Torque and Rotational Motion - JEE Important Topic

The definition of torque in Physics states, “Torque is a unit of measurement for the force that can cause an item to revolve about an axis”. Torque is what causes an angular acceleration. As a result, torque and rotational motion are analogues to force and translation motion. Torque in Physics is only a force's tendency to turn or twist. It is referred to using a variety of terms, including moment and moment of force. Torque is a vector quantity. The direction of the force acting on the axis determines the direction of the torque vector.

Terminologies Used in Relation to Torque

• Axis of rotation: The straight line about which the object rotates is called the axis of rotation.

• Moment arm: The distance of the point of application of force from the axis of rotation is sometimes called the moment arm or lever arm.

• Cross product: A binary operation on two vectors in three dimensions is called a cross product. A vector that is perpendicular to both vectors is produced as a result.

• Right-hand Rule: The direction of the resultant vector, for the cross product of two non parallel vectors, is represented by the right-hand rule. The direction of the resulting vector is shown by the thumb if we point our index finger along vector A and middle finger along vector B.

• Moment of Inertia: The moment of inertia is the term used to describe the rotational equivalent of mass (inertia) in linear motion.

What is Rotational Motion?

The movement of an object in a fixed orbit around a circular route is referred to as rotational motion. Only rigid bodies are taken into account in rotational motion. A massed object that maintains a rigid shape is referred to as a rigid body. The object rotates about an axis, which we will call the pivot point, and will label '$O$'. We will call the force '$F$'. The moment arm is the distance between the pivot point and the point at which the force acts, and is denoted by '$r$'. Note that this distance, '$r$', is also a vector, and points from the axis of rotation to the point where the force acts.

Force Acting on a Rigid Body

Note that the force applied, $F$, and the moment arm, $r$, are independent of the object. Additionally, since the moment arm would be zero, applying a force at the pivot point will not result in any torque ($r=0$).

Translational equilibrium, in which the sum of the forces is equal to zero, is equivalent to rotational equilibrium. The sum of the torques is equal to zero in rotational equilibrium. In other words, the object experiences no net torque.

$\sum{\tau =0}$

An object may be under the influence of many forces, each of which may act at a different position on the item. Then, each force will cause a torque. The total of the individual torques is the net torque.

Rotational Motion Examples

1. Rotation About a Fixed Point

• Ceiling fan rotation

• Minute and hour hands in the clock in motion.

• Opening and closing of the door 

2. Rotation About An Axis of Rotation

• Pushing a ball off a hill

• Motion of the earth

Difference between Circular Motion and Rotational Motion

The primary distinction between circular motion and rotational motion is that the axis of rotation's distance from the centre of mass of the body does not change in circular motion. The axis of rotation and the centre of mass may shift during rotational motion. The axis of rotation and the centre of mass remain constant while moving in a circle. The direction of the rotational axis may shift during rotational motion. The direction of the axis of rotation remains constant while the object is rotating in a circle.

Formula for Torque

The radius vector (which extends from the rotational axis to the location of force application) and the force vector are multiplied to create torque. $\tau $ is used to symbolise the torque.

$\tau =F\times r$ 

$\tau =F\times r\times \sin \theta $

Here, $F=$  linear force, $r=$  distance between the axis of rotation and the point at which linear force is applied and $\theta =$ angle between $r$ and $F$. This is known as the torque formula, which is one of the important rotational motion formulas. From the above discussion, we can say that the SI unit of torque is Nm.

The SI unit of torque is the Newton-metre. Conventionally, the right hand grip rule is used to determine the torque vector's direction. The direction of the resulting vector is shown by the thumb if we point our index finger along vector A and middle finger along vector B.

Types of Torque

Torque can either be static or dynamic.

• Static Torque: Any torque that does not result in an angular acceleration is static. For example, someone pushing on a closed door is applying a static torque to the door because the door is not rotating about its hinges, despite the force applied.

• Dynamic Torque: A racing car's drive shaft transfers the dynamic torque because it must cause the wheels' angular acceleration if the vehicle is to accelerate along the track.

Factors on Which Torque Depends

• The perpendicular separation between the point and the action line about which the torque is calculated. Moment Arm is another name for this.

• The magnitude of the force applied is another factor on which torque depends.

Conclusion

Torque is a unit of measurement for the force that can cause an item to revolve about an axis. It is also known as a moment of force. Torque in rotational motion is equivalent to force in translation motion. Torque can be either static or dynamic. The moment of inertia is the term used to describe the rotational equivalent of linear motion mass. 

Similarly, the opposite of linear acceleration in rotational motion is angular acceleration. Rotational motion dynamics are identical to linear or translational dynamics in every way. The motion equations for linear motion share many similarities with the equations for the mechanics of rotating objects.