PHYSICS 103 FINAL EXAM€¦ · Physics 103 Last In-Class Exam, 18 Dec 07 Problem 2, Page 10 b) A particle of mass m 1=100 g moves along the x-axis with velocity v 0 = 2:0^i m/s. The

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PHYSICS 103 FINAL EXAM

Instructions: When you are told to begin, check that this examination bookletcontains all the numbered pages from 5 through 25.

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This examination is administered under the Princeton University Honor Code. Studentsshould sit one seat apart from each other, if possible, and refrain from talking to otherstudents during the exam. All suspected violations of the Honor Code must be reported tothe Honor Committee Chair at [email protected].

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FURTHER INSTRUCTIONSThe exam contains SIX problems with multiple parts, worth varying numbers ofpoints, which total to 300 points.

Do not panic or be discouraged if you cannot do every part of every problem; thereare both easy and hard parts in this exam. If a part of a problem depends on aprevious answer you have not obtained, assume it and proceed. Keep moving andfinish as much as you can!

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Physics 103 Last In-Class Exam, 18 Dec 07 Problem 0, Page 3

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Problem 1. Conservation Laws

The spring in the figure has a spring constant k = 1000.0 N/m. It is compresseda distance x0 = 15 cm, then launches a block of mass m = 0.20 kg. The horizontalsurfaces and the inclined plane are frictionless. The second horizontal surface is aheight h = 2.0 m above the first, and the angle of the inclined plane is θ = 45◦.

a) (12 pts) For each question below, first answer yes or no, then write 1-2 sentencesexplaining your reasoning.

i) Is the energy of the block conserved after it leaves the spring?

ii) Is the momentum conserved after the block is released by the spring, while theblock is on the lower horizontal surface?

iii) Is the momentum conserved while the block is on the inclined plane? Commenton both the horizontal and vertical components of the momentum.

iv) Is the momentum conserved while the block is sailing through the air? Commenton both the horizontal and vertical components of the momentum.


b) (15 pts) What is the block speed vt at the top of the incline? What is the blockspeed v2 just before landing? What distance d (see figure) on the upper horizontalsurface does the block travel before landing?

vt: v2: d:


c) (10 pts) The frictionless surface on the incline is replaced by one such that thecoeffcient of kinetic friction between the block and the surface is µk = 0.20. Find thework done on the block by the spring (Wspr) and by friction (Wf ). What is the workWg,1 done by the gravitational force while the block is on the inclined plane? Whatis the work Wg,2 done by the gravitational force while the block is flying?

Wspr: Wf :

Wg,1: Wg,2:


d) (13 pts) What is the distance d in this case (with friction on the inclined plane)?

d:


Problem 2. Collisions and Kinematics

a) (20 pts) A particle of mass m = 10.0 g moves with a velocity v0 = 60 m/s alongthe x-axis. The particle hits a body of mass M = 2.0 kg and sticks to it. Determinethe velocity vf of the body plus particle after the collision. Determine the amount ofenergy ∆E lost in the collision.

vf : ∆E:


b) A particle of mass m1=100 g moves along the x-axis with velocity v0 = 2.0i m/s.The particle hits a second particle of mass m2 = m1, which is at rest. The collisionis elastic. After the collision m1 has velocity v1 and m2 has velocity v2.

i) (10 pts) Prove that v1·v2 = 0 in this case. (Hint: use conservation of momentumand energy, and the definition of the dot product.)


ii) (20 pts) After the collision v1 makes an angle θ1=45◦ with respect to v0. De-termine the angle θ2 that v2 makes to v0, and find the magnitudes v1 andv2.

θ2:

v1: v2:


Problem 3. Binary Star System

Two stars each have mass M and radius R. One has its center on the origin ofan xyz coordinate system, and the center of the other is at x2 = 10R. A spacecraftof mass m (with m << M) moves along the x-axis under the force of gravity. (Itsengines have been destroyed.)

a) (12 pts) Below, sketch a graph of the space capsule’s potential energy as it moveson the x-axis. Don’t forget to label the axes. Indicate the value of the potentialenergy at any extrema, and the positions of any extrema, and how the potentialenergy varies with distance from each star when it is very near the star.


b) (6 pts) Suppose the spacecraft is at rest exactly midway between the stars. Is thisa point of equilibrium? If so, is it a stable or unstable equilibrium point?

c) (4 pts) Suppose at t = 0, the spacecraft is at rest at x = 4R. What is the net forceF on the spacecraft?

F = Direction:


d) Suppose instead that at t = 0, the spacecraft is at rest at x = 5R. The captainfires a huge cannonball of mass mc = m/10 in the −y direction. After the cannon isfired, the spacecraft has net mass (9/10)m and moves off in the +y direction.

i) (8 pts) Is angular momentum conserved before and after the cannonball is fired?If vc is the speed of the cannonball after the cannon is fired, what is the mag-nitude Lc of the angular momentum of the cannonball about the z-axis?

yes or no: Lc:


ii) (20 pts) With must vc be in order to permit the spacecraft to escape the grav-itational pull of the binary star system? As always, show all the steps of yourwork clearly for full credit.

vc:


Θ

Problem 4. Banked Curves

An engineer designing the bank on a curved road has to take friction into account.Suppose the road is curved into a circle at a certain point. The radius of the circleis R (measured to the position of the car). Assume the driver wants to negotiate thecurve while maintaining some speed v. The coefficient of friction between the wheelsand the surface of the road is µs.

a) (15 pts) Draw a freebody diagram of the car. Label every force clearly. You donot need to resolve the forces on your diagram, but make it clear where the angle θcomes in.


b) (20 pts) For what speed v is the angle θ a perfect bank? (On a perfect bank, thecar is able to go around the curve, but the road exerts no sidewise frictional force onthe car.) Express your answer for v in terms of θ, g, and R.

v:


c) (15 pts) At what speed v will the car just begin to slide up a bank of angle θ?Express v in terms of θ, g, R, and µs.

v:


Problem 5. Oscillations and Pulleys

In the system shown in the figure, a spring with spring constant k is attachedto a massless string which passes over a pulley and is attached to a mass m. Thepulley is a homogeneous disk of mass mp and radius R. (It is supported by a rigidrod extending out from the table, but the details of that do not matter.) The diskrotates without friction around its axis perpendicular to the page. The string doesnot slip on the pulley.

a) (10 pts) First, take mp=0. The mass is at rest in its equilibrium position when itis at y = 0 and the spring is elongated by d = 0.100 m. Find the tension T0 in thestring when the mass is at y = 0. Express your answer in terms of variables given inthe problem and g as needed.

T0:


b) (10 pts) Find the period P of oscillations of the mass m, assuming that the am-plitude A of the oscillations satisfies A < d so that the string never goes slack. Afterfinding an algebraic expression for the period, also provide a numerical answer inseconds. (Hint: yes, you have all the information you need to do so.)

P , algebraic:

P , numerical:


c) (10 pts) What is the direction of the net torque on the pulley when the mass m isbelow the equilibrium position and moving downward? You must defend your answerwith 1 or 2 sentences and a sketch for full credit.

direction:


d) (20 pts) Determine the period P of oscillations of the mass m in its motion alongthe y axis. As before, the spring is elongated by d = 0.100 m in the equilibriumconfiguration, and you should consider only oscillations with amplitudes smaller thand. In this case, after finding an algebraic expression, use the following values to finda numerical result: mp=1.0 kg and m= 0.10 kg. The moment of inertia for the diskis I = (1/2)MR2.

P , algebraic:

P , numerical:


Problem 6. Spiral String Wave

A string with tension T is found in the lab. Instead of vibrating the string,someone spirals it and produces a spiral wave, as sketched in the figure. A spiralwave on a string, traveling along the z axis, can be thought of as a superposition oftwo independent sinusoidal waves undulating along the two orthogonal directions, iand j. Mathematically, this spiral wave is described by

R(z, t) = R sin(kz − ωt)i +R sin(kz − ωt+ φ)j.

Here, R is the wave amplitude, and φ = π/2. The values k and ω are constants.

a) (5 pts) Based on the R(z, t) given above, in what direction is the wave traveling(+k or −k)?

direction:


b) (5 pts) Consider the small chunk of string in the xy plane (that is, with z = 0).As the figure on the right indicates, this chunk is executing circular motion as thewave travels through. Since φ = +π/2, what is the direction of the circular motion(clockwise or counterclockwise)?

direction:

c) (15 pts) Consider that small chunk to have length δz, with δz small enough thatyou can treat the chunk as a point mass. What is its kinetic energy, K? What is itsangular momentum L about the origin? Express your answers in terms of k, ω, T ,R, and δz as needed.. (Hint: Remember that the velocity of the wave is

√T/µ, and

note that the mass of the chunk is m = µδz. Don’t leave µ in your answers.)

K:

L = Direction:


d) (15 pts) Considering that the chunk of string is executing circular motion, what isthe net force F on that chunk of string? Express your answer in terms of k, ω, T , R,and δz as needed.

F = Direction:

e) (10 pts) You should have found in the previous part that the force F on the chunkof string is a function of R. Find the potential energy U of the chunk, taking thezero point to be at the origin. (Hint: you might need to evaluate an integral along aradius.) Express U in terms of k, ω, T , R, and δz.

U :









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PHYSICS 103 FINAL EXAM€¦ · Physics 103 Last In-Class Exam, 18 Dec 07 Problem 2, Page 10 b) A particle of mass m 1=100 g moves along the x-axis with velocity v 0 = 2:0^i m/s. The