Problem 11: Find the amplitude, frequency and time period of an object oscillating at the end of a spring, if the equilibrium for its position at any instant t is given by x=0.25 cos〖π/2 t〗. Find the displacement of the object after 2.0 seconds.
The given equation of the SHM of this particular object is that at any instant of time, t, and distance x from the mean position is given by
If x0 is the maximum distance from the mean position (also called amplitude), then the general equation of simple harmonic motion is that the instantaneous distance x from the mean position at any time t is
Compare the given equation of SHM (equation 1) with the general equation of SHM (equation 2),
Amplitude: Amplitude of the oscillating body = 0.25 m
Frequency: We know that
Put the values,
Time period: Since T = 1/f, therefore,
Frequency after 2 seconds:
Now put these values in equation (2) for t = 2 seconds and solve for x.