Question

An object moves up and down the y-axis with an acceleration given as a function of time t by the expression a = a sin ω t, where a and ω are constants. what is the period of this motion?

Answer 1

Answer: T = 2*pi/w

Explanation: We know that the acceleration of the object is:

A = a*sin(wt)

Now, if we want to obtain the velocity, we must integrate over time, then we get:

V = -a*cos(wt)/w

We integrate again to get the movement:

P = -a*sin(wt)/w^2

(you can see that the trigonometric equations never vanish)

Now, as you may know that sin and cos are 2*pi (where pi = 3.141592653...) periodic, this means that:

sin(0) = sin(2*pi)

Then the period of the function: P = -a*sin(wt)/w^2

sin( 0 ) is when t = 0, and sin(2*pi) is when t = 2*pi/w

Then the period of motion is T = 2*pi/w .

Answer 2

$w = \frac{2\pi}{t} \\ \\ t = \frac{2\pi}{w}$
So here the time period ; 2pi/omega !!