forces acting on the
ladder should be called N2. The ladder does not slip, but it is not necessarily at the angle for which it is on
the verge of slipping. The weight of the ladder is spread uniformly throughout the length of the ladder.
a) Using the bottom end of the ladder as the pivot point, calculate the torque exerted by the weight.
b) Set up this problem to solve for all of the unknown forces. Solve for them.
[37N·m, 49N, f=N1=14N]
19. A 3m long board has a mass of 50kg, with the center of mass at the center of the board. The board is
supported 0.5m from the lefthand side by a support.. There is another support lm from the righthand end. A
700N person stands lm from the lefthand end of the board.
a) Draw a free-body diagram of the board.
b) Calculate the torque which the person exerts on the board, if we use the lefthand support as the pivot
point.
c) Using the lefthand support as the pivot point, set up all the appropriate equations for solving for the
forces between the board and the supports. Solve for the forces.
[350N·m, 630N, 560N]
20. A 2m ladder leans against a wall, making an angle of 60° with the floor. The wall is smooth and
therefore exerts no friction. The floor is not smooth.
a) Draw a free-body diagram.
b) If the ladder's weight is 10kg, use the floor-end of the ladder as a pivot point to calculate the force
exerted by the wall.
c) Calculate all the other forces exerted on the ladder.
[28N, 28N, 98N, 98N]
21. A 6m beam of 50kg mass is balanced lm from its end if a person stands right at the edge of one of the
ends.
a) Draw a free-body diagram.
b) What must the mass of this balancing person be?
c) What is the magnitude and direction of the force exerted by the support on the beam?
[100kg, 1500N straight up]
22. A 10m long, 500N bar is attached at one to a wall and at the other end by a wire that attaches to the wall
and makes an angle of 30° with the vertical.
a) Draw a free-body diagram.
b) Calculate the magnitude of the tension in the supporting wire.
c) Calculate the horizontal and vertical supporting forces at the wall.
[289N, 144N, 250N]
23. A machinist is trying to loosen a nut from a bolt on a 747 airliner. The machinist puts a 5m
lead pipe around the handle of her wrench and places her weight at the end of the pipe to try and
loosen the nut. The lead pipe makes an angle of 30  with respect to the horizontal.
a) What is the torque she is exerting on the nut?
b) If the bolt has a radius of 1 inch, (2.54 cm), then if it still doesn’t budge, what is the minimum
frictional force between the nut and bolt?
24. A 3m long picnic table of 10 kg mass is placed on top of two supporting boards as shown in
Fig. F. A 5 kg picnic basket is placed 1m from one of the ends.
a) Draw all of the forces acting on the table top and briefly describe each (its name, what causes
it).
b) Solve for the contact forces.
25. Two guys hold opposite ends of a 4m long board of mass 20 kg, with a center of mass 2m
from the ends.
a) Draw a free-body diagram for the board.
b) If “Lefty” can lift no more than 40 kg, and Dexter can lift only 60 kg, how far from “Lefty” should
one place an 80 kg mass on the board so that both workers can lift it?
26. Bobby (m =25 kg) sits at one end of a 4m seesaw of weight 150N, which has its center of
mass 2m from either end, and which pivots about this center of mass. Sally (m =20 kg) sits at the
other end of the seesaw.
a) How far from the pivot must Sally’s kid brother (m =10 kg) sit in order to balance the seesaw?
b) What contact force must be exerted on the seesaw at its pivot to hold it up?
c) Draw a free-body diagram for the seesaw.
27. Two sawhorses support a practically massless 5m long board which supports a 35kg mass
2m from its left end.
a) Draw a free-body diagram for the board.
b) Calculate the support forces that each sawhorse exerts on the board.
28. SET UP, but do not solve the following problem. Write down all of the appropriate equations
you need to start this problem, WITHOUT any actual numbers in these equations. Draw a box
around each equation you want me to consider, and below each box list the quantities that you
would plug into the boxed equations to solve them:
A 4m, 85 kg horizontal steel beam supports a 250N sign at one end of the beam. The other end
of the beam is attached to a vertical wall. In addition, an iron cable, at 30  above the horizontal,
also helps support the sign. Show how you would solve for the tension in the cable.
WORK AND ENERGY
1. A skier of 650N weight is travelling down a 20º slope at a constant speed. Calculate the forces acting on
this person, and the work done by each as the skier skis 100m down the slope.
[650N, 222N, 611N, 0J, 22kJ, -22kJ]
2. The total annual energy consumption of the US is 83 ‘quads’ (quadrillion BTU’s or 8.8 10
a) If there are 250 million Americans, what is the average American’s power consumption?
b) If this power were used to lift a 1000 kg (1 ton) mass two meters in height at a constant
velocity, how long would it take?
3.
a)
b)
c)
d)
J.
A 2.4-kg mass slides 2m down a 30 incline against a coefficient of friction of 0.33.
What is the frictional force?
How much work is done by friction?
How much work is done by gravity?
What is the net work done on the block?
4. A l000kg car has a fuel efficiency of 30mpg, meaning that it uses 6MJ of gasoline energy to go one mile.
a) How much energy is needed (ignoring friction) to accelerate the car from 0 to 13m/s (30mph)?
b) If you find yourself stopping and starting (accelerating from rest to 13m/s, then decelerating back to zero)
four times per mile in city driving, by what percentage does this effect your fuel efficiency?
[85kJ, 6%]
5. The Aston Martin Lagonda gets 11 miles per gallon in highway driving, let us assume at
constant velocity, on flat road of even elevation. One gallon of gasoline contains 1.76 10 J, so
that this is the work done by the engine in going 11 miles. One mile equals 1609m. Assume that


the only forces acting on the car in the x-direction are E , the force exerted by the engine, and f ,
the force of friction, which opposes the car’s motion. Further, assume a mass of 1800 kg for the
car.
a) What work is done by each of the following forces in going 11 miles: friction, the normal force
of the road, and the car’s weight?
b) What is the force of friction on the Lagonda as it cruises down the highway?
c) The GEO Metro gets 58 mpg on the highway. What is the force of friction on it?
6. An 80 kg passenger reads a scale while on a descending elevator. The scale reads 600 N for
the entire time while the elevator travels 10 m.
a) What is the acceleration of the passenger?
b) What work is done by gravity when the passenger travels 10 m?
c) In travelling the 10 m, the elevator goes from 0 m/s to 9.6 m/s. What is the net work done on
the passenger?
7. A janitor is sweeping the floor with a power cleaner, pushing with 100N of force in the direction
of its handle, which makes a 30  angle with the floor. The machine has a mass of 45 kg, and the
janitor pushes it at a constant speed in a straight line.
a) Draw a free-body diagram.
b) Calculate the normal and frictional forces acting on the machine.
c) How much work does the janitor do in cleaning a 5m length of floor? (If you don’t have the
numbers to plug in for this part, at least set up the problem to show how you would do it.)
8. An automobile that gets 30 miles (48km) per gallon is about 13% efficient at converting the 180MJ in a
gallon of gas into mechanical energy.
a) How much work does the engine do in travelling 30 miles?
b) What is the average force that the engine exerts on the wheels? This is equivalent to the frictional force
that the car experiences from the outside world.
c) A bicyclist expends 7000J to pedal 1km at constant speed on level ground. How large a frictional force is
she battling as she pedals?
[23MJ, 480N, 7N]
9. A 60kg skier travels down a slope, for a vertical drop of 200m. The skier starts at rest and ends up at
20m/s at the bottom.
a) What is the total work done on the skier? (Is it positive or negative?)
b) How much energy is lost? This is the work done by nonconservative forces.
c) What is the average friction on the skier?
[12kJ, 106kJ, 106N]
10. You push a 5kg block up a wall by exerting a force of F1=100N at 50° above the horizontal.
a) Draw a free-body diagram.
In pushing the block 3.5m up the wall, find the work done by
b) you
c) the normal force
d) gravity
e) friction
[268J, 0, -172J, -96J]
11. A class of physics students discovers that it takes 100lb (445N) to push the instructor’s car, which is in
neutral.
a) If the students push the car at 3.5m/s for 2 seconds, how much work have they done?
b) What power have the students exerted?
c) If the same 100lb force is exerted on a car travelling at 55mph (25m/s), how much power is exerted? One
horsepower (1hp) is equivalent to 746W. Express the power in both Watts and horsepower.
[3.1kJ, 1.6kW, 11kW, 15hp]
Springs
12. We attach a 1kg mass to a vertically hanging spring, and it stretches 0.20m to reach its new equilibrium
length.
a) What is the spring's spring constant?
As we let the mass drop freely from its unstretched equilibrium point to the new equilibrium point,
b) What work is done by gravity? (Remember to include whether it is positive or negative.)
c) What work is done by the spring?
d) What work must be done by friction in order for the spring to come to rest at its new equilibrium?
[49N/m, 2.0J, -1.0J, -1.0J]
13. An 80kg bungee jumper jumps from the top of a 23m tall building, at the top of which a 200N/m cord is
tied.
a) If no energy is lost during the fall, and the jumper narrowly misses slamming into the pavement, by how
much does the cord stretch when the jumper is at the pavement?
b) After a minute or so, the cord and the jumper come to equilibrium. How far above the pavement do they
come to rest?
[13.4m, 9.5m]
14. A mass on a spring is stretched 1m from equilibrium and released. Its motion is periodic with
a period T. What is the period if the amplitude is only 0.5m for the same system?
a) 0.5 T
b) 0.707 T
c) T
d) 1.414 T
e) 2 T
15. A force F applied to a spring stretches it by an amount y. How much will a force of 3F stretch
it?
a) _____ 9 y
b) _____ 6 y
c) _____ 3 y
d) _____ 1.7 y
e) _____ 1.4 y
16. A bungee jumper is attached to a 50m cord. When she reaches the ‘end of her rope’, having
fallen vertically from rest, what is her speed? How long does it take to reach this point? (We will
ignore for now the fact that the bungee cords will continue to stretch.)
17. A spring-mass system with an amplitude of 25 cm has an energy of 53 J. What is its energy
if the amplitude is only 20 cm?
MOMENTUM/IMPULSE/COLLISIONS
1. A professor throws three erasers per second at the lecture hall wall to illustrate impulse. If
each eraser has a mass of 0.05 kg, each hits the wall with an initial speed of 3 m/s, and each one
falls down to the bottom of the wall after hitting the wall, what is the average force that the wall
exerts on the erasers?
2. A 15g bullet with a speed of 182 m/s hits a 3 kg block of wood and is lodged inside.
a) What is the final speed of the block?
b) If the block slides up a frictionless inclined plane, what will its final height be when it comes to
rest?
3. SET UP, but do not solve the following problem. Write down all of the appropriate eq uations
you need to start this problem, WITHOUT any actual numbers in these equations. Draw a box
around each equation you want me to consider, and below each box list the quantities that you
would plug into the boxed equations to solve them:
Two masses, one a 3 kg lead ball, and the other a 0.5 kg hollow plastic sphere of the same
diameter, approach each other and undergo an elastic collision. The lead ball’s initial speed is 5
m/s, the plastic ball’s is 7 m/s. Set up this problem to solve for the final velocities.
4. Two 2.5 kg masses collide. The first mass comes from the left with a speed of 3 m/s, and
continues, after the collision, with a speed of 1.2 m/s. The second mass has zero velocity before
the collision.
a) What is the final velocity of the second mass?
b) What impulse does the second mass exert on the first?
5. Two masses collide in a totally inelastic collision. Mass one, with a mass of 0.75 kg, starts with
a speed of 2 m/s in the direction of mass two. Mass two, with a mass of 1.2 5 kg, starts with a
speed of 3 m/s in the direction of mass one.
a) Calculate the final velocities of the two masses.
b) What percentage of energy is lost by the two masses in collision?
6. A northbound truck of mass 3300 kg, travelling at 100 km/hr, collides into an 1800 kg
eastbound car travelling at 50 km/hr.
a) What is the final direction of motion of the two vehicles?
7. A small mass and a large mass have identical kinetic energies. Which has the larger momentum? Support
your answer.
[the larger mass]
8. A block slides down a frictionless incline which is at an angle of 35° above the horizontal. The incline
has a length of 5m and the block has a mass of 5kg.
a) Is mechanical energy conserved? Momentum?
b) Draw a free-body diagram for the block.
c) Solve for the acceleration that the block experiences.
9. Two identical 0.2kg billlard balls collide. One is initially at rest and the other is initally moving at 3m/s.
The ball that was initially moving is at rest after the collision.
a) Show that the other ball is now moving at 3m/s.
b) Find the kinetic energy of the ball that was moving before the collision.
c) Is mechanical energy conserved? Momentum?
10. Two equal masses collide. One hits the other, which is originally at rest, head on with a velocity of vo,
so that the collision takes place in one dimension. We shall assume that there is no net external force on the
system.
a) Show that one physically acceptable outcome of the collision involves the incoming particle coming to
rest and the other particle moving. What is that particle's final velocity?
b) Show that such a collision is perfectly elastic.
[same as the other particle’s intial velocity]
11. A 100g ball hits the ground at 6m/s, and rebounds at 4m/s. The Earth is 6.0×1024kg.
a) In a frame of reference in which the Earth was at rest before the collision, what is its final speed?
b) Is mechanical energy conserved?
c) If not, how much energy is lost or gained? (And is it lost or gained?.)
[1.710-25m/s, No, 1J lost]
12. Under what circumstances is an object’s momentum conserved? Under what circumstances is its
mechanical energy conserved?
13. A 2-ton pickup truck, travelling due East at 50mph, slams into a 1-ton compact passenger car, travelling
due North at 20mph. What is the magnitude and direction of the two vehicles directly after the collision if
they wind up traveling together with the same velocity?
[34mph at 11oN of E]
14. A 20kg mass travelling in the positive-x direction at 15m/s collides into a stationary 10kg mass. The
10kg mass travels on at a speed of 20m/s in the positive-x direction after the collision.
a) Find the final velocity of the 20kg mass.
b) Is this collision perfectly inelastic, perfectly elastic, or neither. Justify your answer.
[5m/s, perfectly elastic]
15. A 1500kg car, travelling at 20m/s in the negative x-axis collides into a 2500kg truck, travelling at 17m/s
in the positive y-direction.
a) Give the magnitude and direction of the velocity of the two vehicles right after the collision.
b) How much energy is lost in this collision?
[13m/s at 125o, -320kJ]
Center of mass
16. A soft drink can is 12.5cm tall and has a mass of 17.2g and contains 362g of soda when full. Assuming
the can is symmetric, its center of mass is halfway up the can. Find the center of mass of the can and its
contents when it is one-third full.
[2.6cm]
CIRCULAR MOTION/GRAVITY
1. I’m on a roller coaster ride, upside down at the top of a loop. The loop has a radius of 25m.
The mass of the cart and its contents (me) is 250 kg. The cart’s speed is 20 m/s on the
frictionless tracks. We will make the simplifying estimate that the cart is 50m above the ground.
a) What is the potential and kinetic energy of the cart at the top of the loop?
b) At what height, h , could I have started this ride if I had wanted to travel at the speed I am now
at, assuming no frictional forces?
c) What’s the cart’s speed at the bottom of the loop?
2. A pelican (mass = 10 kg) dives from rest at a height of 20m to catch a fish at sea level. The
pelican is travelling at 10 m/s when it reaches the sea right before it catches the fish.
a) What is the pelican’s original potential and kinetic energy?
b) What is its final potential and kinetic energy?
c) What is the total net work done on the pelican?
d) What work is done by gravity?
e) Explain the difference between the answers to (c) and (d).
3. An elevator starts from rest with a constant acceleration. A 100-lb passenger has an apparent
weight of 180 lb.
a) Describe the motion of the elevator. Do we know whether it is going up or down?
b) Draw a free-body diagram for this problem.
c) What is a/g, the ratio of the elevator’s acceleration to the acceleration due to gravity?
d) If the elevator starts from rest, how far does it travel in 5 seconds?
e) Calculate the average power exerted by the normal force and by the weight. (your answer
need not be in MKS units.)
4. A 2000kg racecar rounds a level, lkm radius circular racetrack at 100m/s.
a) What is the magnitude of the minimum coefficient of friction required for the car to stay on the
track?
b) If the engine of the car exerts 1600N to keep the car at a constant speed, what work does it do in
rounding the track?
c) What work is done by the normal force?
d) What work is done by the weight?
[1.02, 10MJ, 0, 0]
5. Physics class is adjourned to an amusement park for lab, and all students bring along a set of bathroom
scales. You usually weigh 140lb (620N), but at the top of the Ferris wheel, your scale reads a smaller value.
a) Draw a free-body diagram for this problem.
b) If the wheel spins once every 30s, and it has a diameter of 20m, what is your speed at the top?
c) What does your scale register at the top?
[2.1m/s, 590N]
6. Your instructor twirls an empty bucket around his head in a horizontal circle (No one knows why.), with
the bucket completing a circle once every 0.5s. The bucket has a mass of 300g, and it circles with a radius
of 0.9m.
a) What is the velocity of the bucket?
b) What is the centripetal force acting on the bucket?
c) What work does this force do on the bucket during one revolution?
7. A roller coaster car approaches a hill with a speed of 20m/s.
a) For what radius of curvature will a 55kg passenger just barely leave her seat?
b) If the 1000kg car started from rest at an elevation 25m higher than the top of the hill, how much work has
friction done on it?
[41m, -45kJ]
8. A roller coaster car of 2000kg mass is travelling in a vertical circle of radius 25m. It is traveling just fast
enough that, at the top of the circle, its wheels are on the verge of losing contact with the track.
a) Draw a free-body diagram of the car at the top of the circular track.
b) Calculate its speed at the top.
c) Calculate its total mechanical energy at the top of the curve, if the car’s potential energy is zero at the
bottom of the circle.
d) What is the minimum height (relative to the bottom of the circular piece of track) from which the car
could be launched (at zero initial speed) if it must complete this circle without losing contact with the track?
[16m/s, 1.2MJ, 60m]
9.
a)
b)
c)
d)
The earth takes about 365.243 days to circle the sun.
What is the tangential speed of the earth around the sun?
What is the earth’s momentum?
What is the earth’s angular momentum?
What is the earth’s centripetal acceleration?
10. Doubling the angular frequency of a merry-go-round will change each of the following
quantities by how much?
________ the speed of the passengers?
________ their centripetal acceleration?
________ the centripetal force on them?
________ their period of rotation?
11. Two planets of mass m and m have orbits of r and r around the Sun. If the force of the
Sun on a planet is proportional to the planet’s mass, and inversely proportional to the square of its
orbit, then calculate
________ the ratio of planet A’s centripetal acceleration to planet B’s?
________ the ratio of planet A’s orbital speed to planet B’s?
________ the ratio of planet A’s period to planet B’s?
12. The Moon circles the Earth with an orbit of radius 383 Mm. The Moon’s mass is 7.4 10
and it takes 27.32 days to orbit.
a) What is the speed of the Moon as it circles the Earth?
b) What is the force of attraction between Earth and Moon?
kg
13. A car traveling on a circular track halves its speed. What is the ratio of the new values of t he
following quantities to the original values?
________ centripetal acceleration?
________ centripetal force?
________ angular frequency?
________ radius?
________ period?
14. The radius of Saturn around the Sun is 1.43  10 m, and it has a period of about 10760
days. The radius of Uranus is 2.87  10 m, and its period is 30690 days. Calculate
a) the ratio of Saturn’s velocity to Uranus’s ________
b) the ratio of Saturn’s centripetal acceleration to Uranus’s ________
c) the ratio of Saturn’s angular frequency to Uranus’s ________
15. A child is whirling a toy plane (mass=1 kg) around its head. The plane has a speed of 5 m/s
and is moving in a horizontal circle with a radius of 1m. The string makes an angle of  with the
horizontal direction.
a) Draw a free-body diagram of the plane, labelling all of the relevant forces and describing in
words what all of these forces are (names, what causes them).
b) Write the appropriate forces equations.
c) Solve for all of the forces and the angle .
ROTATION
1. A 100 kg snowball of radius 1.1m rolls down a 30 slop to point 20 m lower in elevation. If the
snowball rolls without slipping,
a) what is its final kinetic energy?
b) what is the ratio of rotational to translational kinetic energy?
c) what is the final speed of the snowball?
2. A skater of mass 50 kg begins her final spin with her arms outstretched, for a moment of
inertia of 1.3 kg m . She is spinning 3 times per second.
a) What is her angular momentum?
b) What is her rotational kinetic energy?
c) Someone in the audience tosses her a 0.5 kg bouquet of roses which she catches
without moving her arms. As she spins with the roses held 1m from her body, what is her new
angular velocity?
3. A turntable has a moment of inertia 0.5 km m . It is turning at 33 revolutions per minute. An
0.001 kg m record is dropped on top of the turntable.
a) Give the frequency and angular frequency of the turntable in MKS units before the collision.
b) At how many revolutions per minute does the turntable spin right after the record lands
on it?
4. A roller coaster car of mass 1000 kg begins with zero speed at a height of 100m. It is
designed to round a vertical circle of radius 25m, the bottom of which rests on the ground.
a) If there is no friction between the cart and the track, what is the cart’s speed at the top of the
loop?
b) What is the resulting centripetal acceleration of the cart at the top of the loop?
c) What normal force is exerted by the track on the cart at the top of the loop?
d) If the car were a disc of radius 1m, calculate its speed at the top of the loop.
5. A snowball, with mass m, moment of inertia I, and radius r, starts rolling down a slope, starting
from rest. Show how you would start to solve this problem to calculate th e speed of the snowball
at the bottom of the slope, after it has travelled a distance l. (Remember, you are not given W.).
5. A grinding wheel with a moment of inertia 2 kg m turns at 100 revolutions per second.
a) What is the angular momentum of the wheel?
b) What is its rotational kinetic energy?
c) If a 3 kg magnet, initially at rest, attaches itself to the rim of the wheel (r=0.2m) while the wheel
is in motion, how quickly does the wheel now spin?
6. A turntable turns at 33+(1/3) revolutions per minute. The turntable has a moment of inertia of 0.3kg · m2.
A vinyl record of 50g mass and 30cm diameter, intially at rest is lowered onto the turntable.
a) What is the moment of inertia of the record in MKS units?
b) What is the inital rotational kinetic energy of the turntable (also in MKS)?
c) What is the frequency with which the turntable spins the instant after the record falls on it?
d) Is rotational kinetic energy conserved? Angular momentum?
[0.00225kg·m2, 1.83J, 33.1rev/m, no, yes]
7. What are the units for each of the quantifies in the following equations?
a) ac=r2
b) KE=½12
C) =  o+avgt
2
[m/s , m, rad/s; J, kg·m2, rad/s; rad, rad/s, sec]
8. A 5kg, 0.lm radius solid cylinder of 0.5m length rolls without slipping at 2m/s.
a) What is the angular velocity of the cylinder?
b) What is its moment of inertia?
c) What is the rotational kinetic energy?
d) What is the translational kinetic energy (the non-mtational KE)?
[20rad/s, 0.025kg·m2, 5.0J, 10J]
9. Write the rotational "equivalents" of the following:
a) Newton’s Second Law
b) x=xo+vot+½at2
c) v2=vo2 + 2ax
[ = I,  = o+ vot+½at2,  2= o2 + 2 ]
10. A sphere of 3kg mass and 10cm radius rolls down an inclined plane which makes an angle of 40° with
the horizontal.
a) Draw a free-body diagram of the sphere, showing the location at which each force is applied.
b) What is its velocity at the bottom of the slope if it rolls a distance of 2m down the length of the incline?
[4.2m/s]
11. A 75kg painter is at the top of a uniform 30kg, 4m long ladder when it tips over. Assume that the
bottom of the ladder does not move, and use the bottom as the pivot point for the following questions.
a) What is the moment of inertia of the painter plus ladder about the pivot?
b) When the ladder is 20° from the vertical, what total torque does gravity exert on the painter plus ladder,
relative to the pivot?
c) What is the angular acceleration of the ladder about the pivot at that time?
[1360kgm2, 1200Nm, 0.9rad/s2]
SIMPLE HARMONIC MOTION, WAVES
Masses and springs
1. An 80kg stunt diver is dangling at the end of a 20m long bungee cord.
a) If the cord is 16m long without the diver, what is its spring constant?
b) If the diver swings up and down with a distance of 8m between her highest and lowest points in the
swing, what is the maximum amount of energy she stores in the cord?
c) With what frequency does she bob up and down?
[200N/m, 1600J, 0.25Hz]
2. A 350g mass is attached to a 35N/m horizontal spring, and the mass is pulled 12 cm from
equilibrium.
a) What is the maximum speed of the mass?
b) What is the maximum acceleration of the mass?
3. A mass suspended at the end of a spring stretches the spring from its original 33cm length to
35 cm. The spring constant is 500 N/m. What is the potential energy stored in the spring when
this mass stretches the spring?
4. A 35cm spring loaded with a 10 kg mass stretches an extra 49 cm.
a) Calculate the spring constant of the spring.
b) What is the period of this system if we lift the mass to its original, unstretched position and
then release it?
c) If another spring, identical to the first, is attached to the end of the first, and then the 10 kg
mass is hung from the bottom of the seond spring, what is the period of oscillation of the mass?
5. A mass at the end of a spring is pulled down 15 cm from its equilibrium p osition. What is its
position a time T/8 later?
6. A mass on a spring is stretched 1m from equilibrium and released. Its motion is periodic with a
period T. What is the period if the amplitude is only 0.7m for the same system?
The pendulum
7. A grandfather's clock is sent from one part of the country to another, where the gravitational constant, g,
is larger by a factor of 1.00I.
a) By what factor does the period of the clock's pendulum change? (Use a number greater than 1 if it
increases, less than 1 if it decreases.)
b) By what factor must the length of the pendulum be changed to correct for the difference in gravity?
[0.995, 1.001]
8. A 0.2kg mass is attached to a 3m long string and, later, to a 1m long spring. The period of the mass when
it is swinging as a pendulum is identical to its period oscillating at the end of the spring.
a) What is the stiffness of the spring?
b) What is the new equilibrium length of the spring when the mass hangs from it?
c) By what factor does the period of the mass-plus-spring oscillator change if it is moved to a location
where gravity decreases by 0.1%?
[0.65N/m, 4m, T doesn’t change.]
9. If appropriate holes were drilled in the floors, a pendulum with a period of 6 seconds could be installed in
Bewkes Hall for demostration purposes.
a) What would be the length of such a pendulum?
b) What would be its period if it were hung on the Moon, where g is smaller by a factor of 67
c) What would be the period on the Moon of an oscillator consisting of a mass and spring which has a 6s
period on the Earth?
[8.9m, 15s, 6s]
10. An entertaining physics demonstration consists of a spring with a mass attached to the end, which is
swung like a pendulum. When the period of the pendulum is close to twice the period of the mass-spring
oscillator, the system will oscillate periodically between back-and-forth pendulum motion and up-and-down
oscillator motion.
a) What is the length of the stretched spring, in terms of the spring constant and mass?
b) What is the length of the unstretched spring, in terms of the spring constant and mass?
11. A pendulum of length L has a period of 1.000 seconds. What is the period if the length of the
pendulum is increased to 1.01 L?
12. A 350g pendulum bob hangs on a 25 cm string. The string is pulled to an angle from the
vertical so that the bob is 10 cm higher than if it hung straight down.
a) If the bob is now released from rest, what will its speed be at the bottom of the swing?
b) What is the centripetal acceleration of the bob at the bottom of its swing?
c) Draw a free-body diagram of the bob at the bottom of its swing, and calculate the tension in the
string at that point.
13. A physics student is equipped with a 1m long pendulum and asked to measure g, the
acceleration due to gravity. We will assume that the pendulum is exactly 1m long.
a) How much time will it take for the pendulum to complete 100 swings if g=9.804 m/s ?
b) How much time will it take for the pendulum to complete 100 swings if g -9.805 m/s ?
c) How accurate must the clock be for the student to determine whether g is closer to 9.804 m/s
or 9.805 m/s ? (Include enough significant figures in parts (a) and (b) so that you can answer part
(c).
14. A simple pendulum has a period T here on the surface of the earth. What would its period be
if carried to the moon where the acceleration due to gravity is only one-sixth that on earth?
15. You are designing a pendulum to regulate a ‘Grandfather’s clock’. You want the pendulum to
have a period of exactly 2 seconds.
a) What length should the pendulum have?
b) If the clock is taken to the Moon, where ‘g’ is about one-sixth as large as on Earth, what will the
period of the clock be?
16. A grandfather clock is designed to work properly in Canton, where g=9.804m/s . How long
does it take for the clock to tick off one second in New York, where g=9.802m/s ? Include
enough significant figures to distinguish your answer from 1 second.
17. A spring with a 14 kg mass attached to it has a [stretched] length 0.3m and spring constant
50 N/m. It is allowed to swing from side to side, so that it behaves like both a pendulum and a
spring. What is the ratio of the period of the spring motion to the period of the pendulum motion?
Waves
18. A crowd at Rich Stadium is doing ‘the wave, when someone starts a second, third and fourth
wave, each 100m behind the preceding wave. If someone in the crowd notices that she is raising
her arms about 2 times per minute,
a) express wavelength, period, and frequency in MKS units.
b) What is the speed of the wave in MKS units?
19. Microwaves are electromagnetic waves, like visible light, but with much larger wavelengths.
What is the frequency of microwaves in a beam with wavelength one inch?
Wave equation
20. If the amplitude of pressure waves due to a 10 kHz sound in air is 10 mPa, and at t=0 and
x=0 the disturbance is zero, then
a) calculate the wavelength and period of the sound wave.
b) calculate the sound wave disturbance at x=3.4 cm, t=0.000025s.
c) calculate the sound wave disturbance at x=0.567 cm, t=0.0037s.
21. A lighthouse attendant notices that at midnight the ocean is at high tide (maximum height)
and the water is 0.3m above its average daily height. If the period of oscillation of the water height
is 12 hours, what is the height of the tide at 4:30 pm is the oscillation is harmonic?
22. 440-Hz sound waves are being produced by a tuning fork 2m from your ear. The amplitude
of the disturbance is 15 10 Pa. (You don’t need to know what the units mean.)
a) How long does it take the waves to travel to your ear?
b) What is their wavelength?
c) Setting time equal to zero at a time when the disturbance due to the tuning fork is zero at the
tuning fork (but after the sound begins), calculate the disturbance at a point halfway between the
fork and the ear at t=0.0005 sec.
Harmonics of a string
23. A guitar string is 647mm long. Middle A is 440Hz.
a) If my guitar string beats at 2.5Hz against a calibrated 440Hz tuning fork, and I need to tighten the string
to bring it in tune, what frequency is coming out of my guitar?
b) Calculate the velocity of a wave on that guitar string both tuned and untuned.
c) Calculate the wavelength of the 440Hz 'A' in air and on the tuned string.
[437.5Hz, 569m/s, 566m/s, 0.773m, 1.294m]
24. Consider a stretched lm string, fixed at both ends.
a) What is the fewest number of nodes that a standing wave pattern can have on this string? The fewest
number of antinodes?
Consider a 440Hz oscillation on this string which produces a four-node standing wave pattern.
b) Which harmonic is this pattern? Which overtone?
c) What is the wavelength of this sound wave on the string? In air?
[Two nodes, one antinode, third harmonic, second overtone, 0.67m, 0.77m]
25. A 647mm unfretted guitar string produces a 660Hz fundamental frequency.
a) What is the wavelength of the fundamental?
b) What is the speed of sound on the string?
c) How many audible harmonics (f<20kHz) does the string produce?
d) How many audible harmonics would an organ pipe, which was open at one end and closed at the other,
and had the same fundamental frequency, produce in the audible range?
[1.3m, 850m/s, 30, 15]
26. The second string on a guitar, A, is about 65cm long, and emits a 220Hz pitch.
a) What is the speed of sound in this guitar string?
b) What would be the beat frequency between this string's 15th and 16th harmonics? Between its 16th and
17th?
c) Are any of the A string's harmonics E's (165Hz)? That is, are any of the A's harmonics one, two, three, or
more octaves above 165Hz? If so, what is the frequency of the first E in the A string's harmonics?
27. A string fastened at both ends has a fundamental frequency of 1000 Hz. What is the
frequency of its second overtone?
a) 2000 Hz
b) 500 Hz
c) 3000 Hz
d) 333 Hz
e) 5000 Hz
Doppler Effect
28. A car approaches a stationary observer. The car has a vanity horn that plays a musical ditty. The slowest
speed (other than zero) for which the car can travel and the ditty can be in tune (although in a different key)
is one for which the pitch goes up by a factor of 2 1/12 = 1.0595.
a) How fast must the car travel?
b) If the car is stationary and the observer moves, how fast must the observer move for the ditty to be in
tune?
c) If you approach an AM radio station at 55mph (25m/s), the radio signal will sound different than if you
were standing still. AM signals travel at 3.00 x l08m/s. By how much will 440Hz signal be shifted?
[19m/s, 20m/s, 0.000037Hz]
29. A radar gun in a stationary police car is fired at your car which is approaching the gun at 55mph
(25m/s), which would be okay, except that you are on Main Street downtown. Radar travels at 3.00 x
108m/s.
a) Being well-equipped you have a frequency detector on your dashboard that records a frequency of
exactly 1300MHz for the radar's frequency. What frequency was emitted by the police radar gun? (Keep
nine significant figures in your answer.)
b) The radar bounces off your car back to the police as if your car was a moving source of 1300MHz radar
waves. What frequency do the police detect? (Keep nine significant figures.)
c) The electronics in the radar gun produce a beat frequency between the emitted signal and the reflected
signal. What is the-frequency of this beat (in Hz)?
[1299.99989MHz, 1300.00011MHz, 220Hz]
30. One of the three tenors, who enjoys listening to his own voice, is listening to his echo while driving at
23m/s toward a cliff. He is singing an ‘A’ at 440Hz, and the speed of sound is 340m/s this afternoon.
a) What frequency would an observer at the cliff hear?
b) The echo that the moving tenor hears is equivalent to the sound produced by a ‘ghost tenor’ at the cliff
singing at the frequency calculated in part (a). What does the tenor in the car hear for the frequency of his
echo?
c) An observer at the cliff is also singing an ‘A’. What beat frequency does his picnic companion on the
cliff hear?
[472Hz, 504Hz, 32Hz]
31. Your car and a police car are approaching each other. The police car, travelling at 30mph (13.4m/s),
emits an ultrasonic 40kHz tone in your direction. The police "mix" the tone that echoes off your car with the
original 40kHz tone, and measure the beat frequency between the two signals. This gives them enough
evidence to ticket you for speeding, since you were travelling 55mph (24.6m/s) in a 30mph zone.
a) What was the beat frequency that they measured?
b) If the police car had been stationary, would the recorded beat frequency have been larger or smaller?
[10kHz, smaller]
32. Two cars are tuned to the same radio station and have their stereos cranked really loud. One
is parked and the other is approaching the parked car at 55 mph (24 m/s). An observer standing
miday between the two cars hears a note from a guitar solo which has a frequency of 10 kHz.
a) What frequency does the observer hear from the stationary car?
b) What frequency does the observer hear from the moving car?
c) What is the beat frequency that is heard?
33. A police car speeds along the street with its sirens blaring and lights flashing. Its siren emits
a 1000-Hz pitch.
a) What is the wavelength of the siren’s sound?
b) If the car is travelling at 80 km/hr, what frequency will the pedestrians in front of the siren hear?
c) If a pedestrian in front of the police car has a 1000-Hz siren as well, what frequency will the
police hear from that siren?
34. In order for a 440 Hz siren to sound like an 880 Hz siren, the source and observer must be
moving quickly relative to each other.
a) Must the source and observer be approaching each other, receding from each other, or does it
depend on which one is moving?
b) How fast must the siren be moving if the observer is stationary? The speed of sound is about
340 m/s.
c) How fast must the observer be moving if the siren is stationary?
35. A jet passes you, travelling at 600 mi/hr. If there were a siren of frequency 1200 Hz attached
to the outside of the jet,
a) What is the frequency that you hear as it approaches?
b) What is the frequency that you hear as it recedes?
c) What frequency would the pilot hear if you shrieked at her at 1200 Hz as she approached?
FLUIDS
Pressure
1. If you have a very large glass of water, how long a straw would you need to ensure that no one could
drink
out of it, provided the straw is held vertically?
[10.3m]
2. What is the total force exerted by the atmosphere on the exterior sides of a can of soda of radius 2.5cm
and height 10cm? (Ignore the top and bottom of the can.)
[1600N]
3. You are 30m below the surface of the water.
a) What is the pressure of the water surrounding you?
b) If your chest is 1m around by 0.2m tall, and if we assume it to be cylindrical, what outward force do your
muscles need to exert to allow you to breathe underwater?
c) If you expand your chest to 1.03m around, what work have you done?
[400kPa, 80kN, 630J]
4. The text says that the micturition reflex (the urge to urinate) occurs when the pressure inside your bladder
exceeds 25mmHg.
a) If this is the pressure that drives the urine out of you, with what speed does the liquid leave your body?
b) If the urine leaves someone’s body horizontally at this speed at a height of 0.8m above the ground, how
far will it travel?
c) Does the answer to part (b) make sense?
[71m/s, 29m, nope.]
5. A balsa wood glider has a mass of 100g. Its two wings are triangles which each have a base of
5 cm and a height of 20 cm. What difference in pressure between the top and the bottom of the
wing is needed to keep the glider aloft? (Ignore the buoyant force.)
6. A column of water is used as a barometer to measure air pressure. What is the minimum
length of the glass tubing that must hold the water on an average day at sea level?
7. Atmospheric presure can be thought of as the effect of the combined weight of the air above
the Earth. Assuming that the air has a constant density of 1.29 kg/m (of course, it doesn’t),
estimate the height of the atmosphere.
Archimedes (Buoyancy)
8. A 6m x 7m raft has a weight of 3000N and carries a 900N passenger.
a) What mass of water must it displace to stay afloat?
b) How far below the water's surface is the bottom of the raft when it is in equilibrium?
[400kg, 0.009m]
9. A hot-air balloon has a volume of 2000m3. The canvas skin of the balloon has a weight of 200lb (890N),
and it carries a passenger basket of 501b weight (223N) and two passengers whose combined weight is
2501b (1110N).
a) Draw a free-body diagram of the balloon, assuming that it is in equilibrium, above the ground.
b) What is the weight of the air displaced by the balloon? Let's assume that the temperature outside is 0 oC,
so that its density is 1.29kg/m3.
c) For the balloon to lift, what must be the weight of the air inside the balloon?
d) What is the necessary density for the air inside the balloon to achieve liftoff?
10. A balloon lifts three 1501b passengers inside a 751b wicker basket, with the fabric skin of the balloon
weighing at most 501b.
a) Draw a free body diagram.
b) The density of helium is 0.179kg/m3. What is the minimum volume of balloon needed to lift the
passengers? (1 kilogram weighs 2.21b.)
[260m3]
11. Consider a cubic box with edges of 2.3m length.
a) Calculate the buoyant force acting on the box in air.
b) Calculate the difference between atmospheric pressure at the bottom of the cube and at the top. Which is
larger?
c) What is the difference between the upward force exerted by the outside air on the bottom of the cube and
the downward force exerted by the outside air on the top. What is the direction of this net force?
12. Consider a spherical helium-filled balloon of radius 10 cm.
a) Assuming that the gas is at atmospheric pressure, what is the weight of the gas?
b) What is the buoyant force acting on the balloon?
c) How large a wieght does one need to keep this balloon from flying away?
d) What is the radius of the biggest balloon that a 20 kg child could hold onto before the child was
lifted skyward?
13. A child is holding a helium balloon of volume 0.52 m while riding in an elevator. Ignoring the
mass of the balloon’s covering, but not ignoring the mass of the helium, calculate the tension in
the string by which the child holds the balloon if the elevator is travelling with a co nstant speed.
Continuity
14. The heart pumps blood at a rate of about 5 L/min.. Blood travels at 30 cm/sec in the aorta,
which has a cross-sectional area of about 2 cm . If blood flows at 0.4 mm/sec in the capillaries,
what is the total cross-sectional area of all the capillaries?
Bernoulli
15. Tornados and hurricanes are notorious for picking up heavy objects and flinging them
violently. Imagine a typical family car of 1800 kg mass, which is 2m wide by 6m long.
a) What is the pressure difference between the air above and the air below the car necessary to
lift it with no acceleration?
b) What is the pressure differential needed to lift the car with 15 m/s upward acceleration?
c) What are the wind speeds necessary to accomplish part (b)? How does your answer compare
to the highway 55 mph speed limit?
16. The pressure in a tornado can be as low as 20% of atmospheric pressure.
a) What is the net force that will be exerted on a window pane 25cm x 25 cm as the tornado
reaches it?
b) Calculate the speed of the winds that make up the tornado.
17. Assume a liquid is flowing through a pipe of cross-sectional area A at pressure P and velocity
v. If the area decreases then
a) _____ v will increase and P will decrease.
b) _____ v will increase and P will increase.
c) _____ v will increase and P will remain the same.
d) _____ v will remain the same and P will increase.
e) _____ v will remain the same and P will decrease.
18. Water flows at 5 m/s through a 10 cm diameter pipe before going into a narrower pipe,
through which it flows at 20 m/s.
a) What is the diameter of the second pipe?
b) What is the pressure difference between pipes if they are at the same height?
19. A 0.145kg baseball thrown as a 'curveball' can accelerate as much as 6 inches to the side as it
approaches home plate, for an acceleration of 1.2m/s2. The cross-sectional area of the ball is just the area of
a circle of radius 3.75cm.
a) What difference in pressure on two sides of a baseball which gives rise to its curve?
b) This difference is the result of air travelling faster around one side of the spinning ball than around the
other side. If air along the slower path travels at 40m/s (90mph), what is the speed of air on the 'faster' side?
[39Pa, 41m/s]
20. Water flows at lm/s into a swimming pool through a 1m diameter pipe, which comes from a 0.5m
diameter pipe inside the house at the same height.
a) What is the speed of the water in the inside pipe?
b) What is the volume rate of flow into the pool?
c) What is the pressure difference between the two sections of pipe? Which one is at a higher pressure?
[4m/s, 0.79m3/s, 1500Pa higher at the pool]
21. A hypothetical 0.5kg bird has a wingspan of 0.8m0.4m.
a) What pressure difference between the top and the bottom of the wing does the bird need to fly? Which
pressure is higher?
b) Assuming the bird flies at 10m/s, what is the difference in speed between air flowing over and under the
wing? Which is faster?
[15Pa greater below, 1.3m/s faster above]
22. Winds of 120mph (55m/s) are blowing outside a house that can be considered a cube with 10m long
sides.
a) What is the net force on each wall if the windows are all closed?
b) Is this force inward or outward?
c) If this house were built 50m underwater, what outside water velocity would be necessary to produce the
same net force?
d) What would be the (absolute) water pressure outside the underwater house?
[197kN outward, 2.0m/s, 590kPa]
THERMODYNAMICS
Ideal gas, microscopic picture of heat
1. What is the total internal energy of the air in this room if its dimensions are 10.3m x 10.3m x 2.6m and
T=23 degrees Celsius?
[42MJ]
2. How many moles of gas are there in this room (at T=23 ) if its dimensions are 10.3m 10.3m 
2.6m?
3. If the temperature of a gas is increased by a factor of 2, the average speed of an individual gas
molecule will increase by a factor of
a) _____ 2
b) _____ 4
c) _____ 1.41
d) _____ 1.26
e) _____ It will not increase.
4. The average speed of gas molecules in an ideal gas is halved. If the initial temperature of the
gas is 212F, what is its final temperature?
5. The atmosphere varies in density with altitude, but, assuming that it doesn’t...
a) Estimate the height of the atmosphere, given P and a density of 1.2 kg/m .
b) Calculate the volume of the atmosphere, given that the Earth’s radius is 6.4 10 .
c) Calculate how far the ionosphere (the region just above the atmosphere) moves when night
turns to day and the temperature goes from 55 F to 72F.
Thermal processes applied to gases
6. If an amount of gas at atmospheric pressure and 68°F is sealed inside a glass bottle, and the bottle is
placed in a kettle of boiling water, what is its final pressure?
[129kPa]
7. The volume of air in your classroom is about 160m3. Normally, it is at atmospheric pressure and about
21°C. Write down all the state variables which appear in the ideal gas law for this air for these conditions
and after each of the following operations:
a) Isobaric heating to 37°C.
b) Isochoric heating to 37°C.
c) Isothermal compression to half its initial volume.
Please circle your answers and make it c/ear which part of this problem each answer adresses.
[P(kPa),V(m3),n(mole),T(K)=(101,160,6610,294),(101,168,6610,310),(106,160,6610,310),(202,80,6610,2
94)]
8. A 4-liter cylinder of gas at atmospheric pressure is adiabatically cooled from 273°C to 190°C.
a) Calculate the change in the gas' internal energy.
b) Calculate the work that the gas does on its surroundings during this process.
c) Does the volume increase, decrease, or stay the same?
d) What is the change of entropy of the gas?
e) What is the total change in entropy of the gas and its surroundings during this process?
[-92J, +92J, increases, 0, 0]
9. A cylinder of gas at standard temperature and pressure (0°C, P atm) filling 1m3 undergoes an isobaric
expansion to 2m3 volume.
a) How many moles of gas are we talking about here?
b) What is the final temperature?
c) How much work does the gas do?
d) What is the change in the gas' internal energy?
e) How much heat flows into the gas?
10. 3m3 of an ideal gas occupy a cylinder at atmospheric pressure. For isobaric expansion of the gas to 5m3,
calculate
a) the work done by the gas,
b) the change in the internal energy of the gas,
c) the heat absorbed by the gas.
11. An empty 12oz can of soda has a volume of 355mL=0.355cm3.
a) How many moles of gas would it contain at standard temperature and pressure? If the gas is heated to
100oC, What is its
b) final volume if heated at constant pressure?
c) final pressure if heated at constant volume?
d) Which of the last two processes requires more heat? Why?
[0.0158mole, 485cc, 1.37atm, isobaric]
12. The canvas skin of a 4190m3 (l0m radius) balloon lies on the ground, untilled. A propane torch is used
to fill it with hot air.
a) How much energy does it take to fill the balloon with air at standard temperature and pressure?
b) The air inside the balloon will be at 50°C, for a density of 1.09kg/m3. If the balloon skin and gondola
weigh 1200N combined, how large a mass can the balloon lift?
[420MJ, 720kg]
13. Consider a cylindrical balloon of 10m height and 6m diameter, filled with air at 22C temperature.
a) How many moles of air are there if it is at normal atmospheric pressure?
b) If the air is heated isobarically to 65C, what is the final volume of the gas?
c) How much work does the gas do while being heated? (And is it positive or negative?)
[1.2104mole, 320m3, 4.1MJ]
14. A 2L volume of gas at atmospheric pressure is cooled at constant pressure to 1.5L. In so doing, the
magnitude of its internal energy changes by 80J.
a) What is W? Is it positive, negative, or zero?
b) What is U? Is it positive, negative, or zero?
c) Calculate the net heat flow into the gas.
[-51J, -80J, -131J]
15. Thirty-seven moles of air at 22C fills a 0.95m3 balloon.
a) If this air is heated isobarically to 62C in order to get it to float, how much work does this air does as it
expands and pushes colder air out of its way?
b) If we regard the air as an ideal gas, what is its change in internal energy?
c) What heat must be applied to the balloon?
[13kJ, 18.5kJ, 31.5kJ]
16. An airtight cylinder of gas has a movable, frictionless, massless, airtight piston inside it. The
gas outside of the piston is at atmospheric pressure. The gas inside the piston is at 23 C and
takes up 2200 cm . The cross-sectional area of the piston is 0.01 m .
a) What is the internal energy of the gas in the cylinder?
b) If the gas undergoes isobaric contraction to 1000 cm , what is the final value of U?
c) If the work done on the gas is equal to P V, where V is the change in the volume of the gas,
then what is the net amount of heat absorbed or given off by the gas in the process?
17. A hot air balloon is filled with 4 10 moles of 23 C air at atmospheric pressure.
a) What is the volume of the balloon?
b) If the balloon’s volume is increased isobarically to 110% of its original volume, what work is
done by the gas?
c) What is the final temperature of the gas?
d) How much heat flows into the gas during heating?
18. If the pressure of the gas inside a helium balloon remains the same when it is taken from
room temperature (68 F) and is placed in a container of liquid nitrogen (77K), what is the ratio of
its final to initial volume?
19. An airtight cylinder of gas has a movable, frictionless, massless, airtight piston inside of it with
a cross-sectional area of 0.004 m . Outside the cylinder is air at normal atmospheric pressure.
The temperature of the gas inside is 100 C, and it occupies 2000 cm .
a) If there is no weight on the movable piston, what is the internal energy of the gas in the
cylinder?
b) If a mass is placed on top of the piston that doubles the pressure pushing down on the gas,
and if the temperature is raised to 200 C, what is the volume of the gas?
c) Under the conditions of part (b), what is the new internal energy of the gas?
20. A hot air balloon is filled with 3 10 moles of 27C air at atmospheric pressure.
a) What is the volume of the balloon?
b) If the balloon’s contents are heated isobarically to 50 C, what is its new volume?
c) If air has a density of 1.2 kg/m , what is the maximum load that the balloon can lift in part (b)?
Temperature measurement
21. What is the lowest Fahrenheit temperature that could ever be measured?
22. Is it possible to reach a temperature of -400F? (Show why or why not.)
23. A change in temperature of 18 C is the same as a change of
a) _____ 18F
b) _____ 10F
c) _____ 32 K
d) _____ 32K
e) _____ 10 K
24. A 55C change of temperature is the same as a change of
a) _____ 31F
b) _____ 55F
c) _____ 31 K
d) _____ 99K
e) _____ 99 K
Thermal expansion
25. The thermal expansion of a material is proportional to all of the below except
a) _____ the length of the material
b) _____ the change in temperature
c) _____ the original length of the material
d) _____ the specific heat of the material
e) _____ The expansion is proportional to all of the above.
26. A machinist wants to thread of nut onto a bolt so tightly that it will be nearly impossible to get
off. She can do this by threading the nut on the bolt at a different temperature, and then letting
the temperature return to room temperature. The two possible materials she can use are brass
(=1910 /C) and steel ( =1210 /C). She plans that the nut’s interior diameter and the bolt’s
outer diameter both equal 1.00 cm at room temperature.
a) Which object should be made of which materials?
b) Should the two be heated or cooled together to thread the nut on?
c) By how much should the temperature be changed in screwing on the nut in order for the nut’s
internal diameter to be 0.0001 cm bigger than the outer diameter of the bolt?
Calorimetry with change in phase
27. It takes 5 minutes to bring a pot of 25°C water to 100°C. If the stove continues to heat the water at the
same rate, how much longer will it take for the water to completely boil away?
[36min]
28. You leave a 1L pot of 22°C water on the stove. Five minutes later, it starts to boil.
a) What is the rate at which heat is being deposited in the water?
b) If heat is applied at the same rate, how much longer will it take for the water to boil?
[1100W, 34min]
29. 50g of ice at 0ºC is placed in 1L of water at 22ºC. Find the final temperature of the system (17.2 ºC),
provided that no heat is transferred between the system and its surroundings.
[17.2oC]
30. Four 16g ice cubes are dumped into a glass of 250mL water at 15C.
a) What is the original temperature of the water in Fahrenheit?
b) If the glass is sufficiently well insulated, does all of the ice melt?
c) How much ice melts and what is the final temperature of the drink when it has come to thermal
equilibrium?
[59oF, no, 47g]
31. If someone drops a snowball (at T=0 C) off the Empire State Building (h  400 m), what
fraction of the snowball melts when the snowball hits the sidewalk?
a) If you wish to throw a snowball against a wall and have it completely melt as it hits the wall,
what is the minimum speed it must have when it hits?
32. 100 kg of snow (at 0 C) falls 400m in an avalanche from the top of a massive cliff. If all of the
snow’s energy goes into melting itself when it hits the ground, how much snow melts?
33. Let’s say that spaghetti takes 10 minutes to cook, once it is placed in boiling water. If we start
with a 100-gram copper kettle containing 10 kg of water at 68 F ...
a) What is the inital temperature of the water in C ? (Assume the kettle and water are both at
68F.
b) The specific heat of copper is 0.0920 cal/g/ C and the heat of vaporization of copper is 1150
cal/g. How many minutes will it take to heat the water to boiling if the stove puts out 3500 J/s of
heat?
c) If I forget to put in the spaghetti, how long will it take for all of the water to boil off?
34. A 30g ice cube at 0 C is put into a 250g cup of coffee at 85 C. If the specific heat of coffee is
the same as water’s, what is the final temperature of the drink?
Second law, entropy
35. A 3kg block of 0°C ice melts in the 45°C desert.
a) How much heat goes into melting the ice?
b) What is the entropy change of the ice as it melts?
c) What is the net entropy change of the ice and its surroundings as the ice melts?
36. A l09kg iceberg (approximately 100m to a side) finds itself in 15°C waters.
a) How much heat goes into melting the iceberg, and then into warming it up to 15°C?
b) What is the net entropy change in the iceberg? In the seawater?
c) Would it violate any laws for the process to have taken place in reverse? Why?
[41014J, 1.441012J/K, -1.371012J/K, Yes.]
37. A Carnot engine has the maximum efficiency for any engine operating between two specified heat
reservoirs. Consider a Carnot engine operating between 0ºC and 100ºC, which does 100J of work each
cycle.
a) What is the efficiency of this engine?
b) How much heat does it take to run this engine?
c) How much heat does it dump into its surroundings?
d) Calculate the entropy change of the 100ºC reservoir and of the 0ºC reservoir each cycle.
e) Is your answer to part (d) consistent with the Second Law? Why or why not?
[27%, 370J, 270J, -1J/K, +1J/K, yes.]
38. You are lying on the beach during Spring Break. It is 30C. Your instructor is suffering through yet
another “final snow storm of the season” back in Canton. At your side is a glass containing 100ml of fluid
which, for the purposes of this question we will assume can be considered to be mostly water. This fluid is
in thermal equilibrium with an additional 100g of ice in the glass. In the course of the afternoon, the glass
and its contents warm up to 30C. (You have been so distracted by conversation that you forgot all about
your drink.)
a) What is the entropy change of the ice and fluid?
b) Confirm that the Second Law of Thermodynamics is not violated.
[+209J/K, S=+16J/K]
39. A heat engine produces 50W of power while dumping 100J of heat at 100C. The efficiency of the
engine is 50%.
a) What is the coldest that the ‘hot’ reservoir of this engine can be?
b) How long is each cycle of the engine?
[746K, 2s]
40. 480 g of -5C ice cubes in a tray are pulled out of a freezer and allowed to melt in a 27 C
kitchen.
a) How much heat is transferred to the ice?
b) What is the entropy change of the surroundings of the ice cube?
c) Show whether the Second Law of Thermodynamics holds in this case.
41. Which of the following is not correct for an ideal gas?
a) _____ PV is proportional to the total kinetic energy of the gas molecules.
b) _____ nRT is proportional to the internal energy of the gas.
d) _____ PV/T is proportional to the total number of gas molecules
e) _____ PV/nRT is proportional to the entropy of the gas.
42. Consider a glass of 250 g of liquid water into which we have placed 50g worth of ice cubes of
0C.
a) If all of the ice melts but the final temperature of the drink is only 0 C, calculate the initial
temperature of the water.
b) How much heat is transferred between the water and the ice?
c) What is the entropy change of the ice as it melts?
d) Calculate the entropy change of the cooling water and the net entropy change of the system.
43. A 0.35 kg snowball is thrown from the top of a very tall building of 500m with an initial speed
of 100 m/s.
a) What heat goes into melting of the snowball the instant it is stopped by the pavement?
b) What fraction of the snowball is melted?
c) Show, without solving explicitly for Q, that the melting of the rest of the snowball on a warm
June afternoon (T=30 C) is consistent with the Second Law of Thermo.