Torque

What the exam asks

Expect a beam, a plank, a see-saw or a limb, with one unknown force and a diagram. The commonest items are: find the force at one of two supports, find where a mass must sit to balance, work out the muscle force holding a limb against a load, and decide the effect of moving a pivot or lengthening an arm. The move that solves nearly all of them is choosing the pivot to kill the unknown you were not asked for. If a question gives you two support forces and asks for one, pivot at the other and it vanishes from the arithmetic. The trap that catches most candidates is measuring d along the beam when the force is not perpendicular to it: d is the perpendicular distance to the line of action, so an angled force needs the sin θ. The second trap is including the reaction at your chosen pivot in the torque equation, where it contributes exactly nothing.

Torque is the turning effect of a force, and it is one of the few pieces of Section III physics where a formula is worth carrying in: torque = Fd. The whole topic lives in what d means. It is the perpendicular distance from the pivot to the force's line of action, not the distance to the point where the force is applied, and those two agree only when the force is perpendicular to the arm.

A body in rotational equilibrium has zero net torque, and the useful part is that this holds about every point, not just about the real hinge. That is a licence. You get to choose the pivot, and the right choice is the one that deletes the force you were not asked about, because a force whose line of action passes through your chosen pivot has zero torque and never enters the equation. Nearly every beam question is one well chosen pivot away from a single-unknown problem.

Full equilibrium needs two conditions, not one: the forces sum to zero and the torques sum to zero. Questions about a beam on two supports need both. Levers are the same physics wearing a biological hat: the body is full of third-class levers, where the muscle inserts close to the joint and pays for its speed and range of movement with a force several times the load.

What to hold

• Torque = Fd, where d is the perpendicular distance from the pivot to the line of action of the force.
• A force applied at angle θ to a bar at distance L from the pivot gives a torque of F L sin θ, because only the perpendicular component turns anything.
• A force whose line of action passes through the pivot exerts zero torque, however large it is.
• For a body in equilibrium the net torque is zero about every point, so you may choose the pivot freely and pick the one that is most convenient.
• Choose the pivot at the point of application of an unknown force you do not want, and that force drops out of the equation entirely.
• Full equilibrium is two conditions: the net force is zero and the net torque is zero. A beam on two supports usually needs both.
• A uniform beam's weight acts at its centre of mass, which is its midpoint, so it exerts no torque about that midpoint.
• Rotational equilibrium is a balance of clockwise against anticlockwise torque, so a small force far from the pivot can hold a large force close to it.
• The mechanical advantage of a lever is the effort arm divided by the load arm, so a longer effort arm means less force for the same job.
• In a third-class lever the effort acts between the fulcrum and the load, so the effort arm is the shorter one and the effort force exceeds the load. Most skeletal muscles work this way.
• A third-class lever's poor mechanical advantage buys something real: a small muscle contraction moves the load a long way, and quickly.
• Torque is zero when a force is parallel to the arm, and maximal when it is perpendicular, which is why sin θ and not cos θ appears in the general expression.