A block holding a ringing tuning fork oscillates back and forth,attached to a wall by a spring of spring 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 spring constant for the spring ? The speed of sound at the time of the experiment is 340 m/s.
A) 390 N/m
B) 203 N/m
C) 1590 N/m
D) 290 N/m
E) 100 N/m

Question

Quizplus · Accepted Answer

The correct answer is A.The Doppler effect causes the frequency of the sound to vary as the block oscillates. The maximum frequency is heard when the block is moving towards the observer, and the minimum frequency is heard when the block is moving away from the observer. The difference between the maximum and minimum frequencies is given by:$$\Delta f = 2f_0 \frac{v}{v_s}$$where:* $\Delta f$ is the difference between the maximum and minimum frequencies* $f_0$ is the original frequency of the tuning fork* $v$ is the speed of the block* $v_s$ is the speed of soundWe can use this equation to find the speed of the block:$$v = \frac{\Delta f v_s}{2f_0}$$Once we know the speed of the block, we can use the equation for the period of a mass-spring system to find the spring constant:$$T = 2\pi \sqrt{\frac{m}{k}}$$where:* $T$ is the period of the oscillation* $m$ is the mass of the block and tuning fork* $k$ is the spring constantWe can use the period to find the spring constant:$$k = \frac{4\pi^2 m}{T^2}$$Plugging in the given values, we get:$$k = \frac{4\pi^2 (0.75 	ext{ kg})}{(0.5 	ext{ s})^2} = 390 	ext{ N/m}$$

A Block Holding a Ringing Tuning Fork Oscillates Back and Forth,attached