6.7.1. Moment of force (Torque)

We have learnt that the motion of a rigid body, in general, is a combination of rotation and translation. If the body is fixed at a point or along a line, it has only rotational motion. We know that force is needed to change the translational state of a body, i.e. to produce linear acceleration. We may then ask, what is the analogue of force in the case of rotational motion?

In rotational motion, just as a force is needed to change the motion of an object moving in a straight line, a torque (or moment of force) is necessary to cause a change in the rotational motion of a body. Torque considers not just the size of the force applied but also where and how it is applied in relation to the pivot point. This concept is analogous to the linear concept of force, which directly causes linear acceleration.

Think of opening a door: If you push directly at the hinges (the pivot point), the door won't move. However, if you push at the outer edge of the door, it swings open easily. This demonstrates how the point of application of a force relative to an axis of rotation determines the effectiveness of the force in producing rotational motion.

To look into the question in a concrete situation let us take the example of opening or closing of a door. A door is a rigid body which can rotate about a fixed vertical axis passing through the hinges. What makes the door rotate? It is clear that unless a force is applied the door does not rotate. But any force does not do the job. A force applied to the hinge line cannot produce any rotation at all, whereas a force of given magnitude applied at right angles to the door at its outer edge is most effective in producing rotation.

The torque τ (tau) on the door when a force F acts at a distance r from the axis of rotation can be expressed as τ = r × F. This means the torque depends on both the force applied and the perpendicular distance from the axis of rotation to the point where the force is applied. Torque is a vector quantity, meaning it has both a magnitude and a direction, which is determined by the direction of the applied force and its distance from the pivot.

Consider using a wrench to tighten a bolt. The longer the wrench (greater r), the more torque you apply for the same force. If you use a short wrench, it’s much harder to tighten the bolt because you don’t have enough leverage. This helps visualize how torque works—longer distances or angles of application result in more effective rotational force.

The moment of force has dimensions M L² T -². Its dimensions are the same as those of work or energy. The SI unit of moment of force is newton meter (N m). The magnitude of the moment of force may be written τ = r F sinθ, where θ is the angle between the position vector r and the force vector F.

When calculating the magnitude of torque, you consider not only the force exerted but also the distance from the pivot (the line of action of the force) and the angle at which the force is applied. For maximum torque, the force should be applied perpendicular to the lever arm, meaning θ = 90°, which gives sinθ = 1. If the force is applied parallel to the lever arm, sinθ = 0, and therefore no torque is produced.

If you’re trying to unscrew a tight lid, pushing down straight doesn’t help—there’s no torque if you apply force directly along the line connecting your hand and the center of the lid. But, if you apply a force at an angle, or at the edge of the lid while using a bigger lever like a spoon, you effectively increase the torque.

The moment of a force is a vector quantity. The symbol τ stands for the Greek letter tau. Just as force leads to linear acceleration (F = ma), torque leads to angular acceleration. Therefore, for rotational motion about a fixed axis, we can relate torque τ with angular acceleration α through the moment of inertia I of the body, represented as τ = Iα.

This relationship shows us how much torque is needed to produce a certain angular acceleration in a rigid body. The moment of inertia, which depends on the mass and its distribution with respect to the axis of rotation, plays a crucial role in determining how easily an object can be rotated. A higher moment of inertia means more torque is required to achieve the same angular acceleration.

Think of two different objects: a flywheel and a solid sphere. The flywheel, which has most of its mass distributed away from the center, requires more torque to change its rotational speed compared to a solid sphere with mass evenly distributed. This is why flywheels are used in machinery to maintain steady speeds.

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