# Velocity w/axis vs without

1. ### clm222

38
Hi, quick question
I've just started to learn circular motion and i tried a basic force problem.

Imagine there is a ball on a string with a string length 'r', and a ball mass 'm'.
You apply a force 'F' for one second. Can the arc length of the ball's motion be calculated the same as linear motion? $d=\frac{F{t^2}}{2m}$ (replacing acceleration with F/m, vi=0)
Can I also go as far as the find the angular velocity by dividing by the radius?
$$ω=\frac{F{t^2}}{2mr}$$

Thanks.

### Staff: Mentor

Where, and in which direction, do you apply the force?
If the force is always in direction of the circular motion (as forced by the string), that works. However, angular velocity after the acceleration is not the total distance divided by the radius. That is the angle, not its velocity.

3. ### clm222

38
oh ok,
so if i were to have the force constantly be applied perpendicular to the ball than i CAN use the distance formula?

if so, lets say the ball has initial velocity 'vi', and a force is applied, the final velocity can be calculated using ${v_f}={v_i}+\frac{Ft}{m}$, could i also calculate the distance travled: $d={v_i}t+\frac{F{t^2}}{2m}$ ?

4. ### HallsofIvy

40,678
Staff Emeritus
If the force is applied perpendicular to the motion of the ball then its speed is constant. The distance around the circle in time t is $v_it$.