IB Physics Sub-topic A4 Notes

Some objects, such as balls and planets, exhibit rotational motion around their center of mass when they roll or spin on their axis. Spinning around a central axis means that:

1. Particles move at different points from the central axis. However, all particles cover the same angle (θ), meaning particles further from the axis cover a greater distance per rotation.
2. Thus, particles have the same instantaneous angular velocity (ω), but do not have the same instantaneous linear velocity.
3. Particles exhibit centripetal acceleration (ar) towards the axis, called the radial acceleration. Remember that particles have a velocity perpendicular to the acceleration, called the tangential velocity (v). The formulae relating to centripetal acceleration are:

   $a_{r} = \frac{v^{2}}{r} = r \omega^{2}$

4. We have always considered objects with uniform circular motion and thus uniform angular velocity. However, if angular velocity changes, particles are said to exhibit angular acceleration (α). This subsequently causes the tangential velocity to change, called tangential acceleration (at). The formula for this is:

   $a_{t} = r \alpha$

5. The total acceleration of the particles is thus the vector sum of the centripetal and tangential acceleration. The formula for this is:

   $a = r\sqrt{\omega^{4} + \alpha^{2}}$