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October 29, 2004 Version A

PHYSICS 207: Mid-Term Exam II

Print your name and section clearly on all seven 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 box 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 5 and 15 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 magnitudes. You are allowed one 8½ x 11" sheet of notes and no other references. The exam lasts exactly 60 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 $= 2/5 MR^2$,                    Solid Cylinder $=1/2 MR^2$
Thin Hollow Cylinder $=MR^2$
Long thin rod (axis through center) $= 1/12 ML^2$
Thin rectangular Plate (axis through center) $= 1/12 M(a^2+b^2)$

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:

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PROBLEM 1: (6,6 pts.)

A mass, $m= 11$ kg, slides down of a frictionless circular path of radius, $R= 5$ m, as shown in the figure at right. Initially it moves only vertically and, at the end, only horizontally (1/4 of a circle all told). Gravity, $g= 10$ m/s$^2$, acts along the vertical.

If the initial velocity is -2 m/s k (down is negative),

(a) what is the work done by gravity?

(b) what is the final speed of the mass when it reaches the bottom?

$W=$  


speed $=~$

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PROBLEM 2: (6 pts.)

An object is acted upon by only two forces, one conservative and one nonconservative, as it moves from point A to point B. The kinetic energy of the object at points A and B are equal if

(Circle the correct answer.)

(A)

the sum of the two forces' work is zero.

(B)

the work of the nonconservative force is zero.

(C)

the work of the conservative force is zero.

(D)

the work of the conservative force is equal to the work of the nonconservative force.

(E)

None of these will make them equal.

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

Two uniform density solid cylinders of equal mass M= 3.0 kg and differing radii $R=4.0$ m and $r=2.0$ m are mounted so that the smaller cylinder is held directly above the larger radius mass. Initially the bottom mass is spinning at 2.5 rad/sec while the top mass is a rest. The top mass is very gently dropped onto the bottom so that it remains centered and, afterwards, they stick such that the two masses rotate at the same angular velocity, $\omega_f$.

(a) What is the intial kinetic energy, $K_i$, of the system?
(b) What is $\omega_f$ (the final angular velocity)?

$K_i=$  


$\omega_f=$    

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PROBLEM 4: (6,6 pts.)

You are observing the motion of a very small mass of weight 40 N sitting on a turntable that is situated horizontally. The mass is located 8 m from the rotation axis. Initially the angular velocity is zero but then the turntable is switched on and it accelerates angularly at 1.5 rad/sec$^2$ k.

(a) After 5 seconds what is the turntable's angular velocity?
(b) If the coefficient of static friction, $\mu_s$, between the mass and the turntable is 0.2 then at what angular velocity $\omega_s$ does it begin to slip?

$\omega=$  


$\omega_s=$  

PROBLEM 5: (6 pts.)
A torque can be exerted on a body with a fixed axis of rotation

(Please circle the appropriate letter.)

A.

only by a centripetal force.

B.

only by a force directed radially outwards.

C.

only by a force with components perpendicular to the radius vector and parallel to the rotation axis.

D.

only by a force with a component directed radially outwards.

E.

only by a force with components perpendicular to the radius vector and the rotation axis.

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PROBLEM 6: (10 pts.)

An object, initially at rest, explodes into three fragments. Two of the fragments are equal in mass (each of 2 kg) and leave the center of mass with equal speeds (of 6 m/s). There is an angle of 120$^\circ$ between their respective velocity vectors. (Hint: Take advantage of the symmetry.)

If the mass of the third object is 3 kg, then what is its speed (relative to the center of mass)?

$\sin 30^\circ = 0.500$, $\cos 30^\circ=0.832$

$\sin 60^\circ=0.832$, $\cos 60^\circ = 0.500$

$\sin 120^\circ=0.832$, $\cos 120^\circ=-0.500$

speed$=$

PROBLEM 7: (6 pts.)

Two boys in a canoe toss a ball back and forth. What effect will this have on the canoe? Neglect all frictional forces with water or air.

(Please circle the correct letter.)

A.

None, because the ball remains located in the canoe.

B.

The canoe will drift in the direction of the boy who throws the ball harder each time.

C.

The canoe will drift in the direction of the boy who throws the ball with less force each time.

D.

The canoe will oscillate back and forth always moving opposite to the ball.

E.

The canoe will oscillate in the direction of the ball because the canoe and ball exert forces in opposite directions upon the person throwing the ball.

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PROBLEM 8: (3,3,6,6 pts.)

A thin hollow cylinder of mass $M$ and radius $R$ is positioned so that half the cylinder hangs over the edge of a horizontal table. There is a mass $M$ attached to a string that is wrapped around the circumference and hanging as shown. Gravity, with acceleration $g$, acts downward and there is friction with the table to prevent sliding.

(a) If the weight of the string and hanging mass are said to be zero identify the nature of the cylinder's equilibrium (e.g. stable) towards
(1) displacement to the right?
(2) displacement to the left?

(b) What is the moment of inertia of the cylinder for rotation about point P?

(c) If the hanging mass is said to have the same mass $M$ (asxdd that of the cylinder), what is the initial linear acceleration, $a$, at the cylinder's center of mass in terms of $R$, $g$, and/or $M$? (Hint: Write down an equation for the torque about point P and an equation for forces on the hanging mass.)

(1)  


(2)  


$I_p=$  


$a=$  

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PROBLEM 9: (6,6,6 pts.)

A hammer is aimed so that it imparts an impulse of 4 kg m/s along the $x$-dir to a mass, $m= 2$ kg, initially at rest and attached to a Hooke's Law spring, spring constant of 40.0 N/m, at its equilibrium position (i.e., the initial force by the spring on the mass is zero). Gravity, $g= 10$ m/s$^2$ acts downward.

(a) If the impulse acts over a time of 0.01 seconds, what is the average force $\vec{F}$ that acts on the block (from the hammer).

(b) What is the velocity of the mass immediately after being struck by the hammer?

(c) If the initial velocity is said to be 3.0 m/s and the mass travels on a uniform but rough surface a distance of 0.50 m before reaching zero velocity, then what must be the coefficient of kinetic friction?

    

$\overline{\vert\vec{F}\vert}=$  


$\vec{v_i}=$  


$\mu_k =$