Two identical taut strings under the same tension $\boldsymbol{~F}$ produce a note of the same fundamental frequency $f_0$ - The tension in one of them is now increased by a very small amount $\Delta F$. (a) If they are played together in their fundamental, show that the frequency of the beat produced is $f_{\text {beat }}=f_0(\Delta F / 2 F)$. (b) Two identical violin strings, when in tune and stretched with the same tension, have a fundamental frequency of $440.0 \mathrm{~Hz}$. One of the strings is retuned by increasing its tension. When this is done, $1.5$ beats per second are heard when both strings are plucked simultaneously at their centres. By what percentage was the string tension changed?

Sometimes, even with a wrench, one cannot loosen a nut that is frozen tightly to a bolt. It is often possible to loosen the nut by slipping one end of a long pipe over the wrench handle and pushing at the other end of the pipe. With the aid of the pipe, does the applied force produce a smaller torque, a greater torque, or the same torque on the nut?

Two identical taut strings under the same tension $F$ produce a note of the same fundamental frequency $f_0$. The tension in one of them is now increased by a very small amount $\Delta F$. $(a)$ If they are played together in their fundamental, show that the frequency of the beat produced is $f_{\text {beat }}=f_0(\Delta F / 2 F)$. $(b)$ Two identical violin strings, when in tune and stretched with the same tension, have a fundamental frequency of $440.0 \mathrm{~Hz}$. One of the strings is retuned by increasing its tension. When this is done, $1.5$ beats per second are heard when both strings are plucked simultaneously at their centers. By what percentage was the string tension changed?

A merry-go-round at a playground is a circular platform that is mounted parallel to the ground and can rotate about an axis that is perpendicular to the platform at its center. The angular speed of the merry-go-round is constant, and a child at a distance of 1.4 m from the axis has a tangential speed of 2.2 m/s. What is the tangential speed of another child, who is located at a distance of 2.1 m from the axis? (a) 1.5 m/s (b) 3.3 m/s (c) 2.2 m/s (d) 5.0 m/s (e) 0.98 m/s

A rock is suspended by a light string. When the rock is in air, the tension in the string is 39.2 N. When the rock is totally immersed in water, the tension is 28.4 N. When the rock is totally immersed in an unknown liquid, the tension is 21.5 N. What is the density of the unknown liquid?

A space probe is traveling in outer space with a momentum that has a magnitude of

$$7.5 \times 10 ^ { 7 } \mathrm { kg } \cdot \mathrm { m } / \mathrm { s }.$$

A retrorocket is fired to slow down the probe. It applies a force to the probe that has a magnitude of

$$2.0 \times 10 ^ { 6 } \mathrm { N }$$

and a direction opposite to the probe's motion. It fires for a period of 12 s. Determine the momentum of the probe after the retrorocket ceases to fire

A volleyball is spiked so that its incoming velocity of +4. 0 m/s is changed to an outgoing velocity of -21 m/s. The mass of the volleyball is 0.35 kg. What impulse does the player apply to the ball?

Unless indicated otherwise, assume the speed of sound in air to be v = 344 m/s. A loud factory machine produces sound having a displacement amplitude of $1.00 \mu \mathrm{m}$, but the frequency of this sound can be adjusted. In order to prevent ear damage to the workers, the maximum pressure amplitude of the sound waves is limited to 10.0 Pa. Under the conditions of this factory, the bulk modulus of air is $1.42 \times 10^{5} \mathrm{Pa}$. What is the highest frequency sound to which this machine can be adjusted without exceeding the prescribed limit? Is this frequency audible to the workers?

Would you expect the pitch (or frequency) of an organ pipe to increase or decrease with increasing temperature? Explain.

Consider a sound wave in air that has displacement ampli- tude $0.0200$ mm. Calculate the pressure amplitude for frequencies of
($c$) $15,000 \mathrm{~Hz}$. In each case compare the result to the pain threshold, which is $30 \mathrm{~Pa}$.

Consider a sound wave in air that has displacement ampli- tude $0.0200$ mm. Calculate the pressure amplitude for frequencies of
(b) $1500 \mathrm{~Hz}$;

Unless indicated otherwise, assume the speed of sound in air to be v = 344 m/s. Two organ pipes, open at one end but closed at the other, are each 1.14 m long. One is now lengthened by 2.00 cm. Find the beat frequency that they produce when playing together in their fundamentals.

Two guitarists attempt to play the same note of wavelength $6.50 {cm}$ at the same time, but one of the instruments is slightly out of tune and plays a note of wavelength $6.52 {cm}$ instead. What is the frequency of the beats these musicians hear when they play together?

Two loudspeakers, $A$ and $B$, are driven by the same amplifier and emit sinusoidal waves in phase. The frequency of the waves emitted by each speaker is $860 \mathrm{~Hz}$. Point $P$ is $12.0 \mathrm{~m}$ from $A$ and $13.4 \mathrm{~m}$ from $B$. Is the interference at $P$ constructive or destructive? Give the reasoning behind your answer.