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Torque Equation

Rohan Bhalerao Apr 28, 2019
Torque is basically a force that tends to cause a rotation. This includes all the information related to it, along with the basic torque equation.
According to Newton's second law of motion, a body of mass 'm', subjected 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 angular acceleration, denoted by the tangential or radial components, in circular motion. Here, the force which tends to change the rotational or circular motion of the object is called torque. It's the influence of the force to rotate the object about its axis, called the pivot point or fulcrum.


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, and it depends not only 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'.
Hope, you are aware of the significance of the scalar and vector quantities in physics. Both these quantities, which make up the 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).


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. It is denoted by the Greek letter 'τ', called tau. Therefore,

τ = r × F = rFsin(Θ)
The direction of the 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 is Newton-meter, which is the same as for energy (though it is called joule).
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 will be the sum of the individual ones. However, in rotational equilibrium, as we know, the sum of all forces is equal to zero. So, the sum is also equal to zero. Hence, Σ(τ) = 0.


We can see the application of this concept in our daily life when we tighten a nut using a wrench. In this example, it is the vector product of linear force applied at the end of the wrench, multiplied by length of wrench, from the 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 transforms into the linear motion of the car.
Many a time, we confuse this term with horsepower, which is actually the work done by the engine per unit time, as against the moment of inertia 'I', according to the equation τ = Ia, where 'a' denotes the angular acceleration of the object.