Torque Equation

According to Newton's second law of motion, a body of mass 'm' subject to a net force 'F' undergoes an acceleration 'a' that has the same direction as the force and a magnitude that is directly proportional to the force and inversely proportional to the mass. Thus, F = ma. This force results in linear acceleration in linear motion and circular acceleration denoted by the tangential or radial components in the circular motion. Here, the force which tends to change the rotational or circular motion of the object is called torque. It is primarily an influence of the force to rotate the object about its axis, called the pivot point or fulcrum. Let us understand this concept in detail.

What is Torque?

Torque is more than just a force which is responsible for an object to rotate, it is more of a twist, making the object rotate known as 'Moment of a force'. Let us take the simplest example to understand this concept. When you push a door applying force to the edge of the door, it is simpler to push. However, if you try pushing the same door somewhere in between the edge and the axis or closer to the pivot, it becomes that much more difficult. It is because of torque. Torque depends not just on the force but also on the distance of the application of force from the pivot point. This distance is called the moment arm and is denoted by 'r'. I am assuming here that you are aware of the significance of the scalar and vector quantities in physics. Both these quantities which make up the torque definition are vectors with 'r' being directed from the pivot towards the point where the force is applied and force 'F' being directed towards the rotation of the object (or in our example, the door). With this information in hand, let us understand the torque equation.

Torque Equation

Torque is the cross product or the vector product of the distance vector 'r' and the force vector 'F' with 'Θ' being the angle between r and F. Torque is denoted by the Greek letter 'τ' called tau. Therefore,

τ = r × F = rFsin(Θ)

The direction of the torque vector can be determined using the right hand rule. When we put our fingers in the direction of r, and curl them to the direction of F, then the thumb points in the direction of the torque vector. From this equation, we can identify that if you apply a force at the pivot of the door or in the direction of the distance vector, the torque produced will be zero and hence the door will be unmoved. The SI unit of torque is Newton-meter, which is the same as for energy (though it is called joule). We distinguish between energy and torque by understanding that energy is a scalar quantity and torque is a vector quantity.

Also, you will understand the significance of the component 'sin(Θ)' from the fact that, if we break up the torque vector in its tangential and radial components, we can see that the radial component is parallel to the moment arm 'r' and hence, it cancels out (sin 0 = 0). And it is the tangential component, which is perpendicular to the moment arm (sin 90 = 1), which actually makes the object move.

There may be multiple forces acting on the object at various points of the object. Each force will cause an individual torque. Thus, the net torque will be the sum of the individual torques. However, in rotational equilibrium, as we know, the sum of all forces is equal to zero. There is no net torque acting on the object. So the sum of all torques is also equal to zero. Hence, Σ(τ) = 0.

Examples of Torque Equation

We can see the application of torque concept in our daily life when we tighten a nut using a wrench. In this example, torque is the vector product of linear force applied at the end of the wrench, multiplied by length of wrench from point of contact with nut to wrench end. Also, in a vehicle, the engine crankshaft revolves in a circular fashion due to the torque generated by displacement of engine cylinders. This torque results in the linear motion of the car. Many a time, we confuse torque with horsepower, which is actually the work done by the engine per unit time. And the torque is related with the moment of inertia 'I', according to the equation τ = Ia, where 'a' denotes the angular acceleration of the object.

I hope this article has helped you grasp the concept of torque. Remember this concept every time you have to open a door. Whenever you feel it is easier to open the door, it is the torque responsible for it, with you applying the force farthest from the pivot axis.