### Torque in physics – definition & formula derivation

##### December 30, 2017

# Torque in physics – definition

**Definition of Torque**: **Torque** in physics is also known as the **Moment** of force. In other words, It is the Rotational analogue of force. As force causes translational motion, the Torque is the cause of rotational motion. If the net torque is zero then a body will not change its rotational state of motion. So first, we will cover **what is torque. **And then will see 2 different ways of** derivation of formula.** We will also cover the** law of conservation angular momentum.
**

1)Torque(T) can be defined as the

**moment of force**.

2) Mathematically, it is also defined as the **rate of change of angular momentum**.

3) Again, T is defined as the **cross product** of the force vector and the distance vector by which the force’s application point is offset relative to the fixed suspension point. [ this is basically expanding the meaning of ‘moment of force’]

The SI unit for torque is the newton metre (**N⋅m**).

Now let’s find out its formula or expression.

## Torque – Derivation of formula (2 ways)

We will derive the equation or formula in 2 different ways. First one as the rate of change of angular momentum and the second one as the moment of Force.

## Torque – Rate of change of Angular Momentum with Time

Rate of change of **Angular Momentum** with respect to time = ΔL/ΔT

Now, ΔL/ΔT = Δ(I ω)/ΔT = I. Δω/ΔT ……. (1) [ Here I is constant when mass and shape of the object are unchanged]

Now Δω/ΔT is the rate of change of angular velocity with time i.e. angular acceleration (α).

So from equation 4 we can write, ΔL/ΔT = I α …………………(2)

Have you noticed this expression **I α** ?

I (moment of inertia) is the rotational equivalent of mass(inertia) of linear motion. Similarly angular acceleration α (alpha) is the rotational motion equivalent of linear acceleration.

As in combined (product) form mass **m** and linear acceleration **a** represents **Force** in linear motion, similarly **I** and **α** in product form represent **Torque (τ)**, which in turn is the **rotational equivalent of Force**.

So from equation 5 we get, ΔL/ΔT = τ ……………………. (6) which states that the rate of change of angular momentum with time is called Torque.

**This represents the relation between angular momentum and torque.**

## Formula of torque – moment of Force

Torque (T) is the moment of force. Τ = **r** X **F** = r F sinθ ……………. (3)

[ F is the force Vector and r is the position vector (a vector from the origin of the defined coordinate system to the point where the force is being applied).

*θ* is the angle between the force vector and the lever arm vector. X denotes cross product. ]

Now expanding this by putting **F = ma** we get: (m = mass of the object, and a = linear acceleration)

**Τ = r F sin θ = r ma sinθ = r m αr sinθ = mr^2. α sinθ = I α sinθ = I X α **……………………… (4)

[α is angular acceleration, I is moment of inertia and X denotes cross product.]

## Conservation of Angular Momentum

T = **I ****α** (from equation 4)

or, T = I (ω2-ω1)/t [ here **α** = angular acceleration = time rate of change of angular velocity = (ω2-ω1)/t where ω2 and ω1 are final and initial angular velocities and t is the time gap]

or, **T t = I (ω2-ω1) ……………………(5)**

when, T = 0 (i.e., net torque is zero), then from the above equation we get,

I (ω2-ω1) = 0

i.e., **I ω2=I ω1 ………….. (6)**

Iω2 represents final angular momentum and Iω1 represents initial angular momentum. When net torque is zero, then both these are same.

So, when net torque is zero on a body, then the net change in angular momentum of the body is zero.

In other words, angular momentum remains constant if net external torque applied is zero. This is known as the Law of **conservation of angular momentum**.

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