A piano tuner uses a tuning fork. If middle C has a frequency of 264 vibrations per second, write an equation in the form for the simple harmonic motion.
In a simple harmonic motion, the equation usually takes the form:
x(t) = A * sin(ωt + φ)
Where:
- x(t) is the displacement at time t.
- A is the amplitude of the motion.
- ω is the angular frequency of the motion.
- φ is the phase angle (representing the initial phase of the motion).
To relate this equation to the given problem, let's consider the tuning fork's frequency as the angular frequency (ω) of the simple harmonic motion.
Given that the frequency of middle C is 264 vibrations per second, we can convert it to angular frequency by multiplying it with 2π. This is because in one vibration (or oscillation), the object completes a full cycle which equals 2π radians.
So, ω = 264 * 2π = 528π radians per second.
Next, let's choose an appropriate phase angle (φ) for this problem, which indicates where the object begins its motion.
Now we can write the equation for the simple harmonic motion related to the middle C frequency:
x(t) = A * sin(528πt + φ)
It's worth mentioning that the tuning fork's frequency is typically used as a reference to adjust the pitch of other notes on the piano. The tuning fork is struck and its sound is compared to the sound produced by the corresponding key on the piano. The piano tuner then adjusts the tension in the piano strings to match the reference frequency of the tuning fork.