A painted tooth on a spinning gear has angular acceleration α = (20-t) rad/s2, where t is in s. Its initial angular velocity, at t = 0 s, is 320 rpm.

I think the question will be to find the angular speed at any time t

Since we are given the initial conditions of the angular speed

Explanation:

Given that,

α = (20-t) rad/s2

Give the initial condition

At t=0 w=320rpm

So modeling a differential equation

α=dw/dt

Therefore

dw/dt = (20-t)

Applying variable separation

dw = (20-t) dt

Integrate Both sides

∫dw=∫ (20-t) dt

w = 20t-t²/2 + C

C is constant of integration

Using initial conditions

At t=0 w=320rpm

320=0-0+C

C=320rpm

Therefore the equation becomes

W=20t - t²/2 + 320

This is the model angular speed at any time t