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?

The period of this motion is T = 2pi/w

Acceleration = A = asin(wt)

To calculate the velocity, integration over time is done which results in

V = -a ₓ cos(wt)/w

Integration is done again to get the movement which results in

P = -asin(wt)/w²

Note that sin and cos are 2×pi where pi = 3.14159 periodic. We can infer

that sin(0) = sin(2pi)

The period of the function: P = -asin(wt)/w²

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

The period of motion is T = 2pi/w .

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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 .

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