Physics 102 Homework Set 6 Solutions Chapter 19: Exercises: 4) A swing is an example of a pendulum. When you stand up on a swing, you are effectively shortening the length of the pendulum (since the distance from the axis of the pendulum to the center of mass is less). If you shorten a pendulum, you decrease its period. You can also think about this from the standpoint of rotational inertia. When you stand up, you get closer to the axis of rotation, decreasing the rotational inertia. This allows the swing to go back and forth in a shorter period of time. 10) The two types of waves in this question are longitudinal and transverse. a) an ocean wave is mostly a transverse wave (but there is some longitudinal component) b) a sound wave is a longitudinal wave c) snapping one end of a stretched rope sends a transverse pulse down the rope (like I did in class) 12) Since the period T = 1/f, if we double the frequency, we halve the period. 24) When you bow the string in the middle, you excite the fundamental which has a wavelength of twice the string length. When you bow at the 1/4 point, you excite the next harmonic which has a wavelength equal to the string length. Thus, you have decreased the wavelength by a factor of two. When you halve the wavelength, the frequency must double (since the velocity is unchanged). v = lambda * f. A violinist can use this technique to double the pitch of the note she is playing, effectively raising it by one octave. Note that there is a lot of extraneous information in this problem. 26) a) When the locomotive is coming toward you, you hear the frequency of the whistle increase because of the Doppler shift. b) the wavelength also decreases (you can see this in figure 19.16 where the wave crests are closer together approaching point B). c) the velocity of sound in air is unchanged. The air does not care how the high pitched sound is made (whether it is made by a high pitched whistle at rest or by a low pitched whistle moving toward you). The speed of sound in air is determined by the thermal velocity of the air molecules. This is not changed by the motion of the locomotive. Note that this is a completely different case from one where you throw a ball from the locomotive. In that case, the velocity of the ball is increased by throwing it from the locomotive (if you throw it forwards). For throwing a ball, the velocities add up. For a sound wave, the velocity does not change, but the frequency and wavelength change instead. 28) No, there is not. The doppler effect occurs only when the source is moving closer to you or further away from you. When the object is moving at right angles to you, its distance to you is not changing. This occurs most obviously when swinging a ball on a string around you. In this case, the distance between you and the ball is not changing and the ball is always moving at right angles with respect to you. I swung the doppler ball around my head in class so that you could hear the doppler shift as the ball moved toward and then away from you. I could not hear the doppler shift since the ball was never moving toward or away from me. 34) The shock wave angle narrows as the plane increases its speed. See figure 19.20 for an example. This is also true for motorboats and jetskis. Look at the wake the next time you go boating and notice its opening angle as you vary the speed of the boat. BTW: an acceptable reason for this answer is: 'I saw it while I was boating.' Problems: 4) The frequency f is 2 Hz or 2 (1/s) since it makes two cycles every second. The period T is 0.5 s since it takes 1/2 second for each cycle. In one cycle, the weight goes from bottom, through the average position, to the top, and back down again. Remember that the amplitude is the distance from the average position to the maximum (or to the minimum). Since it is a distance of 20 cm from min (bottom) to avg (middle) to max (top) to avg (middle) to min (bottom), it is a distance of 5 cm from average to maximum. Thus the amplitude is 5 cm. 6) v = lambda * f so that lambda = v / f = 340 m/s / 600 1/s = 0.57 m -------------------------- Chapter 20: Exercises: 6) The electromagnetic wave travels MUCH faster. The same frequency means that the time between wave crests is the same. In that same time, the electromagnetic wave travels much farther, therefore its wavelength is much bigger. We can do the same thing by equation: lambda = v / f. If v is much bigger, then lambda will be much bigger. 8) Olympic track races are frequently decided by differences of 0.01 second or less. If there are ten runners and each runner is separated by a distance of 1.5 m, then there is a distance of almost 15 m between the runners closest to and farthest from the starting gun. The time it takes sound to travel that far is time = distance / velocity = 15 m / 340 m/s = 0.05 s (approximately). This gives an unfair advantage to the closest runner. When the sound is picked up by a microphone, it is transmitted electrically at almost the speed of light to the speaker near each starting block. This reduces the time lag to nanoseconds (billionths of a second) and makes it irrelevant. 11) It is so quiet after a snowfall because sound waves are absorbed by the snow, rather than reflected. ------------------------------------------------------------- Estimation: The radius of the Earth is about 6400 km. The distance around the world (circumference) is d = 2 pi r = 4*10^4 km = 4*10^7 m The speed of sound is v = 340 m/s. Therefore the time it takes for the sound to travel completely around the Earth is: t = d / v = 4*10^7 m / 3.4*10^2 m/s = 1.2*10^5 s This is t = 1.2*10^5 s * (1 hour/3.6*10^3 s) = 33 hours or t = 1.3 days