# Equilibrium and angular acceleration

## Summary

A body is in rotational equilibrium if the sum of the anticlockwise torques is equal to the sum of the clockwise torques about a pivot:

$$\sum\Gamma_\textrm{anticlockwise}= \sum\Gamma_\textrm{clockwise}$$

A body is in translational equilibrium if no resultant force acts:

$$\sum F_\textrm{upward}=\sum F_\textrm{downward}$$

$$\sum F_\textrm{left}=\sum F_\textrm{right}$$

## Get the basics

#### Rotational equilibrium

We can use these rules in combination to solve problems about objects on a pivot.

In this example, body is in rotational and translational equilibrium if the following apply:

• $$Aa=Bb$$ (using torques)
• $$P=A+B$$ (considering vertical forces)

Note that the beam has no mass. If the beam has mass, it the force $$mg$$ acts vertically downwards in the centre.

Here's an example of a rod in equilibrium.

Placed on two supports, our rod becomes a bridge.

A ladder presents a more complex problem.

## Stretch for 7

#### Angular acceleration

Like torque for force and moment of inertia for mass, there is also a rotational equivalent for acceleration, angular acceleration.

Angular acceleration is defined as the rate of change of angular velocity:

$$a={\Delta \omega\over \Delta t}={{\omega _f-\omega_i}\over t}$$

• $$\alpha$$ is angular acceleration (rads-2)
• $$\omega_f$$ is final angular velocity (rads-1)
• $$\omega_i$$ is initial angular velocity (rads-1)
• $$t$$ is the time taken (s)

We can convert between tangential and angular acceleration:

$$a=r\alpha$$

• $$a$$ is tangential acceleration (ms-2)
• $$r$$ is the radius of the circle (m)
• $$\alpha$$ is angular acceleration (rads-2)

Recall from Oscillations and Circular motion, there are two methods for finding angular velocity:

1. Angular velocity is the instantaneous rate of change of angular displacement (rad): $$\omega={\Delta \theta \over \Delta t}$$
2. For repeated circular motion, angular velocity is the total angle moved through for a complete circle ($$2\pi$$ rad) divided by the time period (s): $$\omega = {2\pi \over T}=2\pi f$$, where $$f$$ is the frequency (Hz).

Note the three forms of acceleration:

• Angular acceleration is the rate of change of angular velocity, or the rate of change in the angle subtended.
• Angular acceleration is is proportional to the tangential acceleration, the rate of change of translational velocity. This requires a resultant force acting along a tangent to the circle.
• These are not related to centripetal acceleration. Centripetal acceleration is towards the centre of the circle and thus requires a resultant force along a radius to the circle, perpendicular to the tangent.

## Test yourself

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