Four children climb on a carousel that is initially at rest. If the carousel accelerates to 0.4 rad/s within 10 seconds, what is the angular acceleration? If the carousel has a radius of 20 meters, what is the linear velocity of the children?

If the angular acceleration is a constant A, then this constant times the time will be equal to the angular velocity.

We know that after accelerating for 10 seconds, the angular velocity is 0.4 rad/s

Then:

A*10s = 0.4 rad/s

If we solve this for A we get:

A = (0.4 rad/s)/10s = 0.04 rad/s^2

This is the angular acceleartion.

Now we assume that the angular speed remains constant at 0.4 rad/s, and we know that the radius of the carousel is 20m, we want to find the linear velocity of the children.

We know that for an angular velocity W and for a radius R, the linear velocity is:

V = R*W

Then in this case the linear velocity is:

V = 20m*(0.4 s^(-1)) = 8 m/s