TORQUE & ROTATIONAL EQUILIBRIUM

Lever and Balance Laboratory

Move a weight. Shift the pivot. Find the balance.

Balance the moments

Setup

Forces Perpendicular arms↺ positive · ↻ negative

↺ Counterclockwise

↻ Clockwise

Speed

NET TORQUE

ANGLE θ

ANGULAR VELOCITY ω

ANGULAR ACCELERATION α

Drag a labelled weight or the pivot; edits return to horizontal. Keyboard: focus a handle, then ← / → to move by 0.01 m (Shift: 0.1 m), Home / End for limits. You can also enter positions in Setup.

Straight arrows show force in N on one shared scale. Curved arrows show torque direction. Weight labels are offset for readability; the small dots on the beam are the attachment points. Step = 0.05 s.

Watch the release

Last 30 s · time (s) →

Angle θ (°)

Net gravitational torque τ (N·m)

Account for every object

r = x − pivot position; right is positive. Positive torque is counterclockwise.

Gravitational torque about the pivot
Object	Mass (kg)	Weight (N)	r (m)	Signed τ (N·m)	Rotation

TRY A CHALLENGE

Four against one.

A 4 kg weight is fixed 0.5 m left of the pivot. Position a 1 kg weight on the right to balance it.

The uniform beam is centered. Only W2’s position can change. Release and keep it approximately horizontal for 2 s.

Find the position, then test your prediction.

Why?

Force × perpendicular arm

Torque is a force’s turning effect: |τ| = Fd⊥. The perpendicular moment arm is the shortest distance from the pivot to the force’s line of action. For vertical gravity, d⊥ = |r cos θ|. On a horizontal beam it equals |r|.

Can a smaller weight balance a larger one?

Yes. Put the smaller weight farther from the pivot. Its smaller force can produce the same torque through a longer arm. Equal masses need equal opposite distances when the beam is centered on its pivot.

What happens at the pivot?

A weight exactly at the pivot has r = 0, so its gravitational torque is zero even though its weight is not. It increases the total supported weight.

Why does moving the pivot matter?

Moving the pivot changes every signed arm, including the beam’s own arm. A uniform beam’s weight acts at its center, so it contributes torque whenever the pivot is off-center. Balance occurs when the combined center of mass lies at the pivot.

Does gravity change the balance?

Changing gravity multiplies all gravitational torques by the same positive factor. It changes the angular acceleration, but preserves every balanced configuration. In m₁gd₁ = m₂gd₂, g cancels, leaving m₁d₁ = m₂d₂.

Why doesn’t it return to horizontal?

If the combined center of mass is at the pivot, the net gravitational torque is zero at every angle. Released from rest, the beam stays where it is. This ideal lever has no spring, friction or other restoring mechanism.

Why does a longer effort arm help?

For a centered beam and opposite effort and load, Fₑdₑ = Fₗdₗ. A longer effort arm reduces the force needed to balance the load. Ideal mechanical advantage is Fₗ/Fₑ = dₑ/dₗ.

Both ends turn through the same angle, but move through arc distances s = d|Δθ|. The longer effort arm moves farther: sₑ/sₗ = dₑ/dₗ. Ideally Fₑsₑ = Fₗsₗ for tangential effort and load; a lever trades force against distance, without creating energy. For vertical forces, the vertical movements share the same arm ratio.