Dimensional formula of torque

Force is an important aspect to study in physics and its application; something that is quite synonymous to force is torque. When we talk about force, we think of it as a method of pushing or pulling an object in a specific direction. The force applied differs when the object is pinned or unpinned to a particular position. 

If the object is pinned at some position or a pivot, the object rotates when a specific force is applied to it, and this force is what we call torque. Torque is responsible for producing a modification in an object’s angular velocity, which is called an angular acceleration. Here we will be elaborating in detail about the dimensional formula of torque and other important aspects for your better understanding.

What is torque?

In simple words, torque is the measurement of the force responsible for making an object rotate. If the object rotates by an axis, that we can name as the pivot point “O”. The force applied to make it move if named as “F”. The distance between the pivot point and the point where the force works is called the moment arm “r”.

So Torque is T = r x F = rFsin(θ).

In this equation, both r and F are vectors, and the angle between r and F is “θ”.

The direction of the torque vector

By using the right-hand rule, we can find the direction of the torque. If we point our fingers towards the direction of r and move them to form a fist to the direction of F, then our thumb will be pointing in the direction where the torque vector works.

Example.

If we imagine pushing a door to either open or close it. The force we imply to push (F) leads the door to rotate via its hinges (indicated as the pivot point O). The amount of force you require to apply depends on the distance between you and the hinges (indicated by r). The closer you are to the hinges, the harder it will be to push the door, and this happens when you try to push the door in the wrong direction. The torque created by the force you have applied is smaller than what it should have been if you have pushed it in the correct direction.

Dimensional formula of torque.

The dimensional formula of torque is M1 L2 T-2. 

Here, M= mass; L= length and T= time.

Derivation.

Torque (T) = Moment of inertia x Angular acceleration _____ (1)

As moment of inertia = radius of gyration2 x Mass.

So, the dimensional formula of the moment of inertia will be M1 L2 T0 ______ (2)

Angular acceleration = angular x Time-2

The dimensional formula of angular acceleration is = M0 L0 T-2 _______ (3)

By putting (2) and (3) in (1), we will get

Torque = moment of inertia x angular acceleration.

I = [M1 L2 T0] × [M0 L0 T-2] = [M L2 T-2].

Analysing the dimensional formula of torque examples will help you properly understand the torque applications.

The SI unit of torque is Newton-meter, which is also the expression for Joule. But torque is not energy. So to avoid any kind of confusion, the utility of unites N.m is advised. This differentiation is done as energy is a scalar quantity, but torque is a vector quantity.

Conclusion.

When studying force, it is essential to know about torque and the dimensional formula of torque. You can go through an innumerable dimensional formula of torque examples to clear your concepts. To clearly understand the meaning of the dimensional formula of torque, make sure you study the basic concepts explained above.