PHYS 2211 Module 14 Self Assessment Practice Problems

Module 14 Self Assessment Practice Problems

14.1
A wave is modeled at time t = 0.00 s with a wave function that depends on position. The equation is y(x) = (0.30 m) sin(6.28 m^-1x). The wave travels a distance of 4.00 meters in 0.50 s in the positive x-direction. Write an equation for the wave as a function of position and time.
Answer: y(x,t) = (0.30 m) sin(6.28 m^-1x – 50.3 s^-1t)

14.2
A swimmer in the ocean observes one day that the ocean surface waves are periodic and resemble a sine wave. The swimmer estimates that the vertical distance between the crest and the trough of each wave is approximately 0.45 m, and the distance between each crest is approximately 1.8 m. The swimmer counts that 12 waves pass every two minutes. Determine the simple harmonic wave function y(x,t) that would describes these waves.
Answer: y(x,t) = (0.23 m) sin(3.5 m^-1x – 0.63 s^-1t)

14.2

A swimmer in the ocean observes one day that the ocean surface waves are periodic and resemble a sine wave. The swimmer estimates that the vertical distance between the crest and the trough of each wave is approximately 0.45 m, and the distance between each crest is approximately 1.8 m. The swimmer counts that 12 waves pass every two minutes. Determine the simple harmonic wave function y(x,t) that would describes these waves.

Answer: y(x,t) = (0.23 m) sin(3.5 m^-1x – 0.63 s^-1t)

14.3
A string with a linear mass density of 48 g/m is stretched with a tension of 6.2 N and fixed at both ends. At one end, the string is attached to a device that oscillates in simple harmonic motion with a frequency of 48 Hz and an amplitude of 2.8 cm. This motion produces a periodic wave in the string.
(a) What is the speed of the wave that travels in this string?
(b) What is its wavelength?
(c) What is the wave function y(x,t) for this wave? Assume that at t = 0, the oscillating device is at its maximum amplitude.
(d) What is the maximum transverse velocity v_y of particles on the string?
(e) What is the maximum transverse acceleration a_y of particles on the string?
Answer: (a) 11.4 m/s (b) 23.7 cm (c) y(x,t) = (2.8 cm) cos(0.265 cm^-1x – 301 s^-1t) (d) 843 cm/s (e) 2.54 x 10⁵ cm/s²

14.4
A string with a mass of 0.30 kg has a length of 4.00 m. If the tension in the string is 50.00 N, and a sinusoidal wave with an amplitude of 2.00 cm is induced on the string, what must the frequency be for an average power of 100.00 W?
Answer: 80.9 Hz

14.5
A thin, 80.0-cm wire has a mass of 16.3 g. One end is tied to a nail, and the other end is attached to a screw that can be adjusted to vary the tension in the wire.
(a) To what tension must you adjust the screw so that a transverse wave of wavelength 3.35 cm makes 630 vibrations per second?
(b) How fast would this wave travel?
Answer: (a) 9.1 N (b) 21.1 m/s

14.6
A 1.50-m string of weight 0.0121 N is tied to the ceiling at its upper end, and the lower end supports a weight W. Neglect the very small variation in tension along the length of the string that is produced by the weight of the string. When you pluck the string slightly, the waves traveling up the string obey the equation y(x,t) = (8.50 mm) cos[(172 rad/m)x – (2730 rad/s)t] Assume that the tension of the string is constant and equal to W.
(a) How much time does it take a pulse to travel the full length of the string?
(b) What is the weight W?
(c) How many wavelengths are on the string at any instant of time?
(d) What is the equation for waves traveling down the string?
Answer: (a) 0.094 s (b) 0.21 N (c) 41 (d) y(x, t) = (8.50 mm) cos[(172 rad/m)x + (2730 rad/s)t]

14.7
For a string stretched between two supports, two successive standing-wave frequencies are 520 Hz and 620 Hz. There are other standing-wave frequencies lower than 520 Hz and higher than 620 Hz. If the speed of transverse waves on the string is 355 m/s, what is the length of the string? Assume that the mass of the string is small enough for its effect on the tension in the string to be neglected.
Answer: 1.78 m

14.8
A 0.800-m-long string with linear mass density µ = 7.50 g/m is stretched between two supports. The string has tension F and a standing-wave pattern (not the fundamental) of frequency 564 Hz. With the same tension, the next higher standing-wave frequency is 705 Hz.
(a) What is the frequency of the fundamental standing wave for this string?
(b) What is the wavelength of the fundamental standing wave for this string?
(c) What is the value of F?
Answer: (a) 141 Hz (b) 1.6 m (c) 382 N

14.9
A wire with mass 40.0 g is stretched so that its ends are tied down at points 80.0 cm apart. The wire vibrates in its fundamental mode with frequency 60.0 Hz and with an amplitude at the antinodes of 0.300 cm.
(a) What is the speed of propagation of transverse waves in the wire?
(b) Compute the tension in the wire.
(c) Find the maximum transverse velocity and acceleration of particles in the wire.
Answer: (a) 96 m/s (b) 461 N (c) 1.13 m/s, 426 m/s²

14.10
You are exploring a newly discovered planet. The radius of the planet is 7.20 x 10⁷ m. You suspend a lead weight from the lower end of a light string that is 4.00 m long and has mass 0.0280 kg. You measure that it takes 0.0685 s for a transverse pulse to travel from the lower end to the upper end of the string. On the earth, for the same string and lead weight, it takes 0.0390 s for a transverse pulse to travel the length of the string. The weight of the string is small enough that you ignore its effect on the tension in the string. Assuming that the mass of the planet is distributed with spherical symmetry, what is its mass?
Answer: 2.5 x 10²⁶ kg

14.10

You are exploring a newly discovered planet. The radius of the planet is 7.20 x 10⁷ m. You suspend a lead weight from the lower end of a light string that is 4.00 m long and has mass 0.0280 kg. You measure that it takes 0.0685 s for a transverse pulse to travel from the lower end to the upper end of the string. On the earth, for the same string and lead weight, it takes 0.0390 s for a transverse pulse to travel the length of the string. The weight of the string is small enough that you ignore its effect on the tension in the string. Assuming that the mass of the planet is distributed with spherical symmetry, what is its mass?

Answer: 2.5 x 10²⁶ kg