A piano tuner listens to a tuning fork vibrating at 440 Hz to tune the string of a piano. He hears beats between the tuning fork and the piano string. Is the string in tune? Explain your answer.

Applying Concepts Imagine that you and a friend are trying to tune the lowest strings on two different guitars to the same pitch. Explain how you could use beats to determine if the strings are tuned to the same frequency.

Imagine that you and a friend are trying to tune the lowest strings on two different guitars to the same pitch. Explain how you could use beats to determine if the strings are tuned to the same frequency.

A tuning fork is known to vibrate with frequency $262 \mathrm{~Hz}$. When it is sounded along with a mandolin string, four beats are heard every second. Next, a bit of tape is put onto each tine of the tuning fork, and the tuning fork now produces five beats per second with the same mandolin string. What is the frequency of the string?
(a) $257 \mathrm{~Hz}$
(b) $258 \mathrm{~Hz}$
(c) $262 \mathrm{~Hz}$
(d) $266 \mathrm{~Hz}$
(e) $267 \mathrm{~Hz}$
(f) This sequence of events could not happen.

A piano tuner hears a beat every 2.00 s when listening to a 264.0-Hz tuning fork and a single piano string. What are the two possible frequencies of the string?

Describe the changes in elastic potential energy and kinetic energy that occur when a mass vibrates on a spring.

Use the kinetic theory to explain the difference in wave speed in solids, liquids, and gases.

Imagine you are shaking the end of a rope to create a series of waves. What will you observe if you begin shaking the rope more quickly?

Drawing Conclusions If you yell or clap your hands while standing at the edge of a large rock canyon, you may hear an echo a few seconds later. Explain why this happens.

Ocean waves are partially reflected from the entrance to a harbor, where the depth of the water is suddenly less. What does this tell you about the speed of waves in water of different depths?

Explain why you hear a sound when you clap your hands together.

If you yell or clap your hands while standing at the edge of a large rock canyon, you may hear an echo a few seconds later. Explain why this happens.

If an object on a horizontal, frictionless surface is attached to a spring, displaced, and then released, it will oscillate. If it is displaced 0.120 m from its equilibrium position and released with zero initial speed, then after 0.800 s its displacement is found to be 0.120 m on the opposite side, and it has passed the equilibrium position once during this interval. Find the frequency.

If an object on a horizontal frictionless surface is attached to a spring, displaced, and then released, it oscillates. Suppose it is displaced $0.120 \mathrm{~m}$ from its equilibrium position and released with zero initial speed. After $0.800 \mathrm{~s}$, its displacement is found to be $0.120 \mathrm{~m}$ on the opposite side, and it has passed the equilibrium position once during this interval. Find
(a) the amplitude,
(b) the period,
(c) the frequency of the motion.