Torque and Angular Momentum: Meaning, Formulas, and Relationship

Before we go on to define torque and angular momentum in detail, have a look at the differences between the two. 

Feature	Torque (τ)	Angular Momentum (L)
Primary Role	Torque is the rotational equivalent of force without being it. It is the cause of angular acceleration and is caused by external forces.	Angular momentum is the rotational equivalent of linear momentum. It is a measure of an object's rotational motion.
Definition	Torque is the tendency of a force to cause or change an object's rotation about an axis.	The quantity of rotation of a body. That is the product of its moment of inertia and its angular velocity.
Formula	τ = r × F (Torque is the cross product of the position vector and the applied force).	L = r × p (for a particle) or L = Iω (for a rigid body).
SI Units	Newton-metres (N•m)	Kilogram metres squared per second (kg•m²/s)
Dependency	Torque always depends on the magnitude of the applied force and the distance of the lever arm from the axis.	Angular momentum always depends on the object's mass distribution and how fast it is spinning.
Conservation	Not a conserved quantity. It is the agent of change for angular momentum.	Conserved when the net external torque on the system is zero. This is the Principle of Conservation of Angular Momentum.
Analogy to Linear Motion	Analogous to Force (F). Just as force causes linear acceleration, torque causes angular acceleration.	Analogous to Linear Momentum (p). It represents the "quantity of motion" for a rotating system.

The meaning of torque lies in the tendency of a force to cause an object to rotate about a point or axis. 

In pure physics terminology, torque is dependent on the magnitude of the force, the point where it’s applied, and the perpendicular distance from the pivot. 

When we consider rotational mechanics, torque is responsible for angular acceleration. In contrast, a regular force causes linear acceleration (which you can learn from our Motion in a Straight Line Class 11 notes).

Key Features of Torque 

• Vector Nature: Torque is a vector quantity. Its direction is perpendicular to the plane of rotation.
• Lever Arm: The effectiveness of a force increases with its perpendicular distance from the pivot.
• Rotational Effect: Torque causes angular acceleration in rigid bodies. You can also find the definition of rigid body in our article on the Centre of Mass. 

Types of Torque

There are two main types of torque. 

• Static Torque: This type of torque is applied to an object at rest. One example is turning a doorknob. 
• Dynamic Torque: This torque type applies to moving objects. A common example is a spinning wheel on a bicycle.   

We can calculate torque about a point and about an axis. 

Calculating Torque about a Point

Torque $\vec{\tau }$  about a point $O$  due to a force $\vec{F}$  applied at a point with position vector $\vec{r}$  relative to $O$ is: $\vec{\tau }=\vec{r}\times \vec{F}$

The magnitude is: $\left|\vec{\tau }\right|=rF\mathrm{s}\mathrm{i}\mathrm{n}\theta =r_{\perp }F=r{F}_{\perp }$

where $\theta$ is the angle between $\vec{r}$  and $\vec{F},r_{\perp }=r\mathrm{s}\mathrm{i}\mathrm{n}\theta$  is the perpendicular distance (lever arm), and $F_{\perp }=F\mathrm{s}\mathrm{i}\mathrm{n}\theta$  is the component of force perpendicular to $\vec{r}$ .

Understanding the Direction of Torque

The direction of torque follows the right-hand rule.

 In case you’re wondering what this rule is, 

1. Point your fingers in the direction of the position vector (from pivot to where the force is applied)
2. Curl your fingers toward the direction of the force
3. Your thumb points in the direction of the torque vector

The SI unit of Torque is N•m, or Newton metres.

Calculating Torque About an Axis

For torque about an axis (e.g., $AB$  ), take the component of $\vec{\tau }$ along the axis. If the force's line of action intersects or is parallel to the axis, the torque about that axis is zero. Note that only forces that can cause rotation about the axis contribute to the torque. 

Angular momentum quantifies the rotational motion of a body. It combines its moment of inertia and angular velocity. It is conserved in the absence of external torques.

Also, the calculation of angular momentum helps in analysing rotational systems, such as spinning tops or colliding objects.

Key Features of Angular Momentum

• Vector Nature: This is also a vector quantity. The direction is along the axis of rotation.
• Conservation Law: Angular momentum remains constant if the net external torque is zero. Additionally, read about the conservation of momentum.  
• System Analysis: Applies to both particles and rigid bodies.
• Moment of Inertia: The amount of angular acceleration produced by a given torque depends on the object's moment of inertia. It measures how the mass is distributed relative to the axis of rotation. 

 

 

We can have the formula for angular momentum for a particle and a rigid body in two different ways.

Angular Momentum of a Particle 

For a particle of mass $m$  with linear momentum $\vec{p}=m\vec{v}$  at position $\vec{r}$  relative to a point $O$ , angular momentum is: $\vec{L}=\vec{r}\times \vec{p}$

Angular Momentum of a Rigid Body

For a rigid body rotating about a fixed axis with moment of inertia $I$  and angular velocity $\vec{\omega }$ , angular momentum about the axis is: $\vec{L}=I\vec{\omega }$

The SI unit for angular momentum is $\mathrm{k}\mathrm{g}\cdot {\mathrm{m}}^2/\mathrm{s}$ .

The rate of change of angular momentum equals the net external torque: $\frac{d\vec{L}}{dt}={\vec{\tau }}_{\mathrm{e}\mathrm{x}\mathrm{t}}$

This leads to the important conservation principle. When no external torque acts ${\vec{\tau }}_{\text{ext}}=0,\vec{L}$ or angular momentum is conserved.  We will cover the relationship between torque and angular momentum below in more detail. 

Vector Representation and Direction of Angular Momentum

Angular momentum is a vector quantity. We find the direction of it using the right-hand rule.

The direction of angular momentum is perpendicular to both the radius and velocity vectors of the rotating object. 

Here we are going to explain the relationship between torque and angular momentum. We will see how torque causes changes in angular momentum. That, in turn, leads to the conservation principle when torque is zero.

Torque in Rotational Dynamics

For a rigid body in pure rotation about a fixed axis, we can notice that the net external torque about the axis is equal to the product of the moment of inertia and angular acceleration. We represent that as ${\tau }_{\mathrm{e}\mathrm{x}\mathrm{t}}=I\alpha$

This is the rotational equivalent of $F=ma$ , we already know as the Second Law of Motion.

Then, we also have to see that the internal torques cancel out due to Newton's third law. It leaves only external torques to drive rotation.

To think of an example, consider a bicycle wheel. When you pedal (external torque from your legs), the wheel spins faster.  But if the wheel's spokes were somehow pushing against each other (internal forces), they'd cancel out and the wheel wouldn't accelerate. 

Angular Momentum in Systems

For a system of particles or a rigid body, total angular momentum is the sum of individual contributions. So when we combine both translational and rotational motions, the angular momentum about a point includes contributions from the centre of mass (COM) and rotation about the COM. We see that as 
$\vec{L}={\vec{L}}_{\mathrm{C}\mathrm{O}\mathrm{M}}+{\vec{r}}_{\mathrm{C}\mathrm{O}\mathrm{M}}\times M{\vec{v}}_{\mathrm{C}\mathrm{O}\mathrm{M}}$

where, LCOM is the angular momentum due to rotation about the COM.

For instance, think of a rolling basketball. It has angular momentum from spinning on its axis (like a ball spinning in place) plus angular momentum from the whole ball moving in a circle around the hoop as you dribble it.

Conservation of Angular Momentum

When no external torque acts ( ${\vec{\tau }}_{\text{ext}}=0$ ), angular momentum is conserved. This principle simplifies problems involving collisions or systems with internal forces. 

For instance, when you're spinning in an office chair, you pull your legs up. You automatically spin faster without anyone giving you an extra push, because the total “spin amount” has to stay constant.

Torque and Force Couples

A force couple consists of two equal, opposite forces separated by a distance. That creates pure rotation without translation. The torque due to a couple is: $\tau =Fd$

where $d$ is the distance between the forces' lines of action.

This is constant regardless of the pivot point.

In short, torque is the same no matter where you choose your pivot point. 

 

Torque and angular momentum have several practical applications in the real world. Some common examples are below. 

• Engineering: Flywheels store rotational energy in engines. Here, torque has its use to make the flywheel accelerate with rotational inertia. At the same time, the angular momentum is conserved to maintain the smooth running of the engine. In physics, engineers can further calculate the moment of inertia and angular velocity to determine the stored energy. 
• Sports: Gymnasts use angular momentum to execute twists and flips. They apply more torque to the body movements, which helps in changing the rotational speed. When in the air, they can conserve angular momentum and also control their angular velocity. 
• Daily Life: While driving a car, you can steer the wheel. The steering wheel applies torque to change the direction of the car. As a driver, you can observe that the torque applied to a steering wheel generates angular momentum, which leads to the rotation of the wheels.