I. Warm-up Exercises
1. A traveling wave has a wave speed of 22 m/s and a wavelength of 0.83 m. What is the frequency of the wave?
2. A string has a length of 2.5 m and a mass of 23 g. What is the linear mass density of the string?
3. What is the wavelength of a wave whose wave number is 3.5 rad/m?
4. A stretched string has a length of 1.3 m. The string has a standing wave on it which consists of 3 bumps between the two ends of the string. What is the wavelength of the wave?
5. A string has a linear mass density of 0.035 kg/m . Waves traveling on this string have a speed of 14 m/s . What is the tension in the string?
6. What is the speed of sound in air on a hot summer day when the temperature is 32 oC?
7. The speed of sound in air at a certain location is 340 m/s. A closed pipe has a length of 1.3 m. What is the frequency of the second overtone produced in the pipe?
II. Some Standards
8. A string is positioned along the x-axis. A wave is imposed on the string which has its motion described by the equation (in MKS units) y(x,t) = 0.0015 sin(kx – wt). The wave is seen to have a wavelength of 1.25 m, and it is found that it takes 0.050 s for one wavelength to pass by any given point. Express all of your answers in MKS units! (a) What is the wave number of the wave? (b) What is its angular frequency? (c) What is the speed of the wave on the string? (d) What is the linear mass density of the string if the tension is 1.5 N? (e) What is the displacement of the point on the string at x = 0.30 m and at t = 0.020 s?
9. A horizontal string of mass 0.0035 kg and total length 1.1 m is attached to a wall at one end. The other end runs over a pulley and then runs vertically down to a 3.5-kg mass which hangs from its free end. The length of string between the wall and the pulley is 0.76 m. (a) What is the tension in the string? (b) What is the speed of waves on the string? (c) What are the lowest three frequencies that will produce standing waves on the stretched string between the wall and the pulley? (d) What are the wavelengths on the string corresponding to the three frequencies in the previous question?
10. A closed pipe contains air at 0 oC. The fundamental frequency of sound produced in the pipe is 102 Hz. (a) What is the length of the pipe? (b) What is the wavelength of the sound at the fundamental frequency? (c) What is the wavelength of the first overtone produced in the pipe?
III. So, you think you’re pretty good...?
11. When two pipes are physically close to one another, vibrations in one pipe can cause pressure waves to enter the second pipe and possibly cause resonant vibrations in the second pipe if the conditions are right. Such induced vibrations are called sympathetic vibrations. A closed pipe of length 0.95 m stands next to an open pipe. Both pipes are in a room in which the temperature is 25 oC. What length of the open pipe will cause sympathetic vibrations in the second overtone when the closed pipe is played in its fundamental mode?
12. Two pulleys are positioned 0.80 m apart in a room at 22 oC. A string having a linear mass density of 0.0031 kg/m runs over one pulley and horizontally to the next. The string has a 2.5-kg mass hanging vertically from each of its ends which hang over each pulley. What wavelength of sound is produced if a standing wave is maintained on the string in its second overtone mode of oscillation? (Hint: The frequency of the wave on the string is the same as the frequency of sound waves produced in the air. This is a general feature of waves – the speed and wavelength can change as the wave goes from one medium to another, but the frequency remains the same! This applies to mechanical waves as well as light waves.)
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