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November 2, 2006 Version C1

PHYSICS 207: Mid-Term Exam 2

Print your name and section clearly on all pages. (If you do not know your section number, write your TA's name.) Show all work in the space immediately below each problem. Your final answer must be placed in the boxes provided. Problems will be graded on reasoning and intermediate steps as well as on the final answer. Be sure to include units wherever necessary, and the direction of vectors. Each problem is worth between 4 and 16 points. In doing the problems, try to be neat. Check your answers to see that they have the correct dimensions (units) and are the right order of magnitude. You are allowed one 8.5 $\times$ 11" sheet of notes and no other references. The exam lasts 90 minutes.

NOTE: for the purposes of this exam you should assume $g=10.$ m/s$^2$.

Possibly useful moments of inertia (about high symmetry axes):
Solid Sphere $= \frac{2}{5} MR^2$,                    Solid Cylinder $=\frac{1}{2} MR^2$
Thin Hollow Cylinder $=MR^2$
Long thin rod (axis through center) $= \frac{1}{12} ML^2$
Thin rectangular Plate (axis through center) $= \frac{1}{12} M(a^2+b^2)$

Do NOT write below this point.

SCORE

PROBLEM 1:

PROBLEM 2:

PROBLEM 3:

PROBLEM 4:

PROBLEM 5:

PROBLEM 6:

PROBLEM 7:

PROBLEM 8:

PROBLEM 9:

PROBLEM 10:

PROBLEM 11:

PROBLEM 12:

PROBLEM 13:

PROBLEM 14:

PROBLEM 15:

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PROBLEM 1: 4 pts.
Two marbles, one twice as heavy as the other, are dropped to the ground from the roof of a building. Air resistance is negligible.
Circle the correct letter.

Just before hitting the ground, the heavier marble has

(A)

as much kinetic energy as the lighter one.

(B)

twice as much kinetic energy as the lighter one.

(C)

half as much kinetic energy as the lighter one.

(D)

four times as much kinetic energy as the lighter one.

(E)

an energy that is impossible to determine.

PROBLEM 2: 4 pts. In part (a) of the figure below, an air track cart attached to a Hooke's Law spring rests on the track at the position x $_{\mbox{\small {equilibrium}}}$ and the spring is relaxed. In (b), the cart is pulled to the position x $_{\mbox{\small {start}}}$ and released. It then oscillates about x $_{\mbox{\small {equilibrium}}}$. Which graph correctly represents the potential energy of the spring as a function of the position of the cart?

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PROBLEM 3: (4 pts.)
A car accelerates from zero to 30 mph in 1.5 s. How long does it take for it to accelerate from zero to 60 mph, assuming the power of the engine to be independent of velocity and neglecting friction?

Circle the correct letter.

(A)

3.0 s

(B)

4.5 s

(C)

6.0 s

(D)

9.0 s

(E)

12.0 s

PROBLEM 4: (4 pts.) A person attempts to knock over a tall can by throwing a ball at it. The person has two balls of equal size and mass, one made of rubber and the other of putty. The balls are thrown identically and the rubber ball bounces back elastically,while the ball of putty sticks to the can. Which ball is most likely to tip over the can?

Circle the correct letter.

(A)

The rubber ball

(B)

The ball of putty

(C)

It makes no difference

(D)

More information is needed

PROBLEM 5: (4 pts.)
If all three collisions in the figure shown here are totally inelastic, which of the three bring(s) the car on the left to a halt while respect to an observer on the ground?
(Note: m reflects a mass and v is a velocity.)
$\parbox{0.35\linewidth}{ \begin{enumerate} \item [(A)] I \item [(B)]... ... I,III \item [(F)] II,III \item [(G)] all three \end{enumerate} \hspace*{.3in}}$$\parbox{0.30\linewidth}{~~ \vspace*{-0.5in} \begin{center} \includegraphics[width=4in]{mt2_f06_f2.eps} \end{center}}$

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PROBLEM 6: (8 pts.) Rain (i.e., water) is falling vertically into an 8 kg open cart at the rate of 0.5 liters/minute. Initially the cart is moving horizontally at 5 m/s. Considering only the water collecting in the cart and assuming the cart is moving on a frictionless surface, what is the kinetic energy of just the cart after water collects for 10 minutes? Assume the density of water is exactly 1 gm per cubic centimeter.

$K_{\mbox{\tiny CART}}$ =  


PROBLEM 7:(8 pts)

A horizontal turntable, radius 2.0 m and rotational inertia of 200. kg-m$^2$, as shown, is turning counter-clockwise at an angular velocity of 2.0 rad/sec. A child (of mass 25 kg) is standing at position B.
The child walks towards the center (to point C).
What is the resulting angular velocity of the turntable when the child reaches the center? (Assume the child may be considered to be a point mass.)

$\omega_c=$  

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PROBLEM 8: (16 pts., 6,6,4 )
A thin cylindrical support disk of radius 8.0 m and zero mass is free to rotate about its center. Rigidly affixed to this first disk is a second smaller solid disk of half the radius and a mass of 6.0 kg as shown in the figure below.

Initially at rest, the smaller disk starts out at the top and then it rotates 180$^\circ$ so that it ends up at the bottom.
(a) How much work is done by gravity?
(b) What is the angular velocity of the assembly at the finish?
(c) If instead you brought the assembly to rest by applying a brake to the edge of outer support, then what was the average torque applied?

 

$~~~W=$  















$~~~~\omega=$  









$\tau_{\tiny avg}=$  

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PROBLEM 9: (4 pts.)

A person pulls a box along the ground at a constant speed. If we consider Earth and the box as our system, what can we say about the net external force on the system?

(A)

It is zero because the system is isolated.

(B)

It is nonzero because the system is not isolated.

(C)

It is zero even though the system is not isolated.

(D)

It is nonzero even though the system is isolated.

(E)

none of the above

PROBLEM 10: (4 pts.)
The diagrams below show forces applied to a wheel that weighs 20 Newtons. The symbol $W$ stands for the weight. In which diagram(s) is the wheel in equilibrium?

(Please circle the appropriate letter.)

(A)

(a)

(B)

(b)

(C)

(c)

(D)

(d)

(E)

(b) and (d)

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PROBLEM 11: (8 pts.)

The only force acting on a 2.0 kg object moving along the $x$ axis is shown. Notice that the plot is force versus time. If the velocity $v_x$ is +2.0 m/s at $t=0$, what is the velocity at 4.0 s?

$v=$  

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PROBLEM 12: (8 pts.)
A 80 kg person, starting at rest, falls vertically ($g=10$m/s$^2$), and lands squarely onto a vertically mounted Hooke's Law spring ($k= 2000$ N/m). The spring compresses 2 m before the person comes momentarily to rest.
What was the initial height, $h$, of the person above the top of the uncompressed spring?



$h=$  

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PROBLEM 13:(8 pts)

A solid cylinder (mass, $M$ = 2.0 kg and radius, $r$ = 2.0 m) as shown, is anchored to a wall by a hinged massless rope. The rope is horizontal and the hinge is attached at the very top of the cylinder. The coefficient of static friction with the horizontal surface is 0.5. A horizontal string is attached to the center of the cylinder as shown. What is the minimum tension, T, necessary to cause the cylinder to move from its position?

  

$T=$  

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PROBLEM 14:(4 pts)

You are using a wrench and trying to loosen a rusty nut. Which of the arrangements shown is most effective in loosening the nut? List in order of descending efficiency the following arrangements:
(Note: It is possible for two arrangements to have the same effectiveness.)

     

Best to Worst:  

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PROBLEM 15: (12 pts., 3,6,3)

A solid 4.0 kg spherical ball starts out rotating an angular velocity, $\omega =$ 16 rad/sec, and a center-of-mass velocity, $v_{\mbox{\tiny CM}} =$ 0 m/s, as shown (i.e., the ball is sliding not rolling). The radius of the ball is 0.50 meters and the coefficients of static and sliding friction with respect to the surface are 0.20 and 0.10 respectively.

(a) What is the direction and magnitude of the frictional force on the ball?
(b) After 2.4 seconds, what will be the magnitude of the angular velocity?
(c) If at 2.4 seconds the magnitude of the angular velocity is 6.4 rad/s and the magnitude of the center of mass velocity is 3.2 m/s, will the ball be rolling? Why or why not?












$F_f=$  














$\omega =$  







(c)