Unit 6
Gravitational Force Questions
1. If the distance between two point particles is doubled, what happens to the gravitational force
between them?
2. At the surface of the Earth, an object of mass m has weight w. If this object is transported to
an altitude that's twice the radius of the earth, then, at the new location what is its mass and
weight?
3. A moon of mass m orbits a planet of mass 100m. Let the strength of the gravitational force
exerted by the planet on the moon be denoted by F1, and let the strength of the gravitational force
exerted by the moon on the planet be F2. What is the relationship between F1 and F2?
4. The planet Pluto has 1/500 the mass and 1/15 the radius of Earth. What is the acceleration
due to gravity (gP) on the surface of Pluto?
5. Two large bodies, Body A of mass m and Body B of mass 4m, are separated by a distance R.
At what distance from Body A, along the line joining the bodies, would the gravitational force on
an object be equal to zero? (This is particularly challenging and requires a solid understanding of
algebra).
Answers:
1. One-fourth of original
2. Mass is unchanged, Weight is ¼ of surface weight.
3. Equal in magnitude, opposite in direction.
4. 9g/20 (leave answer expressed as a fraction in terms of g)
5. R/3 from Body A
76B1 - The two guide rails for the elevator shown above each exert a constant friction force of 100 N each on the
elevator car when the elevator car is moving upward with an acceleration of 2 m/s2. The pulley has negligible friction
and mass.
(a) On the diagram below, draw and label all forces acting on the elevator car. Identify the source of each force.
(b) Calculate the tension in the cable lifting the 400-kilogram elevator car during an upward acceleration of
2 m/s2.
(c) Calculate the mass M the counterweight must have to raise the elevator car with an acceleration of 2 m/sec 2.
79B2 - A 10-kilogram block rests initially on a table as shown in Case I. The coefficient of sliding friction
between the block and the table is 0.2. The block is connected to a cord of negligible mass, which hangs
over a massless frictionless pulley.
a.
Calculate the acceleration of the 10-kg block.
A 5-kg mass is now hung on the bottom of the cord as shown in Case II.
b.
On the diagrams below, draw and label all the forces acting on each block in Case II.
10 kg
10 kg
5 kg
5 kg
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c.
Calculate the acceleration of the 10-kg block in Case II.
d.
Explain why the acceleration of the 10-kg block is different in Case II than in Case I.
87B1 - In the system shown above, the block of mass M 1 is on a rough horizontal table. The string that
attaches it to the block of mass M2 passes over a frictionless pulley of negligible mass. The coefficient of
kinetic friction µk between M1 and the table is less than the coefficient of static friction µs.
a.
On the diagram below, draw and identify all the forces acting on the block of mass M 1 .
M1
b. In terms of M1 and M 2 determine the minimum value of µs that will prevent the blocks from moving.
The blocks are set in motion by giving M2 a momentary downward push. In terms of M1, M2, µk, and g,
determine each of the following:
c.
The magnitude of the acceleration of M1.
d. The tension in the string.
86B1 - Three blocks of masses 1.0, 2.0, and 4.0 kg are connected by massless strings, one of which passes
over a frictionless pulley of negligible mass, as shown above. Calculate each of the following.
a.
The acceleration of the 4-kilogram block
b. The tension in the string supporting the 4-kilogram block
c.
The tension in the string connected to the l-kilogram block
82B2 - A crane is used to hoist a load of mass m1 = 500 kilograms. The load is suspended by a cable from a
hook of mass m2 = 50 kilograms, as shown in the diagram above. The load is lifted upward at a constant
acceleration of 2 m/s2.
a.
On the diagrams below draw and label the forces acting on the hook and the forces acting on the
load as they accelerate upward.
b.
Determine the tension T1 in the lower cable and the tension T 2 in the upper cable as the hook and
load are accelerated upward at 2 m/s2.
17M1 - An Atwood’s machine consists of two blocks connected by a light string that passes over a
frictionless pully of negligible mass, as shown in the figure above. The masses of the two blocks, M1 and
M2, can be varied. M2 is always greater than M1.
a) On the dots below, which represent the blocks, draw and label the forces (not components) that
act on the blocks. Each force must be represented by a distinct arrow starting on and pointing
away from the appropriate dot. The relative lengths of the arrows should show the relative
magnitudes of the forces.
b) Using the forces in your diagrams above, write an equation applying Newton’s second law to
each block and use these two equations to derive the magnitude of the acceleration of the blocks
and show that it is given by the equation:
𝑎=
(𝑀2 − 𝑀1 )
𝑔
(𝑀1 + 𝑀2 )
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The magnitude of the acceleration a was measured for different values of M1 and M2 and the data are
shown below.
c) Indicate below which quantities should be graphed to yield a straight line whose slope could be
used to calculate a numerical value for the acceleration due to gravity g.
Vertical axis: _____________
Horizontal axis: ______________
Use the remaining rows in the table above, as needed, to record any quantities that you indicated
that are not given.
d) Plot the data points for the quantities indicated in part (c) on the graph below. Clearly scale and
label all axes including units, if appropriate. Draw a straight line that best represents the data.
e) Using your straight line, determine an experimental value for g.
Inelastic Collisions
1. A 1550 kg torpedo strikes a 770 kg target that is initially at rest. If the combined torpedo and target
move forward with a speed of 9.44 m/s, what is the initial velocity of the torpedo? Assume that no
resistance is provided by the water.
2. An ice hockey puck with a mass of 0.17 kg collides with a 0.75 kg snowball that is sliding to the left
with a speed of 0.50 m/s. The combined puck and snowball slide along the ice with a velocity of 4.2 m/s
to the right. What is the velocity of the hockey puck before the collision?
3. A 500 kg log collides with a second log with the same mass. These combined logs then collide with a
third log with a mass of 500 kg. The final speed of the three combined logs is 3.67 m/s. If the speed of
the third log before collision was 3.00 m/s, and the speed of the second log before collision was 3.50
m/s, what was the speed of the first log before collision?
4. A railway car with a mass of 8500 kg and a velocity of 4.5 m/s to the right collides with a railway car
with a mass of 9800 kg and a velocity of 3.9 m/s to the left. What is the final velocity of the combined
cars?
5. A 25.0 kg sled carrying a 42.0 kg child is moving with a speed of 3.50 m/s when it collides with a
snowman that is initially at rest. If the speed of the snowman, sled, and child is 2.90 m/s, what is the
snowman’s mass?
6. A remora is a type of fish that uses suckers underneath its head to attach itself to other fish, notably
sharks (for this reason it is also called the “sharksucker”). Suppose a remora swimming with a velocity
of 5.0 m/s to the right attaches itself to a 150.0 kg shark that is swimming to the left with a speed of 7.00
m/s. If the remora collides with the shark, the velocity of the two fish combined is 6.25 m/s to the left.
From this information, calculate the mass of the remora.
1.
2.
3.
4.
5.
6.
14 m/s forward
25 m/s to the right
4.5 m/s
0 m/s
14 kg
10 kg
MOMENTUM PRACTICE PROBLEMS
1. A 10.0 g bullet is fired from a 5.00 kg rifle with a velocity of 300 m/s.
What is the recoil velocity of the rifle?
2. In an ice skating show, a 90 kg man at rest pushes a 50 kg woman away
from him at a speed of 0.75 m/s. What is the man’s final velocity?
3. A 5000 kg cannon fires a shell of 3.00 kg mass with a velocity of 250 m/s.
What is the recoil velocity of the cannon?
4. A 1250 kg car traveling at 20 m/s collides “head-on” with a 9,000-kg truck
traveling toward the car at 15 m/s. The car becomes stuck to the truck
during the collision. What is the final velocity of the car and truck?
5. A 3.00 g bullet is fired at 200 m/s into a wooden block of 10 kg mass that is
at rest. If the bullet becomes embedded in the wooden block, find the
velocity of the block and bullet after impact.
6. A 0.15 kg cue ball moving at 0.15 m/s hits the 8 ball that is at rest. The cue
ball continues moving in the same direction at 0.02 m/s. What is the
velocity of the 8 ball?
7. A 2 kg ball moving to the right at 6 m/s hits a 1.5 kg ball that is moving to
the left at 5 m/s. If the 2 kg ball begins moving at 0.8 m/s, what is the
velocity of the 1.5 kg ball?
8. A 1500 kg pickup truck is traveling at 20 m/s when it begins to rain and
water collects in the back. If the truck slows down to 18 m/s, how much
water collected in the truck bed?
1. -0.6 m/s
2. -0.42 m/s
3. -0.15 m/s
4. -10.7 m/s
5. 0.059 m/s
6. 0.13 m/s
7. 1.93 m/s
8. 167 kg
Conservation of Momentum
1. A student stumbles backward off a dock and lands in a small boat. The boat (with the student in
it) drifts away from the dock with a velocity of 0.85 m/s to the west. If the boat and student each
have a mass of 68 kg, what is the student’s initial horizontal velocity?
2. A coal barge with a mass of 13,600 kg drifts along a river. When it passes under a coal hopper, it
is loaded with 8400 kg of coal. What is the speed of the unloaded barge if the barge after loading
has a speed of 1.3 m/s?
3. A child is riding on a sled and jumps off of it. He jumps at a speed of 2.2 m/s and in the direction
opposite the sled’s motion. The sled continues to move in the forward direction, but with a new
speed of 5.5 m/s. If the child has a mass of 38 kg and the sled has a mass 68 kg, what is the initial
velocity of the sled?
4. A swimmer with a mass of 58 kg and a velocity of 1.6 m/s to the north climbs onto a 142 kg raft.
The combined velocity of the swimmer and raft is 0.32 m/s to the north. What is the raft’s
velocity before the swimmer reaches it?
5. A 50.0 g shell fired from a 3.00 kg rifle has a speed of 400.0 m/s. With what speed does the rifle
recoil in the opposite direction?
6. A twig floating in a small pond is initially at rest. On the twig is a snail, which begins moving
along the length of the twig with a speed of 1.2 cm/s. The twig moves in the opposite direction
with a speed of 0.40 cm/s. If the snail’s mass is 2.5 g, what is the mass of twig?
7. An ice skater at rest catches a bag of sand moving to the north with a speed of 5.4 m/s. This
causes both the skater and the bag to move to the north at a speed of 1.5 m/s. If the skater’s mass
is 63 kg, what is the mass of the bag of sand?
1.
2.
3.
4.
5.
6.
7.
1.7 m/s to the west
2.1 m/s
2.7 m/s forward
-0.203 m/s
6.67 m/s backward
0.0075 kg or 7.5 g
24.2 kg
12B2 - A small and a large sphere, of mass M and 3M respectively, are arranged as shown on the left side of the
figure above. The spheres are then simultaneously dropped from rest. When the large sphere strikes the floor, the
spheres have fallen a height H. Assume air resistance is negligible. Express all answers in terms of M, H, and
fundamental constants, as appropriate.
(a) Derive an expression for the speed vb with which the large sphere strikes the floor.
Immediately after striking the floor, the large sphere is moving upward with speed vb and collides head-on with the
small sphere, which is moving downward with identical speed vb at that instant. Immediately after the collision, the
small sphere moves upward with speed vS and the large sphere has speed vL.
(b) Derive an equation that relates vb, vS, and vL.
In this particular situation vL = 0.
(c) Use your relationship from part (b) to determine the speed of the small sphere in terms of vb.
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(d) Indicate whether the collision is elastic. Justify your answer using your results from parts (b) and (c).
(e) Determine the height h that the small sphere rises above its lowest position, in terms of the original height
H.
06Bb2 - A small block of mass M is released from rest at the top of the curved frictionless ramp shown above. The block slides
down the ramp and is moving with a speed 3.5vo when it collides with a larger block of mass 1.5M at rest at the bottom of the
incline. The larger block moves to the right at a speed 2vo immediately after the collision. Express your answers to the
following questions in terms of the given quantities and fundamental constants.
a.
Determine the height h of the ramp from which the small block was released.
b.
Determine the speed of the small block after the collision.
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c.
The larger block slides a distance D before coming to rest. Determine the value of the coefficient of kinetic friction μ
between the larger block and the surface on which it slides.
d.
Indicate whether the collision between the two blocks is elastic or inelastic. Justify your answer.
80M2 modified - A block of mass m slides at velocity vo across a horizontal frictionless surface toward a
large curved movable ramp of mass 3m as shown in Figure 1. The ramp, initially at rest, also can move
without friction and has a smooth circular frictionless face up which the block can easily slide. When the
block slides up the ramp, it momentarily reaches a maximum height as shown in Figure II.
a. Find the velocity v1 of the moving ramp at the instant the block reaches its maximum height.
b. To what maximum height h does the block rise above its original height?
The experiment is repeated except this time the ramp is fixed and cannot move.
c. Would the maximum height the block rise above its original height be more than, less than, or the
same as in initial trial (when the ramp was able to move)? Explain.
______ More than
Explain.
______ Less than
______ Same (as in the initial trial)
01B2 modified - An incident ball A of mass 0.10 kg is sliding at 1.4 m/s on the horizontal tabletop of
negligible friction shown above. It makes a head-on collision with a target ball B of mass 0.50 kg at rest
at the edge of the table. As a result of the collision, the incident ball rebounds, sliding backwards at 0.70
m/s immediately after the collision.
a. Calculate the speed of the 0.50 kg target ball immediately after the collision.
The tabletop is 1.20 m above a level, horizontal floor. The target ball is projected horizontally and
initially strikes the floor at a horizontal displacement d from the point of collision.
b. Calculate the horizontal displacement d.
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B
B
In another experiment on the same table, the incident ball A again slides across the table at the same
speed but this time makes a glancing collision with the target ball B that is at rest at the edge of the table.
The target ball B has a velocity v when it leaves the table and strikes the floor at point P, which is at a
horizontal displacement of d1 from the point of the collision, and at a horizontal angle of 30° from the
+x-axis, as shown above right.
c.
i.
Would the velocity v of the ball be greater than, less than, or the same as the velocity
determined in part a)?
_____ Greater than
ii.
_____ Less than
_____ Same
Would d1 be greater than, less than, or the same as the distance d determined in part b)?
_____ Greater than
_____ Less than
_____ Same
d. In a coherent, well-organized paragraph explain your answers to part c).
14B1 - Starting from rest at point A, a 50 kg person wings along a circular arc from a rope attached to a
tree branch over a lake, as shown in the figure above. Point D is at the same height as point A. The
distance from the point of attachment to the center of mass of the person is 6.4 m. Ignore air resistance
and the mass and elasticity of the rope.
a) The person swings two times, each time letting go of the rope at a different point.
i.
On the first swing, the person lets go of the rope when first arriving at point C. Draw a
solid line to represent the trajectory of the center of mass after the person releases the
rope.
ii.
A second time, the person lets go of the rope at point D. Draw a dashed line to represent
the trajectory of the center of mass after the person releases the rope.
b) The center of mass of the person standing on the platform is at point A, 4.1 m above the surface of
the water. Calculate the gravitational potential energy when the person is at point A relative to
when the person is at the surface of the water.
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c) The center of mass of the person at point B, the lowest point along the arc, is 2.4 m above the
surface of the water. Calculate the person’s speed at point B.
d) Suppose that the person swings from the rope a third time, letting go of the rope at point B.
Calculate R, the horizontal distance moved from where the person releases the rope at point B to
where the person hits the water.
e) If the person does not let go of the rope, how does the magnitude of the person’s momentum, pC,
at point C compare with the magnitude of the person’s momentum, pB, at point B?
_____ pC > pB
______ pC < pB
_____ pC = pB
Provide a physical explanation to justify your answer.
76B2 - A bullet of mass m and velocity v0 is fired toward a block of inertial mass 4m. The block is initially at rest
1
on a frictionless horizontal surface. The bullet penetrates the block and emerges with a velocity of 𝑣0 .
3
(a) Determine the final speed of the block.
(b) Determine the loss in kinetic energy of the bullet.
(c) Determine the gain in the kinetic energy of the block.
85B1 - A 2-kilogram block initially hangs at rest at the end of two 1-meter strings of negligible mass as shown on the
left diagram above. A 0.003-kilogram bullet, moving horizontally with a speed of 1000 m/s, strikes the block and
becomes embedded in it. After the collision, the bullet/block combination swings upward, but does not rotate.
a.
Calculate the speed v of the bullet/block combination just after the collision.
b.
Calculate the ratio of the initial kinetic energy of the bullet to the kinetic energy of the bullet/block
combination immediately after the collision.
c.
Calculate the maximum vertical height above the initial rest position reached by the bullet/block combination.
The experiment is repeated with an identical bullet (same mass and initial velocity), and a 2-kilogram block which is
shorter such that the bullet passes through the block.
d.
State whether the block will have a maximum vertical height greater than, less than, or equal to the height
determined in part c and explain.
______ Greater than
Explain.
______ Less than
_______ Equal to
90B1 - A bullet of inertial mass m is moving horizontally with speed vo when it hits a block of inertial mass
100m that is at rest on a horizontal frictionless table, as shown above. The surface of the table is a height h
above the floor. After the impact the bullet and the block slide off the table and hit the floor a distance x
from the edge of the table. Derive expressions for the following quantities in terms of m, h, vo, and
appropriate constants:
a.
the speed of the block as it leaves the table
b.
the change in kinetic energy of the bullet-block system during impact
c.
the distance x
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Suppose that the bullet passes through the block instead of remaining in it.
d.
State whether the time required for the block to reach the floor from the edge of the table would
now be greater, less, or the same. Justify your answer.
e.
State whether the distance x for the block would now be greater, less, or the same. Justify your
answer.
92B2 - A 30-kilogram child moving at 4.0 meters per second jumps onto a 50-kilogram sled that is initially
at rest on a long, frictionless, horizontal sheet of ice.
a.
Determine the speed of the child-sled system after the child jumps onto the sled.
b.
Determine the kinetic energy of the child-sled system after the child jumps onto the sled.
After coasting at constant speed for a short time, the child jumps off the sled in such a way that she is at
rest with respect to the ice.
c.
Determine the speed of the sled after the child jumps off it.
d.
Determine the kinetic energy of the child-sled system when the child is at rest on the ice.
e.
Compare the kinetic energies that were determined in parts (b) and (d). If the energy is greater in
(d) than it is in (b), where did the increase come from? If the energy is less in (d) than it is in (b),
where did the energy go?
04M1 - A rope of length L is attached to a support at point C. A person of mass m1 sits on a ledge at position A holding
the other end of the rope so that it is horizontal and taut, as shown above. The person then drops off the ledge and swings
down on the rope toward position B on a lower ledge where an object of mass m2 is at rest. At position B the person
grabs hold of the object and simultaneously lets go of the rope. The person and object then land together in the lake at
point D, which is a vertical distance L below position B. Air resistance and the mass of the rope are negligible. Derive
expressions for each of the following in terms of m1, m2, L, and g.
a.
The speed of the person just before the collision with the object.
b.
The tension in the rope just before the collision with the object.
c.
The speed of the person and object just after the collision.
d.
The ratio of the kinetic energy of the person-object system before the collision to the kinetic energy after the
collision.
e.
The total horizontal displacement x of the person from position A until the person and object land in the water
at point D.