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.

And how about a cantilever?

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.

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