AP Physics 2 Exam Question

A 3-meter long, straight metal rod is clamped at one of its ends and is free to vibrate at its other end. When a tuning fork of frequency 440 Hz is struck, it is placed near the free end of the rod. It is observed that the rod resonates at its fundamental frequency.

a) Calculate the speed of sound in the metal rod.

b) If the surrounding temperature increases, what effect would it have on the speed of sound in the metal rod? Explain your answer.

c) If the mass of the rod is doubled while keeping the length constant, how would it affect the fundamental frequency of vibration of the rod? Justify your answer.

Answer

a) To calculate the speed of sound in the metal rod, we can use the formula:

v = f * λ

where v is the speed of sound, f is the frequency, and λ is the wavelength.

In this case, we are given that the tuning fork resonates at the fundamental frequency, which means that the wavelength is twice the length of the rod:

λ = 2 * length

Substituting the given values:

λ = 2 * 3 m = 6 m

Now we can calculate the speed of sound using the formula:

v = f * λ = 440 Hz * 6 m = 2640 m/s

Therefore, the speed of sound in the metal rod is 2640 m/s.

b) The speed of sound in a solid depends on various factors, including temperature. As the surrounding temperature increases, the speed of sound in the metal rod will also increase. This is because an increase in temperature leads to an increase in the average kinetic energy of the metal atoms/molecules. This increased kinetic energy results in faster vibrations of the metal atoms/molecules, which in turn increases the speed of sound. Therefore, an increase in temperature would cause an increase in the speed of sound in the metal rod.

c) The fundamental frequency of a vibrating rod is inversely proportional to its length and proportional to the square root of the tension applied and the inverse square root of the mass. Therefore, if the mass of the rod is doubled while keeping the length constant, the fundamental frequency of vibration of the rod would halve (reduce to half). This can be justified by the following reasoning:

The fundamental frequency, f, is given by the equation:

f = (1/2L) * sqrt(T/μ)

where L is the length of the rod, T is the tension applied, and μ is the mass per unit length of the rod.

Since we are doubling the mass of the rod while keeping the length constant, μ will also double. Plugging this in the equation:

f = (1/2L) * sqrt(T/(2μ))

Now, let's compare this with the original equation:

f' = (1/2L) * sqrt(T/μ)

Comparing these two equations, we can see that f' is half of f. Therefore, the fundamental frequency of the rod will reduce to half if the mass is doubled while keeping the length constant.