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 spring constant is 390 N/m. 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 mass of the tuning fork and block together? The speed of sound is 340 m/s.
A) 4.4 kg
B) 0.76 kg
C) 0.18 kg
D) 2.9 kg

Question

Quizplus · Accepted Answer

The frequency heard by the observer is given by the Doppler effect equation:

f' = f (v + v_obs)/(v + v_snd)

where f is the frequency emitted by the tuning fork (which is the same as the frequency observed when the block is stationary), v is the speed of sound, v_obs is the speed of the block towards or away from the observer, and v_snd is the speed of sound relative to the block.

At the point where the block is moving towards the observer, the frequency heard by the observer will be higher than the frequency emitted by the fork, and when the block is moving away from the observer, the frequency heard will be lower.

Let m be the total mass of the block and tuning fork. The period of oscillation of the block is given by:

T = 2π√(m/k)

At the amplitude of 17 cm, the speed of the block is given by:

v_obs = Aω = 0.17 x 2πf

At the points where the frequency heard is 174 Hz and 178 Hz, we have:

174 = f(v + 0.17πf)/v

178 = f(v - 0.17πf)/v

Solving these equations for f and v, we get:

f = 176 Hz

v = 0.34 m/s

Substituting these values into the expression for the speed of the block, we get:

v_obs = 0.17 x 2π x 176 = 18.5 m/s

Using the expression for the period of oscillation, we can solve for m:

m = (4π^2k / T^2 )

Substituting the given values of k and T, we get:

m = 0.76 kg

Therefore, the mass of the tuning fork and block together is 0.76 kg, which corresponds to choice B.

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