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Unformatted text preview: Problem #1: Ranking Problems For this proofent (inciuding both Part i and Part if), you snoutd not need to do any
caicuiations, beyond very simpie ones that you can do in your head. Part I: Applied Torques The foliowing diagrams shovv six identicai discs of radius 60 cm, each free to rotate about an axis through its center. A force is applied to each disc at a different
+ ) tocation, as shovvn * Rank the six discs according to the torque they experience from these forces, from
the most positive torque through most negative. (Use the usuai convention:
) counterctockwise torques are positive, clockwise are negative.) _——‘*_— —__——I—II
We+ Poeﬁ‘twe I ___——‘ ““———
ff: NR7 W+ regent ate Five massspring systems are set up as shown below, with varying Spring
constants, masses, and amplitudes.‘ ' 1,: 4o M/M it via
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3*“ ﬁt in»: ' a) Flank the five systems according to their frequenciesfrom highest frequency to 
lowest. indicate any ties. aﬁea‘t"
"‘ F” Is) Now rank the systems according to their total energy (kinetic plus potential),
from highest energy to lowest. jmclmﬂlg any +rggr  r} Wff‘ Wm? “WM 7 QM’W Problem #2: Graphs ‘n' Stuff  (30 points) The following graph shows a wheel's angular speed cu (measured in radians per second, or simply in s1) as a function of time t. The wheei‘s moment of inertia about its axis of spin is I: 3.0 kg m2. 1 a .5 3’ to 11 Mr is t: (Sec) Try to deduce the answers to the feiiowing questions directiy from the graph; iittie or
no catcuiarion should be necessary. a) At t: 4 s, what is the wheel's angular speed Lu ? b) At t: 2 s, what is the wheel's angular acceleration of ?
f c) Between 1‘: 0 s and t: 4 s, what angle 6 (measured in radians) does the wheel
rotate through? a.“ d) Between 1*: Gas and t: 8 s, how many! seemirolutiens does the wheel
make? (9} At t:_ B S, Whﬁi/ls the mat tGFGILJE 0!“. WthlT .1" . a
1. f) At t: it] s] what is the wheel's angular acceleration 0" ’? The f
time. — ._____ [Q ” {3 Lace) Try to deduce the answers to the foiiowing duestions directiy from the graph; iittie or  no caicuiation shouto' be necessary. ‘F—Lrﬂ— Q) What is the period Tof the oscillations? h) What is the amplitude A of the oscillations? i) What is the frequency chond} of the oscillation? i} On the same axes as the existing graph, sketch" a new curve showing an
osciiiation with the same amplitude and the same frequency as the existing” one,
but a different phase constant €35. _ _ (The new phase constant can be whatever you want it to be, as tong as it’s different
from the BXiSfii‘ig one.) 13> Problem #3: TrueiFalse  (18 points) .I' Mark each of the following statements True or False.
if you mark a statement as faise, then modify it, by adding, deieting, or changing a
few words, so that it becomes true. 1) It you were to move the earth five times further away from the sun than it is now,
then the gravitational pull of the sun on the earth would get five times stronger_._ _ 'l 2) If you were to move the earth five times further away from the sun than it is right
now, then the gravitational pull of the. earth on you would become twentyfive times
stronger. '  .   ’ ~, .  . t . IL  3) If a car collides with a brick wall and comes to rest, then some of the car's initial ..[iTtOmentum is transformed into heat. ' x 4) You shoot a cannonoall into the air, at a 45degree. angle above the ground. If
the only force acting on the cannonball is gravity, then the X—component of its kinetic energy will not change during the flight. 5) You set up a pendulum on earth, and find that its frequency is three cycles per
second. it you set up an identical penduium on the moon, it will have a tower frequency. ' / a_ 6) The total momentum of a system will not change, unless an external force does
work on the system. fr Problem #3: Sanity Checks (2o points) As always on a sanity check problem, do not try to caicuiate the answer to this
probiem directly. Instead, think over the physical situation described, examine the
five equations proposed as possible answers, and use various "sanity checks” to "
eliminate those equations which cannot be correct. The problem: Against the advice of your lawyers, you are the proud inventor of a rocketpropelled
merrngoround. The. merrv+gowround is a uniform disc oi mass M and radius Ft,
with a rocket engine attached a distance L from the center, as shown in the
overhead view below. The merrygoround starts from rest. When you fire the rocket, it exerts a uniform
force F, directed tangentially as shown. After you turn on the rocket, how much time
tdoes it take for the merrygoround to complete its tirst revolution? (Neglect ail forms of friction and air resistance, and assume that the rocket
continues to fire with force F throughout the revolution), t :— 7’ a) What should he the units of t? ‘ Which equation(s}, if any, does this eliminate? ls}. if the rne'rrygoround‘s radius Ff were inereased (while all other variables.
Including Its mass, remain the same), how should tchange? 41—: ' Which equation(s), if any, does this eliminate? Briefly justifyr your answer. 0X it the distance L were decreased (while all other variables remain/the same}
how should 1‘ change? Which equatioh(s), if any, does this eliminate? Briefly iustify your answer d) If two or more equations remain, continue to propose additional sanity checks
and explain which equation(s), if any, they eliminate, until at most one equation is
left. e) Which equation, it any, might be the correct answer to the pros "I. Extra credit: (use the back of the facing page en") Set up and solve this problem "for real", using the principles of rotational dynamics
we discussed in Tuesday's lecture. Does your result agree with the equation, it
any, that you chose in partr e? Problem #5 ' I (25 points)  A spaceship of mass m = 50000 kg is in orbit around a planet of unknown mass M.
The planet's radius Ft = 2000 km, and the ship is a distance it = 4000 km from the
planet's surface. The pilot notices that in a time t: 10000 seconds, her ship goes
exactly one quarter of the way around the planet. The onty force acting on theﬁsgaceship is gravity. m 4:
.r’ t H“H\\H
f Q0 H\
't.
ﬁx 'II
/ "ﬁrL T
l mu t=l00005
I I
l’
L r a) Calculate the ship's speed v, and indicate the direction of its velocity on the "
diagram above. b) Calculate the ship‘s acceleration a, and also indicate the direction of this acceleratipn on the diagram above. 1‘... C) Calculate the planet‘s mass M. The pilot now lands her ship on the planet's surface, steps out and starts to explore. d) What will she find to be the local gravitational acceleration g on the planet‘s
surface? (As always, feel free to invent answers to parts a, b, or c, if necessary, so
you can do this part.) e) Describe at least one simple experiment she could do to measure 9. Assume
she has access to any tools and equipment you would find in an ordinary
undergrad physics lat}. 6:
Problem #ﬁ: Where do people in physics problems learn to drive? (25 pts) Two oars collide at an intersection. The instant before the collision, Car #1. whose
mass m1: 3000 kg, was moving due west with speed v1: 10 rnr‘s. Car #2, whose mass m2 : 1500 kg. was moving at an unknown speed v , in a direction 9: 300
east of north. ' The instant after the coiiision, the two cars had locked together and were moving
directly north at an unknown speed to. N bﬁi’po rd" W t; GER} : E
.
__‘a
.F’
————+  Q
WigML _ (Gutier‘xeqci ﬂew) (ﬁﬁr Lead view) a) Calculate the unknown speed v2 of the second car before the collision. b) Calculate the speed vf ef the cars the instant after the eeliisien. After the eellisien, the ears slide nerth fer a distance i. = 20 meters betere eerning te
rest. Assume their brakes are by new ieeked; they are net reiling, but are skidding
aeress thejavernent. (3) Calculate the beeftieient of kinetic trietien/urk between the tires and the
pavement. 7 .
Problem #3: The Hour That Stretches (20 points) Suppose you own a clock that keeps time by means of a simple pendulum of length
L = 25 cm, with a bob of mass m : 100 g. :.______."” I? .. ,‘.._
" /a) What is the period of this pendulum's oscillations? Now you take this Clock with you when you moxie to Mars, where the local
gravitational field is weaker than that of the Earth. (On Mars, 9 z 3.6 rnfs2.) b) On Mars, what is the period of the pendulum's oscillations? How will this affect
the {iogk's usefulness as a timekeeping tool? One of your friends offers to try to get the Clock working the way it did on Earth, by
changing the mass of the pendulum bob. o) Will his plan work? If not, why not? it so, what new mass will he have to use for the pendulum boo?  4 Another of your friends offers to try to get the elect: working the way it did on Earth,
by Changing the length of the pendulum. d) Will her plan work? If not, why not? It so, what new length will she have to use
for the pendulum? Problem tier3 If (25 points) A skier of mass M: 80 kg is at the top of a hill of height h = 5 meters, moving to the
right at speed we : f3 mfs. At the bottom of the hill, a sled of mass m = 40 kg waits at
rest, tethered to an (unmovable) tree by a spring whose spring constant it 2 2000 N!
m. The person skis down the (frictionless) hill, then jumgs onto the sted. as
M.
£33.}, k ‘1: k
k_____ m get sq; a) Calculate the skier's speed v2 at the bottom of the hill, just before iumping onto
the sled. ___ [3) Calculate the speed v3 of the skier and the sled, the instant after he jumps
aboard. (Assume that the skier and the sled are now moving as a single object.) With the skier aboard, the sled new escilietes back and forth, driven bythe spring. a
' 0) Calculate the amplitude of these oscillations. d) How much time passes, from the moment the skier temps onto the sled, until the
sled returns to its starting point (new moving to the left)? ...
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 Fall '07
 Graham

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