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The tip of each prong of a tuning fork emitting a note of 320Hz vibrates in SHM with an amplitude of 0.50mm. What is the speed of each tip when its displacement is zero?

As with any question that lists numbers, the first thing to do is to note down the known variables:

f = 320 Hz

A = 0.50 mm

x = 0 mm

v = ?

This requires use of two of the simple harmonic motion (SHM) equations: x = Asinwt and v = Awcoswt. The second equation is simply the derivative of the first equation. Firstly, substitute the known values into the first equation to find t:

x = Asinwt

0 = 0.50 * sinwt

w = 2*pi*f  (definition of angular frequency)

0 = 0.50 * sin(2*pi*320)t

This gives (2*pi*320)t = k*pi, where k is a whole number. This is because sine of any multiple of pi will give zero, which can be seen from the graph of sine. 

This gives t = k / 2*320 = k / 640.

Substitute this value of t into the equation for v to find the speed:

v = Awcoswt

   = 0.50 * (2*pi*320) * cos((2*pi*320)*(k/640))

   = pi * 320 * cos(k*pi)

The cosine of any multiple of pi is either +1 or -1. Since we only need the speed of the tip, whether it is + or - doesn't matter. Therefore:

v = pi * 320 * 1

   = 320pi mm/s

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