Equilibrium of a rigid body

Two conditions for equilibrium

Narrated video. Transcript:

A book on a table, a ladder against a wall, a bridge over a river. These rigid bodies stay put. They are in mechanical equilibrium.

Two things must be true. First, the total external force is zero, so the centre of mass does not accelerate. This is translational equilibrium.

Second, the total external torque is zero, so the body does not start to turn. This is rotational equilibrium.

Take a light rod and push up on both ends with equal forces F. The torques about the centre cancel, so it does not turn. But the net force is 2F, so it moves up.

Now push one end up and the other end down. The net force is zero, so the centre stays put. But both forces turn the rod the same way, so it spins.

A pair of equal and opposite forces with different lines of action is called a couple. A couple causes rotation without translation.

You apply a couple when you open a bottle cap or turn a steering wheel with both hands. A compass needle in the Earth’s magnetic field feels a couple too.

Principle of moments and centre of gravity

Narrated video. Transcript:

A see-saw is a lever: a rigid bar that turns about a fixed point called the fulcrum.

Put a load F₁ on one side and push down with an effort F₂ on the other. Let d₁ be the load arm, the distance of the load from the fulcrum, and d₂ the effort arm.

For balance, the torques must cancel: F₁d₁ = F₂d₂. Load times load arm equals effort times effort arm. This is the principle of moments.

The ratio F₁/F₂ is the mechanical advantage. It equals d₂/d₁, so a long effort arm lets a small effort lift a big load.

The fulcrum pushes up too. For translational equilibrium, this reaction R equals F₁ + F₂, if we ignore the weight of the bar.

Cut out an odd-shaped piece of cardboard. There is one point where it balances on the tip of a pencil. That point is its centre of gravity.

At the centre of gravity, the total torque of gravity on all the tiny pieces of the body is zero, so it balances there.

When g is the same over the whole body, as it is for everyday objects, the centre of gravity is at the same place as the centre of mass.

Try it: balance the see-saw

Interactive simulation. Things to try:

• Drag the loads along the beam, or use the sliders for the selected load.
• The beam balances when clockwise moments equal anticlockwise moments.
• Add a third load and try to balance all three.
• Watch the pivot force R: it always equals the total weight.

Work it out

A 30 kg child sits 1.6 m to the left of a see-saw’s pivot. Where must a 40 kg child sit to balance it?

1. 1.6 m to the right
2. 1.2 m to the right
3. 2.1 m to the right
4. 1.2 m to the left

Metre-stick problem

A metre stick balances on a knife edge at its 50 cm mark. Two 5 g coins are stacked at the 12 cm mark, and the stick now balances at the 45 cm mark. What is the mass of the stick?

1. 33 g
2. 50 g
3. 66 g
4. 99 g

Summary

• Translational equilibrium: ΣF = 0
• Rotational equilibrium: Στ = 0
• Principle of moments: F₁d₁ = F₂d₂
• Mechanical advantage: MA = F₁/F₂ = d₂/d₁

• A rigid body is in equilibrium only if both the net force and the net torque are zero.
• A couple (equal, opposite forces along different lines) has zero net force but a non-zero torque.
• On a balanced lever, clockwise moments = anticlockwise moments, and the fulcrum’s reaction balances the total weight.
• The centre of gravity is where the gravitational torque is zero. In uniform g it is the centre of mass.