Physics 171.201

Midterm Exam 2

 

November 13^th, 2002

 

Answer all three problems. Be sure that you pace yourself so that you have enough time to work on each question. Partial credit will be given for partially correct answers. Be sure to show your working.

 

 

 

 

 

1.    (40 points) Consider a compound pendulum

consisting of two identical strings and two identical

YB

YA

L

masses that are arranged as shown on the right.

L

 

Assume that the displacements Y_A and Y_B about the

equilibrium position (dotted line) are both much smaller

than L,

M

and that the masses are free to move only in

the plane of the paper.

 

(a) Write down the equation of motion for the system,

making use of small-angle approximations where

appropriate. (HINT: don’t forget that the tension

in the upper string is twice that in the lower string.)

 

M

(b) Determine the frequencies for the two normal modes.

 

(c) Describe the normal modes of the system quantitatively

and sketch them

 

 

 

 

 

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2. (40 points). Consider a damped harmonic oscillator

consisting of a mass on a spring that is immersed in a viscous

liquid. The spring constant is K, the mass is M, and the

viscous force on the mass is – GMv, where v is the velocity.

Y is the displacement relative to the equilibrium position.

 

The mass is driven at resonance by the application of an

external driving force of the form F = F₀ cos (w₀t), where

w₀ = (K/M)^1/2 is the natural oscillation frequency in the

absence of damping.

 

(a) Write down the equation of motion for system

 

(b) Roughly ow long does it take the steady-state solution

to be established, i.e. for any transient solution to die

away? (Simply state the result without deriving it.)

 

(c) Show that the steady state solution for the displacement

can be written in the form Y = Y ₀ sin (w₀t), and determine

the relationship between Y₀ and F₀.

 

(d) For the steady state solution, compute the velocity of

the mass (as a function of time) and the rate of power input

by the driving force. Where does this power go?

 

 

 

 

 

3. (20 points). Write a couple of paragraphs explaining why it sounds very noisy when your head is above water in a crowded indoor swimming pool, but very quiet when you put your head under water.

 

Your explanation should include a qualitative discussion of the laws governing the reflection and transmission of waves at interfaces between media, the impedance of sound waves in air, the impedance of sound waves in water, and the impedance of hard, tiled walls as dashpots.

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