Rotational dynamics

Equations of rotational motion

Using a similar process of rearranging and substitution to the suvat equations of linear motion, we can derive equations for rotational motion.

Translational Rotational
\(v=u+at\) \(\omega_f=\omega_i+\alpha t\)
\(s={(u+v)\over 2} t\) \(\theta = {(\omega_i+\omega_f)\over 2} t\)
\(s=ut+{1\over 2}at^2\) \(\theta=\omega_i t+{1\over 2}\alpha t^2\)
\(v^2=u^2+2as\) \({\omega_f}^2={\omega_i}^2+2\alpha \theta\)
  • \(\theta\) is angular displacement (rad)
  • \(\omega_i\) is initial angular velocity (rads-1)
  • \(\omega_f\) is final angular velocity (rads-1)
  • \(\alpha\) is angular acceleration (rads-2)
  • \(t\) is the time taken (s)

These equations apply if angular acceleration is constant.


Rotational motion graphs

We can also display these quantities on rotational motion graphs.

In this example:

  • the blue line with increasing gradient represents the angular displacement the gradient of this graph is the angular velocity, which is shown as the green line increasing linearly with time
  • the angular accleration is constant with zero gradient (red line)

Newton's second law

An object experiences angular acceleration if it is not in rotational equilibrium.

By substition into Newton's second law, \(F=ma\):

\(\Gamma=I\alpha\)

  • \(\Gamma\) is resultant torque (Nm)
  • \(I\) is moment of inertia (kgm2)
  • \(\alpha\) is angular acceleration (rads-2)

It is important to consider how you might increase angular acceleration. In simple terms, \(\alpha = {\Gamma \over I}\), so angular acceleration increases with torque and decreases with moment of inertia.

A closer examination of the situation below reveals that \(\alpha={Mgr\over Mr^2}={g\over r}\), so only the perpendicular distance has an effect.

A graph of torque vs time would be proportional in shape to the equivalent angular acceleration graph.