Physics 195 Spring 2010 Homework #5

Due by 5 pm Wednesday March ^3rd.

1) 1) Draw several pictures to show what happens as these waves pass through each other. Assume that sharp corners are possible in a wave and use them in your drawings.

a)

b)

2) A student measures the period of a pendulum three times and gets the answers 1.6s, 1.8s, 1.7s.

a) What are the mean and standard deviation?

b) If the student decides to make a fourth measurement, what is the probability that this new measurement will fall outside the range 1.6s to 1.8s?

3) a) Show that f(x,t) = 0.1sin[2π(0.5x - 20t)] satisfies the wave equation

= 1600.

b) What is the propagation velocity?

4) A mechanical oscillator connected to the end of a stretched string creates a transverse displacement of the end of the string given by

ξ = 0.01sin(20t) where ξ is in metres. If the tension in the string is 10 N and the string has mass density 20 g/m, find

a) the velocity of the transverse waves

b) the frequency, f, of the waves, and

c) the wavelength.

5) A nylon guitar string has a linear density of 7.20 g/m and is under a tension of 150 N. The fixed supports are a distance 90.0 cm apart and string oscillates in its third mode.

a) Find the speed of waves on the string.

b) Find the frequency of the normal mode.

c) Find the wavelength of the mode.

6) A string of length L = 1.20m with linear density μ = 1.6g/m is driven at one end by a sinusoidal oscillator with frequency f = 120Hz. The amplitude of the oscillation is small enough that the end of the string can be considered as fixed. The other end goes over a pulley (the 1.20 m is between the driver and the pulley) and is attached to a mass m that hangs down.

a) What mass is needed for the string to oscillate in its fourth mode (n = 4)?

b) What mode can be set up if the mass m = 1.50kg?

7) Two guitar strings have the same length and mass per unit length. The tighter string has a fundamental frequency of 440 Hz. When the two strings are played together you hear beats at 2 Hz. What is the fundamental frequency of the looser string?

8) Find the Fourier series representation for a wave with speed v on a string of length L started from rest so that,

y(x,0) =

You may assume that the ends of the string are fixed.

9) a) What is the fastest transverse wave that can be sent along a steel wire, given that the volume density of steel is 7800kg∕m³ and that the tensile stress must be limited to no more than 7.00 × 10⁸N∕m².

b) Does your answer depend on the diameter of the wire? Explain.

Note: Look carefully at the units of the quantities you are given!

10) If a set of N measurements, x_i, has mean = ∑ _i=1^Nx_i then the deviation of x_i is defined to be d_i = x_i -. Prove that the mean of the deviations is always zero.