Questions

1

A 0.10-kg block attached to a spring slides back and forth along a straight line on a smooth horizontal surface. Its displacement from the origin is given by

$$x = ( 10{~\rm cm}) \cos [ (10 {~\rm rad/s}) t+\pi /2 {~\rm rad}]~.$$

(a)

What is the oscillation frequency? (5 points)

(b)

What is the maximum speed acquired by the block? At what value of $x$ does this occur? (5 points)

(c)

What is the maximum acceleration of the block? At what value of $x$ does this occur? (5 points)

(d)

What external force must be initially applied to this block-spring system to give it this motion? (5 points)

(e)

What is the total mechanical energy of this system? (5 points)

2

A siphon is a device for removing liquid from a container that cannot be tipped. It operates as shown in Fig. 1. The tube must initially be filled, but once this has been done the liquid will flow until its level drops below the tube opening at $A$. The liquid has density $\rho$ and negligible viscosity. The atmospheric pressure is $P_0$ and the gravitational constant is $g$.

Figure 1: Prob 2

(a)

With what speed does the liquid emerge from the tube at $C$? (5 points)

(b)

If the cross-sectional area of the tube is uniform, is the speed of the liquid inside the tube constant throughout the length of the tube? If so, why? If not, why not? (5 points)

(c)

Assuming uniform cross-sectional area of the tube, what is the pressure in the liquid at the topmost point $B$? Express your answer in term of $\rho$, $P_0$, $g$, $d$, $h_1$ and $h_2$. (10 points)

(d)

What is the greatest possible height $h_1$ that a siphon can lift the liquid? Express your answer in term of $\rho$, $P_0$, and $g$. (5 points) Hint: The maximum $h_1$ occurs at $h_2=d=0$.

(e)

If the whole system-the tube, the container, and the liquid-shown in the figure is in an elevator which is accelerated upwards by $a$, how does the answer to part (d) change? (5 points)

3

The linear density of a vibrating string is $1.6\times 10^{-4} ~\rm kg/m$. A transverse wave is propagating on the string and is described by the equation

$$y_1=(0.021 {~\rm m}) \sin [ (2.0{~\rm m^{-1}})x + (30 {~\rm s^{-1}})t] ~.$$

(a)

What is the direction and magnitude of the velocity of this travelling wave? (5 points)

(b)

What is the tension in the string? (5 points)

(c)

Another travelling wave described by the equation

$$y_2=(0.05 {~\rm m}) \cos [ (2.0{~\rm m^{-1}})x + (30 {~\rm s^{-1}})t]$$

is superimposed on top of $y_1$. What is the time-averaged power transferred along the string by the resultant wave $y_1+ y_2$? (15 points)

Hint: $2\sin \theta \cos \theta = \sin 2\theta$, $\sin^2 \theta = {1\over 2}(1-\cos 2\theta )$, $\cos^2 \theta = {1\over 2}(1+\cos 2\theta )$, and $\int_0^T {dt\over T}~\sin ({4\pi t\over T}+\phi)= 0$.

(d)

A certain length $L$ of the above string is used to generate standing waves. Assuming that the string is stretched with both ends fixed, obtain the expression for the frequency of the lowest ($n=1$) mode (fundamental frequency) of this string, in terms of the wave-speed $v$ and a length $L$. It is known that humans can hear frequencies of 20 Hz to 20,000 Hz. What is the desired range of the length $L$ of the above string that can generate audible fundamental frequencies, assuming that the value of the wave velocity is the one obtained in part (a)? (5 points)

Note: In part (d), for practical reasons, the frequencies near 20,000 Hz will be difficult to generate using the above string. This is just meant to be a theoretical question.

4

An acoustic burglar alarm consists of a source emitting waves of frequency 28 kHz. Assume a speed of acoustic waves $v= 340 ~\rm m/s$.

(a)

An intruder is walking at 0.95 m/s directly away from the alarm. Naturally, the acoustic waves emitted by the alarm will reach his body. What frequency of acoustic waves will the intruder's body receive? (5 points)

(b)

The acoustic waves are reflected from his body and travel back to the alarm. Assume that you can model the reflected acoustic waves as if they were generated by the body of the intruder. Then what is the frequency of the reflected acoustic waves that are detected by the alarm? (5 points)

(c)

What is the beat frequency which results from superposition of the emitted and reflected acoustic waves at the burglar alarm? (5 points)