Practice Problems
Moments of Inertia about the center of mass: Disk I = 1 mr 2 , Solid sphere I = 2 mr 2 , hollow
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sphere I = 2 mr , Thin Rod of length L rotating around the end I = 1 mL2
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1. Batman steps off a window ledge 20 meters off the ground, holding onto the end of a massless cable
attached to something 40 meters off the ground. The cable is perfectly straight, and initially makes an
angle of 59° with respect to the vertical, as shown. He swings on the cable, and when he is at his
lowest point, he grabs Robin who was about to be trampled by a rampaging rhinoceros. Robin was
running at 4 m/s away from Batman when he grabbed him. As Batman continues to swing on the cable
(now with Robin swinging with him), what is the maximum angle they will swing up to (θ in the
diagram below)? Batman has a mass of
90 kg, and Robin is 40 kg.
θ
59°
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2. OTE: This problem is pretty hard – more complicated than a problem you could be given on
an exam. So if you are pressed for time, you might be better off skipping this one for now, and
coming back to it later if you have time.
A 1500 gram block of wood has a fan attached to it, which provides a constant thrust of 20 Newtons.
The block has a coefficient of friction of 0.4 between it and the floor. The block is released from rest,
and after it travels 1 meter, it smacks onto the bottom of a thin rod of length 0.5 meters and 2 kg (the
rod is initially hanging perfectly vertically as shown). The block of wood has superglue on the front, so
it sticks to the rod. The rod can pivot around the top. What is the maximum angle, measured from
vertical, that the rod (with the block attached) swings to if the fan shuts off the moment the block hits
the rod? (This problem has a couple tricky steps (some that we haven’t talked about much), so I will
break it down into parts for you to help you solve it. When you are comfortable solving it with me
breaking it into parts, come back to the problem later and see if you can figure out the steps for
yourself, without the questions below that help guide you through it)
a.) The picture to the right shows the
initial condition – both the block and
rood are stationary (before the fan starts
accelerating the block). Draw below
pictures of the two other important times
in this problem.
pivot
L
Wood
block
Thrust
b.) What is the speed of the block just before it hits the rod?
c.) In the Batman and Robin problem, you could treat both Batman and Robin as point masses, and
thus use conservation of linear momentum. In this problem, the wood block can presumably be viewed
as a point mass – but can you do that with the rod? In other words, can we treat the rod as if the entire
thing is moving with some particular velocity v? Based on this – should we say that it is linear
momentum that is conserved during the collision, or angular momentum?
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d.) What is the angular velocity of the rod (and block together) just after the collision?
e.) After the collision, the block and rod rotate together. We could view the block as a point mass
moving with some translational velocity, or we could instead treat it as part of the rod-block system,
figuring out their combined moment of inertia, and treat the block’s kinetic energy purely as rotational
kinetic energy due to it rotating around the pivot (along the with rest of the rod). Let’s do the latter. So
– what is the combined moment of inertia of the block and rod together? (note: you should have
already calculated this for the previous problem, when doing angular momentum conservation)
f.) We want to know the angle the rod swings to, but the angle itself won’t show up in our energy
relationship – instead height does (in gravitational energy, for the rod and for the block). So, let’s first
focus on how high they swing. The block can be treated as a point mass, so all of its mass is at one
spot. The rod though has its mass distributed along the entire rod. Do we need to worry about that
though for potential energy? Or can we treat the mass as if it is at one spot for that purpose – and if so,
where?
g.) Put both the final height of the rod’s center of mass and the final height of the block in terms of the
final angle, and use conservation of energy to solve for that maximum angle.
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3. Two blocks are connected by a massless rope hung over a pulley as shown to the right. The pulley
has a mass of 2 kg and diameter of 10 cm. How long will it take for the 4 kg
block to reach the floor?
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4. Considering the same situation, how fast will the 4 kg block be moving when it hits the floor? First
solve by using your answer to the previous problem and kinematics. Then use work-energy to solve,
and make sure the answers agree.
5. A solid sphere rolls down a hill (h1=2 meters), around a 30 cm loop, and off a 30 cm tall ramp (h2)
inclined at 20°. The ball will then fly across a treacherous moat filled with miniature alligators, and
smack into a wall on the other side. The wall is 80 cm (L) away from the ramp, horizontally. If the ball
has a mass of 100 grams and a radius of 10 cm, at what elevation (measured with respect to the base of
the ramp) does the ball hit the wall? The ball will start rolling (spinning) as it goes down the hill, but
there will not be any frictional losses (i.e. it’s like a marble – it will roll, but not lose energy due to
friction).
h1
Loop of
radius 30 cm
20°
h2
L
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5. b. For the previous problem, make sure that you would also be comfortable solving it if there were
friction involved. Say for example there is a rough patch with some coefficient of friction µ for a
distance L between the hoop and the ramp – what would change in your analysis?
6. Two objects collide in outer space. Both momentum and mechanical energy are always conserved
for such a collision.
(a) True
(b) False
7. An object moving in the x-direction experiences a force of 2 N, also in the x-direction, for a duration
of 2 seconds. The work done by the force on the object is:
(a) 1 N
(b) 2 N
(c) 4 N
(d) There is not enough information to tell
8. Three cars (F, G, and H) are moving with the same velocity, and simultaneously slam on the brakes.
The heaviest car is F, and the lightest is H. Assuming all three cars have identical tires and braking
systems, which car travels the greatest distance as it skids to a stop?
(a) F
(b) G
(c) H
(d) They all travel the same distance
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9. For the same situation as above, which car takes the longest time to stop?
(a) F
(b) G
(c) H
(d) They all take the same amount of time to stop
10. If the potential energy on a particle at some point in space is zero, then the net force acting on it
must also be zero.
(a) True
(b) False
11. A spring loaded gun has a spring that is compressed 10 cm when the gun is “loaded”. The gun is
held at an elevation of 2 meters (the end of the barrel is 2 m above the ground) and pointed straight
upward. When fired, the 50 gram bullet is in the air for 7 seconds before it lands on the ground (the
gun was pulled out of the way so the bullet could fall all the way to the ground). The
barrel of the gun is completely frictionless. What is the spring constant of the spring in
the gun?
12. A uniform beam of length 7.60 meters and mass 50 kg is carried by two workers, Sam and Joe, as
shown to the right. What vertical force is each worker exerting?
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13. Block 1 is on top of block 2, as shown below. There is friction between the two blocks, and
between block 2 and the plane. You are initially holding the blocks stationary in the position shown,
with the left edge of block 1 a distance x away from the left edge of block 2. When you release the
blocks, block 1 starts to accelerate down the plane (leftward) while block 2 accelerates up the plane.
How long will it take for the left edge of block one to be lined up with the left edge of block 2?
The pulley has a mass of mp and a radius of r. The blocks have masses m1 and m2, and µk is the
coefficient of kinetic friction between the two blocks and block 2 and the plane. Block 1 is heavier
than block 2. (note that this is similar to the last problem on exam 2, but here the blocks are going
different ways, there is friction on all surfaces, and the
pulley has mass)
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14. A 10 kg rectangular door hangs from two hinges, as shown to the right. The door is 1 meter wide
and 2.2 meters tall. The hinges are attached 0.5 m from the top (hinge A) and 0.5 m from the bottom
(hinge B). What is the horizontal force that each hinge exerts on the door (also indicate direction)?
(don’t worry about finding the vertical force each hinge applies to the door)
hinge A
hinge B
15. Stan is riding his big wheel at a constant 5 m/s, heading due east. Just as he passes Kenny, Kenny
ignites a rocket on the back of his own big wheel, giving his (Kenny’s) big wheel an acceleration of 3
m/s2, to the east. If Kenny was initially traveling at 1 m/s to the east, how long will it take Kenny to
catch up to Stan, and how far away from the point at which Stan first passed Kenny will they meet
again?
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16. Cartman is standing on a bridge watching Kenny and Stan race on their big wheels. The bridge is
10 meters due east from where Stan first passes Kenny. Cartman wants to drop a water balloon such
that it hits Kenny just as he passes underneath the bridge. If Cartman drops the balloon from a height 6
meters above the elevation of Kenny’s head, how far to the west of the bridge should Kenny be when
Cartman drops the 400 gram water balloon?
17. A 5 kg block rests on a 40° incline as shown to the right. The coefficients of static and kinetic
friction between the block and the incline are 0.4 and 0.2, respectively. You are exerting a horizontal
force (F) on the block.
a) What is the minimum force you must push with in order to keep
the block from sliding down the incline?
F
b.) How hard do you need to push to get the block to start sliding up the incline?
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18. A 12 kg model rocket is blasting off, zooming straight upwards (in reality, the mass of a rocket
decreases as its engine propels it, since the engine essentially ejects mass out the back. Let’s assume
that lost mass though is negligible compared to the total rocket mass, so we can assume its mass
remains constant). Its engine provides a constant thrust of 300 N. After 7 seconds, a malfunction
causes an internal explosion that blows the rocket into two pieces. Right after the explosion, one piece
has a velocity of 40 xˆ + 160 yˆ . The second piece has twice the mass of that first piece. When the second
piece eventually lands, how far away will it land from the spot it launched from? (yes, this is not an
easy problem. I’ll break it down into parts for you – when you are comfortable solving it with the parts
I lay out, see if you can solve it without that guidance)
a.) Use Newton’s 2nd Law and kinematics to find the velocity of the rocket just before the explosion.
b.) What is the elevation of the rocket when it explodes? (you may have already found this for part a)
c.) What will the velocity of the second piece be just after the explosion?
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d.) How long after the explosion will it take the second piece to fall back to the ground?
e.) How far does it move horizontally (x direction) in that amount of time?
19. A conical pendulum is formed by attaching a 500 gram ball to a 1.0 meter long string, then
allowing the mass to move in a horizontal circle of radius 20 cm.
a.)What is the tension in the string?
b.) What is the ball’s angular speed in rpm?
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20. A 5 kg block is released from rest at the top of an incline that is 3 meters tall and 6 meters long, as
shown below. The coefficient of kinetic friction between the block and the incline is 0.1. How fast is
the block going when it gets to the bottom? Solve in two completely different ways, and make sure
your answers agree.
6m
3m
Don’t forget to review the earlier exams, the activities we have done since the last exam.
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