A little help with Vibrations and Waves?

Question

The motion of an object is described by the equation below.

x = (0.30 m) cos(pi t/7)
Find the frequency of the motion.
Hz
Find the period of the motion.
s

Answer 1

I have to make a lot of assumptions on things you should have provided. For example, I assume t is time, x(t) is the vertical displacement at any time t, and .3 m = A the amplitude on your wave.

L = vt or L/v = t; where v is the velocity of the wave and L is the distance it travels in t time. When any wave travels its length, that's called a cycle. So when L is the wavelength, t is the number of seconds to travel one cycle (one wavelength). t in seconds per cycle is called the period. Thus, 1/t = f is in cycles per second, which is the frequency of the wave.

With all that, once you've found the frequency (f) in cycles per second, just invert that and you'll have the period (t) in seconds per cycle. Or, vice versa, find t, which I show how to do next, and then invert t to get the frequency.

The length of a cyclical sin or cos wave can be graphed from theta = 0 degrees through theta = 360 deg (or the equivalent 0 through 2 pi radians). That is, a wave oscillates through 360 degrees for each wavelength and each 2 pi oscillation is called a cycle.

Theta = wt; where w is the angular velocity and t is the time the wave oscillates at w. In your problem w = pi/7; so from theta = wt; we have t = theta/w = 2 pi//pi/7 = 14 seconds (assuming w was in radians per second...something else you failed to provide). We use theta = 2 pi = 360 degrees because that marks the angular distance of one cycle...one wavelength. And, by definition, a period is the length of time a wave takes to oscillate one cycle of 360 degrees or 2 pi.

As f = 1/t, when t = 14 seconds is the period, f = 1/14 cycles per second, which is the frequency.

Answer 2

It has a frequency of 50 Hz, its period is 1/50 of a second (0.02 sec).