Physics 10310 Final Exam May 6, 2008

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INSTRUCTIONS: Write your NAME and LECTURE SECTION (I: Ruchti, II: Hildreth) on the front of the blue exam booklet. The exam is closed book, and you may have only pens/pencils and a calculator (no stored equations or programs and no graphing). Show all of your work in the blue book. For problems II-VI, an answer alone is worth very little credit, even if it is correct – so show how you get it. Suggestions: Draw a diagram when possible, circle or box your final answers, and cross out parts which you do not want us to consider.

Moments of Inertia for various objects are given on the last page of the exam

Physics 10310 Final Exam May 6, 2008

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I. Multiple Choice Questions (3 points each) Please write the letter corresponding to your answer for each question in the grid stamped on the first inside page of your blue book. No partial credit is given for these questions.

1. A constant horizontal force is applied to an object sitting on a horizontal surface. The coefficient of static friction µs between the object and the surface is µs = 0.4. The coefficient of kinetic friction µk between the object and the surface is µk = 0.3. The magnitude of the force is just sufficient to overcome the force of static friction. After the object begins to move, its acceleration is: (a) 9.80 m/s2 (b) 0.98 m/s2 (c) 2.94 m/s2 (d) 3.92 m/s2 (e) cannot be determined 2. A diagram representing the total energy of a mechanical system is shown in the figure below, where the energy is shown as a function of the particle’s position on the x axis, and the potential energy is given by the curve labeled U(x). The total energy of the system is shown by the horizontal line labeled “Etot”. The particle has mass m = 2 kg. At point A, the particle’s speed is: (a) 1 m/s (b) 4 m/s (c) 1.4 m/s (d) 2 m/s (e) 3 m/s

3. A cannonball is dropped from rest off of a cliff of height H. Simultaneously, an identical cannonball is shot vertically upward with an initial velocity such that it will reach a maximum height H. The cannonballs collide somewhere during their trajectories. The point of collision hcoll is located such that: (a) 0 < hcoll ≤ H/4 (b) H/4 < hcoll ≤ H/2 (c) H/2 < hcoll ≤ 3H/4 (d) 3H/4 < hcoll ≤ H 4. A very sticky gumball moving upwards with velocity v strikes another identical sticky gumball at rest. They stick together and continue moving upward. The maximum height above the collision point that they attain, in terms of the original gumball velocity v, is: (a) v2/(8g) (b) v2/(4g) (c) v2/(3g) (d) v2/(2g) (e) v2/g 5. A constant force is applied perpendicular to the end of a rod of length L that rotates about its center, as shown in the figure. The rod is initially at rest, and the force follows the tip of the rod as it moves, remaining perpendicular to the rod. After a short time interval, the tip of the rod is moving with speed

E (Joules) x

Etot

U(x)

-1 -2 -3

-4

-5 -6 -7

A

Physics 10310 Final Exam May 6, 2008

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M

xi x0 20 cm

vfirst. Then, the length of the rod is changed to L/2, but the force is kept constant, and the force is again applied to the rod starting from rest. After the same time interval, what is the relationship between the final velocity of the tip, vnew, and tip’s speed in the first case? (Irod = 1/12 M2)

(a) vnew = 4 vfirst (b) vnew = 2 vfirst (c) vnew = vfirst (d) vnew = ½vfirst (e) vnew = ¼ vfirst

6. A spring (force constant k = 490 N/m) with a mass M = 10 kg attached, oscillates with simple harmonic motion. The spring is initially compressed 20 cm from its unstretched length as shown in the figure and then released. Assuming the form x = A cos (ωt + δ) for the harmonic motion of M, the numerical values for A, ω, and δ are respectively:

(a) 0.2m, 7 rad/s, π rad (b) 0.2m, 7 rad/s, 0 rad (c) 0.2m, 49 rad/s, π rad (d) 0.2m, 0.045 rad/s, π rad (e) 0.2m, 7 rad/s, π/2 rad

7. In a lecture demonstration, a spinning disk of mass M – attached to a massless rod of length d - is hanging from a cord as shown in the figure. Assume the angular momentum of the disk is L and points at right angles to the cord. The precession frequency of the disk (the frequency with which the disk completes one revolution in the horizontal plane about the cord) is given as:

(a) Mgd/L (b) L/(Mgd) (c) MgLd (d) 1/(MgLd) (e) None of these

8. A ball formed from a hollow circular shell of mass M = 0.5 kg and radius R = 0.2 m is rolling without slipping on a horizontal surface. Its translational velocity is v = 3 m/s. The total kinetic energy of the ball is:

(a) 2.25 J (b) 1.5 J (c) 3.75 J (d) 0.75 J (e) None of these

L d

F

pivot

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Problems (15 points each) II. An unusual amusement park attraction of recent years it the “Bungee-Swing.” Usually known by a more alarming name, like the “Rip Cord” at Cedar Point, it takes the lazy rope swing to a new level. Riders are dropped from a 15-story (150 m) tower while attached to a super-strong bungee cord with spring constant k that is attached to an arch of equal height. The un-stretched length of the cord is 100 m and the rider jumps off of the platform with the cord un-stretched. Our intrepid rider has mass 60kg. To make this ride thrilling, we want the rider to get as close to the ground as possible, and the following steps will lead us to some design considerations for this ride.

a) Write expressions for the initial and final energies of the rider, beginning at the launch platform and ending at the bottom of the swing. Also, write Newton’s Second Law for the rider at the bottom of his swing.

b) What spring constant k must our cord have to allow the extension of the cord to the full 150 meters?

c) Find the speed of the rider at the bottom of the swing, assuming full extension of the cord. d) What is the tension in the cord at the maximal extension?

III. A space module is in a stable circular orbit at an altitude h1 = 2000 km above the earth’s surface. (Assume Mmodule = 104 kg, ME = 6 x 1024 kg, RE = 6.37 x 106 m, G = 6.67 x 10-11 Nm2/kg2.)

a) Find the gravitational potential energy of the module. b) Find the kinetic energy of the module. c) Find the total mechanical energy of the module.

The module is then boosted to a higher stable circular orbit at an altitude of h2 = 4000 km. (For parts (d) and (e) below, ignore loss of module mass due to fuel use.)

d) How much energy is required to boost the module to this new orbit?

100 m

150 m

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IV. A car of mass 1000 kg travelling north with speed VC hits a truck of mass 3000 kg travelling east with speed VT. In the ensuing crash, the truck and car stick together and slide away to the northeast, at a direction 30° north of east, as shown on the diagram. The vehicles are sufficiently mangled that their wheels no longer turn, and they slide along the pavement for 35 meters until coming to rest. The coefficient of kinetic friction between the mangled tires and the pavement is µk = 0.4.

a) Find the speed VF of the car/truck wreck after the collision, just as it begins to slide.

b) Find the initial speeds, VC and VT. c) How much energy is lost in the collision?

V. Physics of the Hula Hoop. In this problem, we consider the rotation of a thin hoop around a cylindrical pole. A thin hoop of mass M and radius R rotates with constant angular velocity in a horizontal plane around a vertical cylindrical pole of radius r, as shown in the figure. As can be seen in the figure, the center of mass of the hoop rotates in a circle about the center of the pole. The pole has a rough surface with coefficient of static friction µs.

a) Copy the “Side View” diagram into your blue book and draw a free body diagram for the hoop. (Hint: there is a force of contact between the cylinder and the hoop.)

b) What is the radius of the hoop’s rotation? c) Write Newton’s Second Law for the components of the hoop’s motion, choosing your axes in the

vertical and radial directions. d) Find the minimum velocity of the hoop’s center of mass such that the hoop will rotate stably and

not fall down the pole. e) Find the critical hula angular velocity, ωhula of the hoop that sustains stable motion.

r

R

Top view Side view

VT

VC

VF

30°

North

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VI. A pendulum is made of a rod of length L=1 m and mass mrod =3 kg, attached to a solid sphere of radius R=0.2 m and mass msphere = 4.5 kg. The axis of rotation is at the end of the rod.

a) What is the moment of inertia of the system about the rotation axis? b) Where is the center-of-mass of the pendulum relative to the axis of rotation? c) Write down Newton’s 2nd Law (for rotational motion) for the system configuration show.

Assume the angular displacement φ is small. d) Find the period of the pendulum for small angular displacement φ.

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L = 1m

R = 0.2 m

φ