Question

initial angular velocity is 3 rad/s , final angular velocity is 25 rad/s , and the time of motion is 10s. find distance travelled?

Answer 1

For the first part, remember that the rotational kinetic equations are nearly exactly the same as their linear counterparts, so just like to find average distance with linear problems
\[ d=\frac{v-v_0}{2}t\]
to find the number of radians the object traveled through it's
\[ \theta = \frac{\omega-\omega_0}{2}t\]
Theta will be in radians. To find out how far that is in meters, you need to know the radius of rotation. You first convert theta into revolutions, then into meters using the circumference of the circle it makes during rotation, giving
\[\text{distance} = \left(\theta \right) \left(\frac{1 \ rev}{2\pi \ rad} \right) \left( \frac{2 \pi R}{1 \ rev} \right)= R \ \theta \ \text{meters}\]

I'm not sure what your second question is asking, however.