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 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 at the time of the experiment is 340 m/s.
A) 4.4 kg
B) 0.76 kg
C) 0.18 kg
D) 2.9 kg

Question

Quizplus · Accepted Answer

We can use the equation for the frequency of a spring-mass system:
$$f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$$
where f is the frequency, k is the spring constant, and m is the mass of the object attached to the spring.
We know the spring constant, so we can rearrange the equation to solve for the mass:
$$m = \frac{k}{(2\pi f)^2}$$
Using the range of frequencies heard by the observer, we can find the maximum and minimum masses of the tuning fork and block together:
$$m_{max} = \frac{390}{(2\pi 174)^2} \approx 2.86 	ext{ kg}$$
$$m_{min} = \frac{390}{(2\pi 178)^2} \approx 2.64 	ext{ kg}$$
The actual mass must lie between these two values. We know that the amplitude of the motion is 17 cm, or 0.17 m. We can use the equation for the maximum kinetic energy of a simple harmonic oscillator to find the maximum velocity of the block:
$$K_{max} = \frac{1}{2}m\omega^2A^2$$
where K is the kinetic energy, $\omega$ is the angular frequency (equal to $2\pi f$), and A is the amplitude.
Solving for the maximum velocity:
$$v_{max} = \sqrt{\frac{2K_{max}}{m}} = A\omega = 2.87 	ext{ m/s}$$
Now we can use the Doppler effect equation to relate the observed frequency to the actual frequency of the tuning fork:
$$f_{obs} = f\left(\frac{v_{sound} \pm v_{obs}}{v_{sound} \mp v_{source}}ight)$$
where $v_{sound}$ is the speed of sound, $v_{obs}$ is the velocity of the observer relative to the source (in this case, equal to the maximum velocity of the block), and $v_{source}$ is the velocity of the source relative to the observer (in this case, zero).
Substituting in the given values and solving for the actual frequency f:
$$f = \frac{f_{obs} v_{sound}}{v_{sound} \pm v_{obs}} = \frac{(174+178)/2 \cdot 340}{340 + 2.87} \approx 174.4 	ext{ Hz}$$
Now we can use the equation we derived earlier to find the mass of the tuning fork and block:
$$m = \frac{390}{(2\pi f)^2} \approx 0.76 	ext{ kg}$$
Therefore, the answer is B) 0.76 kg.

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