Rotational Equilibrium


Equilibrium

A rigid body is considered to be in equilibrium, if there is:

A body in equilibrium experiences NO acceleration and will remain in equilibrium indefinitely, UNLESS it is disturbed by an external force. (Newton’s First Law)

Translational Equilibrium

Translational equilibrium is obtained when the resultant force or vector sum of all forces acting upon the rigid body is zero. In simpler terms, this means that:

  • forces pointing to left = forces pointing to right; AND
  • forces pointing up = forces pointing down

Rotational Equilibrium (Principle Of Moments)

Rotational equilibrium is obtained when the algebraic sum of the torques is zero.

However, it can also be interpreted as:

The principle of moments states that for a body to be in rotational equilibrium, the sum of clockwise torques about any point (which acts as a pivot) must equal to the sum of anti-clockwise torques about the same point.

Sum of clockwise torques = sum of anti-clockwise torques

equilibrium of force

Recap: Torque = moment of force


Summary

In summary, the conditions for object to be in equilibrium:

  1. The sum of moments about any point is zero. (Rotational equilibrium – Principle of moments)
  2. The vector sum of forces on object is zero. (Translational equilibrium)

Self-Test Questions

equilibrium of force

Consider the diagram above, whereby a plank is supported by a pivot. Two forces, 120 N and F are acting on the plank. Given that the plank is perfectly balanced (has rotational equilibrium), find F.

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Recall that moment of force is calculated by:

$$r = F \times d$$

Anti-clockwise moment:

$$\begin{aligned} r_{\text{anti-clockwise}} &= F \times d \\ &= 120 \times 6 \\ &= 720 \text{ Nm} \end{aligned}$$

Clockwise moment:

$$\begin{aligned} r_{\text{clockwise}} &= F \times d \\ &= F \times 8 \\ &= 8F \end{aligned}$$

Since there is rotational equilibrium, we know that $r_{\text{anti-clockwise}} = r_{\text{clockwise}}$ from the principle of moments.

$$\begin{aligned} r_{\text{clockwise}} &= 8F \\ r_{\text{anti-clockwise}} &= 8F \\ 720 &= 8F \\ F &= 90 \text{ N}\end{aligned}$$


State the Principle of Moments and discuss how this principle may be applied to two persons of different weights trying to balance on a see-saw.

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The principle of moments states that for a body to be in rotational equilibrium, the sum of clockwise torques about any point (which acts as a pivot) must equal to the sum of anti-clockwise torques about the same point.

For the see-saw to be balanced, the see-saw must be in rotational equilibrium – which mandates that the clockwise moment about the pivot be equal to the anti-clockwise moment about the same pivot.

Since the two persons have different weights, varying the perpendicular distances to the pivot is the only way for the clockwise and anti-clockwise moment to be equal.

Consider that the two persons are labelled as 1 and 2, with:

  • their weights labelled as $W_{1}$ and $W_{2}$, whereby $W_{1} > W_{2}$.
  • their distance away from the pivot of the see-saw labelled as $d_{1}$ and $d_{2}$

Recall that moment of force is calculated by:

$$r = F \times d$$

For the see-saw to be balanced:

$$\begin{aligned} r_{\text{clockwise}} &= r_{\text{anti-clockwise}} \\ W_{1} \times d_{1} &= W_{2} \times d_{2} \end{aligned}$$

Since $W_{1} > W_{2}$, $d_{1}$ must be smaller than $d_{2}$ for the see-saw to be balanced. (I.e. The heavier person will have to sit closer to the pivot.)


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