Rotational Motion of a Rigid Body

Objects are made up of particles. When in rotational motion, these particles move in a specific manner, and it is important to study how these particles move with respect to each other. A rigid body is a collection of particles moving in sync, and the body does not deform when in motion.

Understanding rotational motion is the key to accomplishing things like putting a satellite into orbit, launching a spacecraft, winning the Grand Prix, etc. It is an integral part of engineering, the automobile industry, and space projects. But we must first understand rotational motion and its nuances.

What Causes Rotational Motion?

We know that when a body moves in circles around a fixed axis or a point, it is said to be in rotational motion. When a body moves in a rotational motion around a given axis or a line, i.e., at a fixed distance and fixed orientation relative to the body, the body is rotating around the axis. For example, when we open a door, it turns around the hinges.

When a body moves such that it rotates around a single point and not an axis such as a spinning top, it is in rotational motion around that point. But what causes rotational motion? Force is responsible for all motion that we observe in the physical world. The force responsible for rotational motion is called torque or the moment of the force.

Torque is described as the measure of any force that causes the rotation of an object about an axis. You experience torque when you open the door to a room. When force is applied, the door rotates. The speed at which the door opens can be controlled by the amount of force applied. In other words, the speed depends on the torque applied to the door.

Torque can be of two types—static and dynamic. When the torque on a body does not produce an angular acceleration, it is called static torque. In contrast, when the torque acting on a body produces angular acceleration, it is called dynamic torque.

The Rotational Variables

The motion of a body is controlled by certain variables, such as velocity, displacement, etc. Similarly, in rotational motion, we have certain variables called the rotational variables. The different types of rotational variables are:

• Angular Displacement

When a body is in rotational motion, we measure the displacement of the body and not the distance covered. A body in rotational motion starts at an initial position. The angle of this position to the axis of rotation is taken as zero radians. As the body moves, the distance between the current and the initial position of the body changes. The distance between these positions is measured in radians and is called the angular displacement of a body.

• Angular Velocity

In rotational motion, a rigid body is moving in a path shaped like a circle. At any given point, the tangent to a specific point denotes the angular velocity of a body. But what is angular velocity? When rotating about a fixed axis, a rigid body constantly changes its angle with respect to its initial position and the fixed axis.

Let us denote the angle between the initial and current position of a rigid body as . Linear velocity is measured as the change in distance with respect to time. Similarly, angular velocity is measured as the change in the angle with respect to time.

It is denoted by =d/dt.

• Angular Acceleration

When torque is applied to a rigid body already in rotation with a fixed angular velocity , the application of the external torque results in a change in the angular velocity of the body. The measure of the change in angular velocity with respect to the time of a rigid body in rotational motion due to the application of an external torque is called angular acceleration. If is the angular velocity of a rigid body, the angular acceleration of the body is given as =d/dt. Angular acceleration also plays a role in the rotational inertia of a rigid body. Rotational motion with constant acceleration is the basis of many important phenomena like car speeding, particle accelerators, etc.

The Rotational Inertia of a Rigid Body

According to Newton’s second law, all bodies tend to resist a change in their current state. A body at rest resists change when it is set in motion, and a body in motion resists change by not coming to a stop immediately. Similarly, when a rigid body is put into rotational motion, the amount of torque required to change the angular velocity of the body is called its rotational inertia.

The rotational inertia of a rigid body is an important concept as it helps us understand the amount of torque required to achieve a certain objective. The rotational inertia of a rigid body is affected by the mass and the distribution of the mass of the body with respect to the axis around which the body rotates. The distance of the centre of mass from the axis of rotation increases or decreases the rotational inertia of a rigid body.

Conclusion

When a body moves in a circular path around a fixed axis, it is said to be in rotational motion. A body in rotational motion can be rotating around a fixed axis or a fixed point. The parameters that govern the rotational motion of a rigid body are angular displacement, angular velocity, and angular acceleration. 

A body in rotational motion opposes a change being introduced in its angular velocity by an external torque. This property of resistance that a body in rotational motion exhibits is called the rotational inertia of a rigid body. The rotational inertia of a body is affected by the mass and the distribution of the mass of the body with respect to the axis around which the body rotates. The distance of the centre of mass from the axis of rotation is also a factor in determining the rotational inertia of a rigid body.