IB Physics Sub-topic A4 Notes

Translational and rotational motion

In the SL syllabus, the motion of objects is only considered as translational, as movement from one place to another. This applies to the center of mass of an object and describes its motion as one unit, such as a box being pushed on a floor as one unit. Moving as one unit in one direction means that -particles move together and have the same instantaneous velocity.

However, some objects, such as balls and planets, also 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:

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

Comparing the types of motion

Whilst it is important to understand how rotational motion functions, you may be asked to compare and contrast translational and rotational motion. Thus, let’s summarize the key points we have learned so far:

Using these new perspectives of rotational motion, the SUVAT equations that apply to translational motion can be changed as well:

$\omega_{f} = \omega_{i} + \alpha t$

$\phi = \omega_{i} t + \frac{1}{2}\alpha t^{2}$

$\omega_{f}^{2} = \omega_{i}^2 + 2\alpha \phi$

$\phi = \frac{(\omega_{f} + \omega_{i})t}{2}$

You will be asked to perform calculations with these rotational SUVAT equations, so make sure you practice questions involving them often!