A 1024 Hz tuning fork is dangled at the end of a string such that its center of mass is 1.2 m below the point at which the other end of the string is attached to door frame, making a pendulum of sorts. The tuning fork may be treated as a point mass for the purposes of this problem. If the pendulum is brought to an amplitude of 22o and released, what is the range of frequencies heard by an observer standing in the plane of the pendulum's motion? The speed of sound is 340 m/s.
Question
A 1024 Hz tuning fork is dangled at the end of a string such that its center of mass is 1.2 m below the point at which the other end of the string is attached to door frame, making a pendulum of sorts. The tuning fork may be treated as a point mass for the purposes of this problem. If the pendulum is brought to an amplitude of 22o and released, what is the range of frequencies heard by an observer standing in the plane of the pendulum's motion? The speed of sound is 340 m/s.
Solution
This problem involves the concept of the Doppler effect, which is the change in frequency or wavelength of a wave in relation to an observer who is moving relative to the wave source.
Here are the steps to solve this problem:
Step 1: Calculate the maximum speed of the pendulum.
The maximum speed of the pendulum can be calculated using the formula for the speed of a pendulum at the bottom of its swing:
v = sqrt(2gh)
where g is the acceleration due to gravity (9.8 m/s^2) and h is the height of the pendulum. In this case, the height is the maximum displacement of the pendulum from the equilibrium position, which can be calculated using the formula:
h = L*(1 - cos(A))
where L is the length of the pendulum (1.2 m) and A is the amplitude in radians. Convert the amplitude from degrees to radians by multiplying by π/180.
Step 2: Calculate the Doppler effect.
The Doppler effect can be calculated using the formula:
f' = f*(v + v0) / (v + vs)
where f' is the observed frequency, f is the source frequency (1024 Hz), v is the speed of sound (340 m/s), v0 is the speed of the observer (0 m/s, since the observer is stationary), and vs is the speed of the source (the speed of the pendulum calculated in step 1).
The maximum observed frequency occurs when the pendulum is moving towards the observer (vs is negative), and the minimum observed frequency occurs when the pendulum is moving away from the observer (vs is positive).
Step 3: Calculate the range of frequencies.
The range of frequencies is the difference between the maximum and minimum observed frequencies.
By following these steps, you should be able to calculate the range of frequencies heard by the observer.
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