For the simple harmonic motion described by the trigonometric function, find the maximum displacement, the frequency, the value of d when t = 4 and the least positive value of t for which d = 0.

Question

For the simple harmonic motion described by the trigonometric function, find the maximum displacement, the frequency, the value of d when t = 4 and the least positive value of t for which d = 0.

d=1/2cos20πt

(a) Find the maximum displacement.

(b) Find the frequency.

_______cycles per unit of time

(c) Find the value of d when t = 4.

d=_____



(d) Find the least positive value of t for which d = 0.

t=______

Answer 1

The general format of a cosine function is d = a cos(bt), where

a is the maximum displacement (aka amplitude). In this problem a=1/2, so the answer is 1/2.

b is used to get the frequency, where frequency = b/2π. In this problem, b = 20π, so the answer is 20π/2π = 10 cycles per unit of time.

To get the value of d at t=4, just plug it in. 1/2 cos (20π x 4) = 1/2. (A cosine wave goes up and down. At t=4, it's at one of the ups.)

To get the least positive value of t when d=0, think of what a cosine wave looks like. It starts at cos(0)=1 and then goes down to 0 at cos(π/2)=0. So in the problem, we need to find the t so that 20π t = π/2. The answer is t = 1/40.