- 17 PHY 121
Section 5:
e
Inclined Plane:
Example 5-a: A 15 N box is sliding down a 20 incline. If μk = .2, find its acceleration.
Ans: -1.51 m/s2
Problem 5-1: Make a sketch showing all forces on
a. a glider sliding down a frictionless incline.
b. a metal block sliding down a ramp.
c. a metal block being pulled up a ramp by a string.
Problem 5-2: A car is being towed by a chain, up a hill at a steady speed.
a. Which is larger: the pull from the chain, or friction, or are they the same size?
b. Which is larger: the normal force, or the car's weight, or are they the same size?
Problem 5-3: The cord pulls the block up the board. Both are
wood. Find the block's acceleration.
Ans: 4.80 m/s2
Uniform Circular Motion:
(Example: driving around a circular track at a constant speed.)
Newton's 1st law says your path is a straight line in the absence of a force. So, there must be a force
to the side (toward center of circle) to pull you off this straight line:
Centripetal force:
m = mass, v = velocity, r = radius
Examples:
a. For a car on a curve, the centripetal force is provided by friction between the tires and the
road.
b. For a cork being twirled on a string, it is provided by the pull from the string (string
tension).
c. For the Moon orbiting the Earth, it's Earth's gravity.
- 18 By F = ma, the acceleration this causes is:
Centripetal acceleration:
In spite of constant speed, you are accelerating, due to changing direction of your velocity vector:
Example 5-b: An unbanked curve has a radius of 80.0 m. If the pavement is dry concrete, what is
the fastest you can drive around this curve without skidding?
Ans: 28.0 m/s
Centrifugal "force":
Seen from above:
2
The eggs appear to feel an outward force of mv /r.
Due to inertia, not a true force. (Such "fictitious forces" or "pseudoforces" come from looking at the
world from an accelerating frame of reference.)
Problem 5-4: A fly clings to
a ceiling fan as it turns
counterclockwise. Draw an
arrow on the fly showing the
direction of its velocity.
Repeat for its acceleration
and the net force on it.
Problem 5-5: You put a dime 10 cm from the center of an old style record player turntable, then turn
it on. The dime goes around in a circle until it reaches a speed of .770 m/s and then it flies off.
What is the coefficient of static friction?
Ans: . 605
Prob. 5-6: A .200 kg set of fuzzy dice hangs from a rear-view mirror. As the car
takes a 250 m radius curve at 30.0 m/s, the “centrifugal force” pulls the dice to the
side. What angle does the cord make with the vertical? (Hint: Use ΣFx = ma to get
the x component from the string. Use ΣFy = 0 to get the y component from the
string. Then find θ from the string tension’s components.)
Ans: 20.2°
- 19 -
2
(Notice mv /r equals force toward center, not tension in cord.)
6
Prob. 5-7: A spaceship moves 7068 m/s in an orbit with an 8.0 x 10 m radius. Find the force of
gravity on a 75 kg astronaut inside.
Ans: 468 N
(A "weightless" astronaut is actually in free fall: The ship drops at the same rate you do, so you
float relative to the ship. The ship doesn't crash because as it moves to the side, the earth curves out
from under it.)
- 20 Section 6: Work, Energy and Power:
Definition of work:
W = (F cosθ) s
Unit: As a unit of work, a Nm is called a joule.
Problem 6-1: You carry a 100 N bucket of water 100 m over level ground. How much work do
you do on it?
Although it has no direction, work can be positive or negative:
Example: W = (25 N)(cos 180°)(100 m)
= – 2500 J
W > 0 if work done on object (energy is added.)
W < 0 if work done by object (energy is removed.)
ENERGY = the ability to do work. (Something that can do work has energy.)
Unit: Same as work (joule).
Potential Energy = amount of work an object could do because of its position.
Gravitational Potential Energy:
PE = mgh
Kinetic Energy = amount of work on object could do because of its motion.
KE = ½ mv2
Problem 6-2: 1250 J of work is done accelerating a 10.0 gram bullet from rest. What is its final
speed?
Ans: 500 m/s
CONSERVATION OF ENERGY:
Total energy of an isolated system is constant.
- 21 THE WORK - ENERGY THEOREM:
Because of conservation of energy,
(KE + PE)i + Wnc = (KE + PE)f
Wnc = Work done by any non - conservative forces. (Conservative forces can be used to store
work as a form of potential energy; non-conservative forces can't. Gravity, for example, is
conservative; friction or a force from your hand are non-conservative.)
Example 6-a: 50 J is lost to friction as the box slides down the
ramp. Find its speed at the bottom.
Ans: 6.23 m/s
Problem 6-3: A ball is dropped from a window. How fast does it strike the street, 2.5 m below?
(no friction)
Ans: 7.00 m/s
Example 6-b: A 1000 kg car is going 30 m/s when it runs out of gas at point A. If the average
force of friction is 550 N, and the car is going 2.0 m/s when it rolls into the gas station at point B,
what is A’s elevation above sea level?
(Ans: 67.2 m)
Problem 6-4: The 65 kg sled slides down the hill and across the level ground. Friction = 115 N.
Find s, the distance it goes.
(Ans: 49.9 m)
- 22 -
Problem 6-5: A 170 gram hockey puck is going 15 m/s when it first leaves the stick. It then slides
80 m before coming to rest. Find
a. the work done by friction.
b. the force of friction.
c. the coefficient of friction.
Ans: -19.1 J, .239 N, .143
POWER – Definition: The rate work is done.
It can also be shown that
P = F|| v
F|| = F cosθ, v = speed
Units: a joule/second is called a watt.
British:
1 horsepower = 550 ftlb  746 watts
sec
Problem 6-6: A 50 hp motor is used to lift an elevator. If it is to rise at 1.40 m/s, what is the
maximum force the motor can exert on the elevator?
Ans: 26.6 kN
- 23 Section 7:
Newton's third law:
If A exerts a force on B, then B exerts an equal and opposite force on A.
(Called an action - reaction pair.)
Notice:
- The reaction does not act on the same body as the action.
- Both are the same kind of force. (For example, the reaction to a gravitational force is another
gravitational force.)
Momentum:
⃑
⃑
Consider two objects pushing or pulling on each other. From the fact that their forces are equal and
opposite, it can be shown that their momentum changes are equal and opposite:
Conservation of momentum:
The total momentum of an isolated system remains constant. (In magnitude and direction.)
Example 7-a: A 20 gram bullet going 190 m/s imbeds itself in a 4.0 kg wooden block, initially at
rest. Find their common speed immediately after the collision.
Ans: .945 m/s
Problem 7-1: A 5.00 kg gun, initially at rest, fires a 5.00 g bullet at 300 m/s. Find the gun's speed
of recoil.
Ans: .300 m/s
(The same effect is responsible for rocket propulsion.)
- 24 (Notice in 7-1, Ei = 0, Ef > 0.)
Elastic collision: One where E is conserved.
Inelastic collision: One where E is not conserved.
(The most extreme case is a perfectly inelastic collision: one where they stick together.)
2
mv is conserved either way, but not 1/2 mv unless it's elastic.
Remember to treat momentum as a vector:
Example 7-b: Two balls of mass m
collide on a pool table, as shown. Find
v1 and v2.
Ans: 3.50 m/s, 2.87 m/s
Problem 7-2: A clumsy astronaut, floating in a
space station, bumps into another astronaut as
shown. Find their final speeds.
Ans: .808 m/s, .620 m/s
Problem 7-3: A 1000 kg car is going in the x direction at 25 m/s, and a 1500 kg car is going in the y
direction at 20 m/s. They have a perfectly inelastic collision at an intersection. (They lock
together.) Find the magnitude and direction of their velocity just after they collide.
Ans: 15.6 m/s, 50.2°
Rotation:
θ = how far
ω = how fast
α = rate it picks up speed
- 25 Units for θ: 1 revolution = 360 = 2π radians
θ in radians is defined as
θ=
(or, s = rθ)
Notice an angle in radians in just a dimensionless number:
Example: θ =
= 2.00
cm over cm cancels out. The 2.00 has no unit.
Written 2.00 rad to avoid confusion with 2.00°.
Definition: Average angular velocity: ωav =
ω: "omega"
Problem 7-4: It takes exactly 3 minutes for a turntable to make 100 revolutions. Find ω in rev/min
and rad/sec.
Ans: 33.3 rpm, 3.49 rad/s
Definition: Average angular acceleration: αav =
α: "alpha"
Example: A fan's rotation rate drops from 4.00 rev/sec to 0 during 10.0 sec. Find α.
α=
= – .400 rev/s2
(Minus means slowing down.)
For uniformly accelerated rotational motion:
5 equations on formula sheet, analogous to those in sec. 1
Δθ = ωavt
ωav = ½(ωi +ωf)
2
Δθ = ωit + ½αt
ωf = ωi + αt
2
2
ωf = ωi + 2αΔθ
Problem 7-5: A wheel originally turning at 50.0 rev/min has a constant angular acceleration of .250
rev/s2. Find how far it turns in 4.00 sec.
Ans: 5.33 rev
- 26 Section 8:
From θ = s/r in previous section,
s = rθ
θ must be in radians
Divide both sides by t. v = s/t, ω = θ/t
vT = rω
ω in rad/s
vT = tangential velocity: v of a point on
wheels edge, of box, etc.
 by t again: aT = rα
α in rad/s2
Tangential acceleration
Example 8-a: The small pulley turns at 500 rev/min .
Find:
a) The belt's velocity.
b) The angular velocity of the large pulley.
Ans: 2.62 m/s, 13.1 rad/s (or 125 rpm)
Problem 8-1: Label each of these as s, v, a, θ, ω or α:
5 m/s
4m
7 rad
2
9 m/s
8 rev/min
12 mi/hr
3 rad/s
2
6 rad/s
Problem 8-2: A wheel rolls down the road at 15 m/s without slipping. If its radius is 40 cm, find its
angular velocity in (a) radians per second, and (b) revolutions per minute.
Ans: 37.5 rad/s, 358 rev/min
Problem 8-3: A car goes from 0 to 30 m/s in 15 s. If its wheels have a 40 cm radius, find (a) the
tangential acceleration of a small stone stuck in the tread, and (b) the angular acceleration of the
wheels.
Ans: 2.00 m/s2, 5.00 rad/s2
Problem 8-4: 30 revolutions after starting from rest, the wheel is turning at 40
rev/min. At this time, find
a) the angle it has turned, its angular velocity, and its angular acceleration.
b) the distance the box has moved, its velocity, and its acceleration.
Ans: (Some of what it asks for is given information. Putting answers
here would give those away.)
- 27 Rotational Kinetic Energy:
A point-like particle on a circular orbit:
½mv2 = ½m(rω)2 = ½mr2ω2 = ½(mr2)ω2
mr2 is called the particle’s moment of inertia, I.
So, KE = ½ I ω2
Anything else that rotates can be thought of as a collection of
particles like that one. To get system's total I, add I's of its parts:
2
I = Σmr
2
What Σmr adds up to depends on the object's shape. See list on formula sheet. Notice that I is
larger when mass is located farther from the axis of rotation.
Example 8-b: In each case, the system
starts from rest. Find how fast the rod is
spinning when the weight hits the floor.
(The rod and small wheel at the center
have negligible mass. The box hits the
floor with a speed small enough to
ignore.)
Ans: 7.00 rad/s, 2.33 rad/s
Example 8-5: Both wheels have the same mass and
radius, and start at rest. The boxes fall from the same
height. Compare the final angular speeds.
-4
Problem 8-6: The mass of a certain yo-yo is 100 g, and its moment of inertia is 1.20 x 10 kgm2.
If it is spinning at 70 rad/s at the bottom of the string, how high will it climb before stopping?
Ans: .300 m
- 28 Review of sections 5 - 8:
As usual, the best 4 out of 5 count, for 25 points each. Questions on the actual test are not
necessarily similar to these.
1. A machine weighing 1100 N, mounted on skids, is being pulled at a constant speed up a 25°
incline by a cable parallel to the incline. If the tension in the cable is 620 N, what is the coefficient
of kinetic friction?
Ans: .156
2. Two objects collide as shown.
What is the final speed of the 3.0 kg
object?
Ans: 1.91 m/s
3. The hammer of a pile driver has a mass of
230 kg, and on impact drives the pile 13 cm
deeper into the ground. If it is released 4.7 m
above its final stopping point, what is the
average force it exerts on the pile? (You may
use any method you want, but this is easiest
using the work - energy theorem.)
Ans: 81.5 kN
4. A carousel accelerates uniformly from rest, with an angular acceleration of 4.1 rev/min2. At the
end of 1.2 minutes, find:
a. its angular velocity in revolutions per minute,
b. the velocity of someone 7.0 meters from the center, in meters per second.
Ans: 4.92 rev/min, 3.61 m/s
5. Short answer, 5 points each:
a. A large, fast moving truck drives into the back of a small, slow moving car. During the
collision, is the force on the truck from the car
i. in a forward direction, or toward the back?
ii. larger, smaller, or the same size as the force on the car from the truck?
b. A record rotates at a steady 33 rpm for 3 minutes. What is its angular acceleration, α?
- 29 c. A solid sphere and a hollow sphere, whose masses and radii are identical, are both spinning
with the same angular velocity about axes through their centers. Which has the greater
rotational kinetic energy, or are both the same?
d. The force from the rope maintains a constant
magnitude as the block slides along. Compare the work
the rope does on the block between A and B to the work
done between B and C.
e. A car is going around a curve at a constant speed. How many times as much is its
acceleration at 60 miles per hour compared to if it went 30 miles per hour?