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PHYSICS 101 MIDTERMOctober 25, 2007 2 hours

Please circle your section1 9 am Nappi 2 10 am McDonald3 10 am Galbiati 4 11 am McDonald5 12:30 pm Pretorius

Problem Score1 /152 /103 /204 /155 /206 /20

Total /100

Instructions: When you are told to begin, check that this examination bookletcontains all the numbered pages from 2 through 15. The exam contains 6 problems.Read each problem carefully. You must show your work. The grade you get dependson your solution even when you write down the correct answer. BOX your finalanswer. Do not panic or be discouraged if you cannot do every problem; there areboth easy and hard parts in this exam. If a part of a problem depends ona previous answer you have not obtained, assume it and proceed. Keepmoving and finish as much as you can!

Possibly useful constants and equations are on the last page, which you maywant to tear off and keep handy

Rewrite and sign the pledge: I pledge my honor that I have not violated the Honor Codeduring this examination.

Signature

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Problem 1: Grab Bag

(a) [6 pts] Three balls, A, B, and C are thrown from the top of a cliff of height h = 100moverlooking a flat land. The speed of the the three balls is identical: vA = vB = vC = 10m/s.The angle in the figure is φ=π/4.

VB

φ

h

VC

VAφ

a.1 [3 pts] Which one of the three balls will have the largest speed at the moment it makescontact with the ground? Box your answer and explain your reasoning on the side.

• Ball A

• Ball B

• Ball C

• They all reach ground with the same speed

a.2 [3 pts] Which of the three balls will reach the ground in the shortest time? Box youranswer and explain your reasoning on the side.

• Ball A

• Ball B

• Ball C

• They all reach ground at the same time

• Ball B and C reach ground at the same time, before ball A.

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(b) [9 pts] A uniform plank of length L=2 m and mass m=10 kg is at rest and leaningagainst a frictionless wall. The horizontal surface is rough, and the coefficient of staticfriction is µs=0.8.

L

θ

b.1 [3 pts] Draw a complete body diagram on the figure, identifying all forces.

b.2 [2 pts] Find the vertical component Fy of the force exerted by the horizontal surfaceon the plank.

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b.3 [2 pts] Find the normal force N exerted by the vertical wall on the plank.

b.4 [2 pts] Find the horizontal component Fx of the force exerted by the horizontal surfaceon the plank.

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Problem 2: Rock the Cart!A cart on wheels of mass M=100 kg is standing still on frictionless rails on a horizontalsurface. Billy is playing on the car, and has a mass m of 50 kg. The cart is initially at rest,and Billy plays while sitting on one end of the cart. All of a sudden, Billy stands up andstarts running towards the other end of the cart. While on the cart, Billy reaches a steadyvelocity v = 10 m/s relative to the cart.

Billy

(a) [4 pts] As a result of Billy’s motion, the cart also gets in motion with a velocity urelative to the ground when Billy has a velocity v relative to the cart. What is the velocityof Billy relative to the ground? Express your answer as a function of u and v.

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(b) [6 pts] Calculate the velocity u of the cart relative to the ground at the moment whenBilly achieves the maximum velocity v.

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Problem 3. Roller Coaster.

A cart of mass m is released with zero velocity at the top of the roller coaster and then runsin the loop of radius R shown in the picture.

θ

RH

a. [7 pts] If the cart has to stay on track in the loop of radius R, at what minimal heightmust it be released? I.e., find the minimum height h that will allow the cart to complete theloop of radius R.

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b. [6 pts] Assume that the cart starts at an height H > h that allows it to complete theloop. Find the expression of the normal force as function of H and of the angle θ from thevertical.

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c. [7 pts] Compute the apparent weight of the cart at the bottom and at the top of theloop. Show that the difference between the two is always 6 mg, where m is the mass and g isthe acceleration of gravity.(To answer this question you do not necessarily need to use youranswer to part b.)

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Problem 4. Target Practice.

An archer is shooting arrows, eachwith a mass m of 165 g, at a tar-get that is a distance l=100 m away.When the bullseye is in the archer’sline of sight, i.e., looking directlydown the shaft of the arrow withthe bow drawn as shown in the fig-ure, the arrow makes an angle of 0◦

relative to the horizontal.

l = 100 m

d =

1.52

m

a. [6 pts] The archer shoots an arrow and it leaves the bow with a speed v0 of 48.5 m/s. Ig-noring air resistance, at what angle relative to the horizontal must the archer loose thearrow so that it hits the bullseye? Hint: you might find the following identity useful:2 sin θ cos θ = sin(2θ)

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b. [3 pts] For how much time was the arrow in flight? (If you did not get an answer to parta, use an angle θ=15.00◦)

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c. [6 pts] A steady wind starts blowing. Assume that the effect of the wind on an arrow

during its flight is as if a constant force ~F=−0.118Nx̂ was applied to the arrow, where the x̂direction is along the horizontal as illustrated in the figure. The archer now shoots a secondarrow with exactly the same speed as before (v0=48.5 m/s) but, anticipating the wind, headjusts the angle to θ=12.8◦. Where, relative to the bullseye, does the second arrow hit?

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Problem 5. The Velodrome.

A velodrome has the form of a circularbowl whose flat bottom is a circle of radiusr=25 m. The side of the velodrome hasthe same radius of curvature r=25 m, asshown in the figure below which representsa vertical cross section of the velodrome,through its central axis.

r r

r r

r

R

a. [7 pts] When on the edge of the flat bottom of the velodrome, riders are moving in acircle with radius r=25 m. If the coefficient of static friction is µs=0.5, what is the largestspeed vA for uniform circular motion (rolling without slipping) around this circle?

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b. [3 pt] What is the angular velocity ωA for the motion in part a?

c. [7 pts] Now suppose a rider goes on the banked curve of radius R that passes throughpoint B on the figure, corresponding to an angle θ=30◦, and that he rides at a speed suchthat no friction is required to maintain uniform circular motion. What is the speed vB forthis motion?

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d. [3 pt] What is the angular velocity ωB for the motion in part c?

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Problem 6. The Rube Goldberg Orange Juice Squeezing Machine.

Milkman takes empty milk bottle (A), pulling string (B) which causes sword (C) to severcord (D) and allow guillotine blade (E) to drop and cut rope (F) which releases batteringram (G). Ram bumps against open door (H), causing it to close. Grass sickle (I) cuts a sliceoff end of orange (J), at the same time spike (K) stabs “prune hawk” (L) who opens hismouth to yell in agony, thereby releasing prune and allowing diver’s boot (M) to drop andstep on sleeping octopus (N). Octopus awakens in a rage and, seeing diver’s face which ispainted on orange, attacks it and crushes it with tentacles, thereby causing all the juice inthe orange to run into glass (O)...

a. [4 pt] If the guillotine blade (E) falls by a distance h=65 cm before cutting rope (F),what is the speed u of the blade just before it cuts the rope?

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b. [4 pts] The battering ram (G) has a mass m of 52 kg. Each of the massless chains thatsupport it have a length l of 1.73 m and make an initial angle θ of 23◦ to the vertical. Whatis the momentum Pram of the battering ram just before it bumps the door (H), if the chainsare vertical at the moment of the collision?

c. [3 pt] The battering ram travels parallel to the wall and bumps into the door at a distanced=67 cm out from the wall. What is the angular momentum of the battering ram about theaxis of rotation of the door? The door is initially at rest and is open by 90◦ with respect tothe wall. For this question, you may consider the battering ram as a point-like object withmass m and velocity v.If you did not get the solution of part b, assume that the velocity of the ram v is 2 m/s.

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d. [3 pt] The door has a mass M of 33 kg and a width w of 75 cm. What is the momentof inertia of the door about its axis of rotation?

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e. [6 pts] What is the angular velocity of the door about its axis of rotation just after ithas been bumped by the battering ram, assuming that the collision is elastic?Note: the reaction force of the hinge acts in a direction perpendicular to the axis of rotationof the door. The external torque is therefore null, and the angular momentum of the system(door + ram) is conserved. Due to the presence of the reaction force of the hinge, linearmomentum is not conserved.

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POSSIBLY USEFUL CONSTANTS AND EQUATIONSYou may want to tear this out to keep at your side

L = Iω I = Σmir2i x = x0 + v0t + at2/2

PE = mgh KE = 12Iω2 KE = 1

2mv2

ω = ω0 + αt ω2 = ω20 + 2α∆θ ∆θ = ω0t + 1

2αt2

v = v0 + at ~F∆t = ∆~p F = −GMm/r2

F = µN s = Rθ τ = F` sin θΣτ = Iα v = Rω ~p = m~vac = v2/r W = Fs cos θ v2 = v2

0 + 2a∆xWnc = ∆KE + ∆PE a = Rα I = 1

2mr2 [disk]

I = 25mr2 [sphere] I = 1

3Mw2 [thin sheet] L = mvr

REarth = 6400 km MEarth = 6.0× 1024 kg G = 6.67× 10−11 Nm2/kg2