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# 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.