### Spring 2009 Physics 211 Hour Exam 3(25 questions)

The grading button and a description of the scoring criteria are at the bottom of this page. Basic questions are marked by a single star *. More difficult questions are marked by two stars **. The most challenging questions are marked by three stars ***.

True-false questions are worth 2 points each, three-choice multiple choice questions are worth 3 points each, five-choice multiple choice questions are worth 6 points each. The maximum possible score is 110. The exam period was 90 minutes; the average score was 71.1; the median score was 74. Click here to see the formula sheet that came with the exam.

### QUESTION 1*

This and the next question refer to this situation:

A mass M hangs from the end of a rod of mass m and length L. The rod is held in place with a hinge attached to a wall on the right, and it has a cable with a tension which is measured to be T.

What is the value of M, in terms of m, L,T and θ ?

(a)   M = (T / (gsinθ)) - m/2
(b)   M = (T sinθ/g) - m/2
(c)   M = m tanθ (T / g - m) / (T/g + m)
(d)   M = T cosθ + mg
(e)   M = m sinθ + T / g

### QUESTION 2*

If θ is decreased, holding m, M and L constant, the tension T

(a)   decreases.
(b)   stays the same.
(c)   increases.

### QUESTION 3**

This and the next two questions refer to this situation:

A solid cylinder (I = 1/2 MR2) and a solid sphere (I = 2/5 MR2), of the same radius and mass, roll down an incline of angle θ = 30°. They both start from rest at a distance D = 2 meters up the incline, as shown in the diagram, and roll without slipping to the bottom of the incline.

The initial gravitational potential energy of the two objects is partially lost in work done overcoming frictional forces as the objects roll down the incline.

(T)   True
(F)   False

### QUESTION 4**

The change in translational kinetic energy (1/2 MV2) between the top and the bottom of the incline is

(a)   larger for the solid sphere than it is for the solid cylinder.
(b)   the same for both objects.
(c)   smaller for the solid sphere than it is for the solid cylinder.

### QUESTION 5**

What is the value of the ratio of the velocities of the sphere and the cylinder Vsphere / Vcylinder at the bottom of the incline?

(a)   0.76
(b)   0.83
(c)   0.91
(d)   1.04
(e)   1.17

### QUESTION 6**

This and the next question refer to this situation:

Four 1 kg-masses are held together in the form of a square by four rigid, massless rods, each of length 2l. The positions of the point masses are designated a, b, c and d, as shown in the diagram. Axis AB lies in the plane formed by the square and passes through its center. Axis CD is parallel to AB and passes through the masses c and d. Axis XY is perpendicular to the plane of the square and passes through its center, as shown on the right.

Compare the three moments of inertia of the square about the three specified axes, IAB, ICD, IXY. Which one of the following statements is correct?

(a)   IAB = ICD > IXY
(b)   IAB < ICD < IXY
(c)   IAB < ICD = IXY
(d)   IAB > ICD < IXY
(e)   IAB < ICD > IXY

### QUESTION 7**

If instead of being massless, the identical, uniform rods have masses of 1 kg each, what is the moment of inertia of the square about the CD-axis? (Assume that the rods have negligible cross section.)

(a)   14.7 kg l 2
(b)   18.6 kg l 2
(c)   19.2 kg l 2
(d)   21.7 kg l 2
(e)   24.9 kg l 2

### QUESTION 8**

This and the next two questions refer to this situation:

A massless rope is holding a solid disk of mass M and radius R (I = 1/2 MR2). The rope is attached to the center of the disk and a frictionless, vertical wall as shown in the figure. The center of the disk is at a distance L below where the rope is attached to the wall.

What is the force exerted on the disk by the wall?

(a)   Mg
(b)   MgL
(c)   MgR/L
(d)   sqrt(2)MgL
(e)   MgR

### QUESTION 9**

If the rope is cut, a net torque will cause the disk to begin to rotate.

(T)   True
(F)   False

### QUESTION 10*

If the disk were replaced by a solid sphere with the same mass M and radius R, how would the tension in the rope holding the disk (Tcylinder) compare to the tension in the rope holding the sphere (Tsphere)?

(a)   Tdisk > Tsphere
(b)   Tdisk = Tsphere
(c)   Tdisk < Tsphere

### QUESTION 11**

This and the next question refer to this situation:

Consider the earth as a uniform, solid sphere of radius Re = 6.4 × 106 m with mass Me = 6.4 × 1024 kg that makes one complete revolution in 1 day (= 24 hours.) As part of a carefully crafted attack, aliens from outer space drop a small neutron star on the equator. The mass of the neutron star is Mn = 6.4 × 1024 kg, the same as the mass of the earth. (Assume the neutron star is a point mass.)

After the mass is dropped, how long does it take the earth to make one complete revolution?

(a)   0.93 days
(b)   1.00 days
(c)   2.00 days
(d)   2.67 days
(e)   3.50 days

### QUESTION 12**

If instead the neutron star was placed in Urbana, how would the time for 1 revolution compare to when the star was placed on the equator?

(a)   It would be less than with the star on the equator.
(b)   It would be the same as with the star on the equator.
(c)   It would be greater than with the star on the equator.

### QUESTION 13***

This and the next question refer to this situation:

A spool with a thread wound around it is pulled with a force T = 30 N as shown below. The total moment of inertia of the spool is I = 1.25 kg·m2, its mass is M = 10 kg, its outer radius is R = 0.5 m and its inner radius is r = 0.1 m. The spool rolls without slipping and starts from rest.

Find the angular acceleration α of the spool.

### QUESTION 14*

The angular acceleration vector points out of the screen.

(T)   True
(F)   False

### QUESTION 15*

A flying saucer in the shape of a disk with a 100 m radius is initially at rest. Two diametrically opposite rockets fire and the saucer begins to rotate on a frictionless surface. The rockets each exert a force of 2000 N for 10 revolutions and then shut off. The saucer and rockets have a combined moment of inertia of I = 20 × 106 kg m2.

After the rockets shut off, what will be the rotational frequency of the saucer?

### QUESTION 16**

Two children are sitting on a "see-saw" as shown. One weighs four times more than the other. If the heavy child sits 3 m from the center of the seesaw, what is the length, L, of the see-saw such that it remains perfectly balanced, that is, perfectly horizontal. Note that the children sit right at the ends of the see-saw. You may assume that the see-saw itself is massless.

(a)   L = 7 m
(b)   L = 12 m
(c)   L = 15 m
(d)   L = 18 m
(e)   L = 20 m

### QUESTION 17**

This and the next question refer to this situation:

A piece of gum of mass m = 0.1 kg, is thrown at a bar of mass M = 1 kg, and length d = 1 m, pivoted about its center and initially at rest as shown. It sticks to the end of the bar and the final angular speed of the bar is measured to be ω = 3 rad/s. Assume that the gum has a size much smaller than d.

What is the initial speed v of the gum?

(a)   v = 3.7 m/s
(b)   v = 4.8 m/s
(c)   v = 5.2 m/s
(d)   v = 5.9 m/s
(e)   v = 6.5 m/s

### QUESTION 18*

If the bar was initially rotating clockwise (CW) instead of being stationary and the final angular speed was still ω = 3 rad/s, the speed of the wad of gum would have been

(a)   smaller than v.
(b)   the same than v.
(c)   larger than v.

### QUESTION 19***

If the magnitude of the applied force F is the same in each case, and there are no other external forces acting, then the magnitudes of the acceleration of the center of mass of the two systems shown below are the same.

(T)   True
(F)   False

### QUESTION 20**

A bar is placed on a frictionless pivot as shown. The bar is composed of two pieces made of different materials. Each piece has a different uniform mass density, as indicated by the dark-shaded and light-shaded regions. The bar is in static equilibrium as shown.

Of the following choices, which best describes the approximate location of the center of mass of the bar?

(b)   At the pivot.
(c)   In the light-shaded region, to the right of the pivot.

### QUESTION 21***

Two point particles having mass M and 2M are attached to the ends of a massless rod of length L. Each mass is supported by a vertical string, and the system is initially horizontally at rest. In case 1, the string holding the heavier particle is cut, and in case two the string holding the lighter particle is cut. Which of the following statements best describes the magnitude of the angular acceleration of the system about the bottom of the remaining string in each case?

(a)   |α1| = |α2|
(b)   |α1| > |α2|
(c)   |α1| < |α2|

### QUESTION 22*

A uniform painting weighing W = 120 N is supported by two ropes that are attached to the ceiling. The first rope exerts tension T1 and is attached some distance x from one end of the picture. The second rope is attached to the other end of the picture and exerts tension T2. The length of the painting is L = 1.5 m.

Calculate the distance x if the tension T1 in the first rope is measured to be 85 N.

(a)   x = 0.44 m
(b)   x = 0.36 m
(c)   x = 0.80 m
(d)   x = 0.60 m
(e)   x = 0.25 m

### QUESTION 23*

This and the next two questions refer to this situation:

A turntable has a mass of 1 kg and a radius of 0.17 m and is initially rotating freely at 78 rpm (ωi,t = 8.168 rad/s). There are no external torques acting on the system. The moment of inertia of the turntable can be approximated by that of a disk (Idisk = MR2/2).

A small object, initially at rest, is dropped vertically onto the turntable and sticks to the turntable at a distance d of 0.10 m from its center as shown in the figure. When the small object is rotating with the turntable, the angular velocity of the turntable ωf,t is 72.7 rpm (7.613 rad/s). What is the mass of the small object that was dropped onto the turntable?

(a)   0.048 kg
(b)   0.070 kg
(c)   0.086 kg
(d)   0.105 kg
(e)   0.123 kg

### QUESTION 24**

Now suppose the same turntable (without the small object) is freely rotating at an initial angular velocity of ωi,t = 8.168 rad/s. A record of mass 0.2 kg and radius of 0.17 m is dropped vertically onto the turntable with no initial angular velocity (ωi,r = 0 rad/s). When the record and turntable begin rotating with the same angular velocity, what was the loss in rotational kinetic energy of the system? Assume that the hole in the center of the record can be ignored.

(a)   0.080 J
(b)   0.095 J
(c)   0.103 J
(d)   0.121 J
(e)   0.137 J

### QUESTION 25**

After the first record is rotating with the turntable, a second, identical record is dropped on top of the first in a similar way and begins rotating with the first record and turntable. The change in angular momentum of the whole system due to placing the second record on top of the first record and the turntable would be

(a)   less than that of placing the first record on the turntable.
(b)   zero.
(c)   more than that of placing the first record on the turntable.

Scoring of True-False (T,F) Questions:
• If you mark the correct answer, you get 2 points.
• If you mark the wrong answer or if you mark neither answer, you get 0 points.
Scoring of Multiple Choice I (a,b,c) Questions:
• If you mark the correct answer, you get 3 points.
• If you mark a wrong answer or if you mark none of the answers, you get 0 points.
Scoring of Multiple Choice II (a,b,c,d,e) Questions:
• If you mark one answer and it is the correct answer, you get 6 points.
• If you mark two answers and one of them is the correct answer, you get 3 points.
• If you mark three answers and one of them is the correct answer, you get 2 points.
• If you do not mark the correct answer, or if you mark more than three answers, or if you mark none of the answers, you get 0 points.