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November 19, 2004 Version A

PHYSICS 207: Mid-Term Exam III

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: (6,6,8 pts.)

A 30 kg child is sitting with his center of mass 2 m from the frictionless pivot of a massless see-saw as shown. The see-saw is initially horizontal and, at 3 m on the other side, there is a massless Hooke's Law spring (constant 750 N/m) attached so that it sits perfectly vertical (but slightly stretched). Gravity acts in the downward direction with $g=$10 m/s$^2$.

Assuming everything is static and in perfect equilibrium.
(a) What force, $F_S$, is provided by the spring?
(b) What is the magnitude and direction of the force, $\vec{F_P}$ applied at the pivot?

$\vec{F_S}=$  


$\vec{F_P}=$  

($b$) Now the child briefly bounces the see-saw (with a small amplitude oscillation) and then moves with the see-saw. What is the angular frequency of the child?

$\omega=$  

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PROBLEM 2: (6 pts.)

Two strings are held at the same tension and driven with the same amplitude and frequency. The only difference is that one is thicker and has a mass per unit length that is four times larger than the thinner one. Which string (and by how much) transfers the most power?

(Circle the correct answer.)

(A)

the thicker string by a factor of 4.

(B)

the thicker string by a factor of 2.

(C)

they transfer an equivalent amount of energy.

(D)

the thinner string by a factor of 2.

(E)

the thinner string by a factor of 4.

PROBLEM 3: (6 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.

(b)

B.

(a)

C.

(d)

D.

(c)

E.

(b) and (d)

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PROBLEM 4: (6,6,8 pts.)

An object of mass 2 kg undergoes an elliptical orbit about a ``star'' (of mass $3.0\times10^{11}$ kg) as shown. At point $A$ the object's velocity is 20 m/s (and perpendicular to the gravitation force on the object). ( $G= 6.67 \times 10^{-11}$ N m$^2$/kg$^2$)

($a$) What is velocity of the object at point B (the furthert point)?

($b$) Assuming the velocity at point C is 100 m/s, what is the change in potential energy from point A?

$c$. Assume that now a force acts at point A to now initiate a circular orbit with a velocity of 5 m/s, what would be the object's orbital period?

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PROBLEM 5: (6 pts.)

A transverse pulse is initially travelling to the right on a string that is joined, on the right, to a thicker string of higher mass per unit length. The tension remains constant T throughout. Part of the pulse is reflected and part transmitted. The drawing to the right shows the before (at top) and after (bottom) the pulse traverses the interface. There are however a few mistakes in the bottom drawing. Identify two things wrong in the bottom sketch assuming the top sketch is correct.

 1:



2:

PROBLEM 6: (5 pts.)

An under damped oscillator which is set into motion but has no additional external driving force after that has an amplitude that

(Please circle the correct letter.)

A.

decays exponentially without oscillation back to its equilibrium position.

B.

oscillates at more than its undamped frequency but its maximum amplitude decays exponentially with time.

C.

oscillates at less than its undamped frequency but its maximum amplitude decays exponentially with time.

D.

oscillates at a gradually diminishing frequency and with an amplitude that decays exponentially with time.

E.

is best described by a non-linear first order differential equation.

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PROBLEM 7: (8 pts.)
A person initially weighs 800 Newtons when standing on the surface of a large spherical planet of uniform density and radius R. If this person were standing on a planet of the same density but with one third the radius, what would his weight, $W_n$, become?

$W_{n}=$  


PROBLEM 8: (5 pts.)

An under damped oscillator which is driven by a periodic external driving force for a long time

(Please circle the correct letter.)

A.

has a large amplitude only if the frequency of the driving force is close to the resonant fequency of the oscillator.

B.

has oscillations with amplitudes that vary from very large to very small.

C.

has an oscillation amplitude that is proportional to the amplitude and frequency of the driving force.

D.

has an oscillation amplitude that grows increasingly larger with time and without limit regardless of the damping.

E.

has as oscillation amplitude that eventually decays with time.

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

A small person is riding a unicycle and is halfway between two posts 200 m apart. The guide wire was originally 200 m long, weighs 1.0 kg and has cross sectional area of 2 mm$^2$. Under the weight of the unicycle it sags down 0.01 m at the center and there is a tension of 5000 Newtons along the wire.

    

(a) What is the Young's Modulus of the wire (to two significant figures)?
(b) How long does it take a transverse wave (a pulse) to propagate from the support to the unicycle?
(c) If the pulse is now said to be a perfectly sinusoidal wave and has a frequency of 100 Hz, what is the angular frequency?
(d) At what transverse amplitude of the wave, in the vertical direction, will the wire's maximum acceleration just reach 10 m/s$^2$ (assuming the wire's oscillatory motion and tension are not affected by the rider's motion).









$Y~=$  








$t~=$  







$\omega~=$  







$A~=$