A block holding a ringing tuning fork is attached to a wall by a spring. The block oscillates back and forth on a horizontal, frictionless surface, with amplitude 17 cm. The mass of the tuning fork and block together is 0.75 kg. An observer who is situated so that the block's direction of motion is either directly towards or away from him notes that the frequency he hears varies between 174 and 178 Hz. What is the period of motion of the block? The speed of sound is 340 m/s.
A) Cannot do without knowing the spring constant.
B) 0.28 s
C) 0.14 s
D) 0.044 s
E) 0.022 s

Question

Quizplus · Accepted Answer

The variation in frequency observed due to the Doppler effect allows us to calculate the velocity of the block and subsequently its period of oscillation. The frequency variation is given as 174 to 178 Hz, indicating a Doppler shift due to the motion of the block towards and away from the observer. The speed of sound is 340 m/s. Using the formula for the Doppler effect, $f' = f \left(\frac{v + v_o}{v}\right)$ for the source moving towards the observer and $f' = f \left(\frac{v}{v + v_s}\right)$ for the source moving away, where $v$ is the speed of sound, $v_o$ is the observer's velocity (zero in this case), and $v_s$ is the source's velocity. Since the observer is stationary, we can solve for $v_s$ using the given frequencies as the shifted frequencies and solve for the velocity of the block. Knowing the velocity and that the motion is harmonic, we can then find the period of the block's oscillation without needing the spring constant. The correct calculation involves using the average frequency to find the maximum velocity of the block at the extremes of its motion and then using this to find the period of the oscillation, which is independent of the spring constant for a simple harmonic oscillator.

A Block Holding a Ringing Tuning Fork Is Attached to a Wall