Rotational Motion of Rigid Body for JEE

Rotational motion of an object or a body is defined as its movement around a fixed orbit circular path. We consider only rigid bodies for rotational motion. Rigid bodies are nothing but objects with a certain mass that hold a rigid shape. In other words, rigid bodies can be defined as bodies that do not undergo a change in size or shape when an external force is applied to them. The motion of a rigid body can be divided into two categories: Translational Motion and Rotational Motion.

As rigid bodies have two types of motion therefore in such a case it is necessary to examine both linear and angular velocities of the body. Thus, the dynamics of rotational motion around a fixed axis need to be evaluated. The dynamics of rotational motion are nothing but a comparison of quantities associated with their linear motion analogous to its rotational motion.

Rotational Motion is the type of motion wherein the rigid body rotates along some axis. The plane along which a rigid body rotates is called its axis of rotation. One of the cardinal examples of this is the motion of the earth about its axis of rotation.

Rotational Motion of a Rigid Body

The rotation of a rigid body along some axis is known as the rotation motion of a rigid body. This motion does not change the shape or size of the rigid body. Also, the rotation of a rigid body could be along with a fixed point or an axis of rotation.

When the rotation of a rigid body is along a fixed axis the forces that lie in the perpendicular plane to the axis are considered.

Rotational Motion Examples

Some examples of rotational motion include the motion of a body about a fixed point or an axis of rotation.

• The rotation of a ceiling fan, opening and closing of the door etc., are some examples of rotation of an object about a fixed point.

• The rotation of the earth about its axis is a classic example of a body with translation as well as rotational motion.

To better understand pure rotational motion, suppose a disc is rotating about an axis passing through the centre. If we consider two points at radius R and r respectively, then at both points the tangential velocity will be different. For radius r the tangential velocity will be $\omega~r$ and for radius R the velocity will be $\omega~R$. Also R > r, therefore r is a nearer point than R.

As the two points are at different points and have different tangential velocities therefore there exists a relative speed between them whose magnitude is equal to $\omega~(R-r)$. The relative speed is non-zero and $\omega$ is the angular speed of the disc. Thus, the example is that of a pure rotational motion.

Torque on a Rotating Rigid Object

Some coupled forces on a rigid body cause it to rotate along its axis. These forces are called torque or moment of force. The figure below depicts the torque on a rigid body.

Torque on a Merry-go-round 

The torque of a rotating rigid body is denoted as $\tau=r . f=r F \sin \theta$

Where, $\tau$ is the torque, F is the force applied, r is the distance between an applied force from its axis of rotation and $\theta$ is the angle between r and F.

In case the direction of F and r are parallel to one another the angle between them is zero, thus making the value of $\sin \theta$ as zero. It makes the net torque acting on a body to be equal to zero.

Work Done in Rotational Motion

Now that we have understood the torque acting on a rigid body, let us dive into some more concepts such as the Kinetic energy of a rigid body and the work required to rotate an object.

According to the work-energy principle, the total work done on a body is the sum of all the forces acting on it which is equal to the change in kinetic energy of the body.

As we know that the work done on an object is equal to the change in its kinetic energy thus work is denoted as

• $W=\Delta K E=K E_{2}-K E_{1}=\dfrac{1}{2} I\left(\omega_{2-}^{2} \omega_{1}^{2}\right)$

Power in Rotational Motion

Power for both linear and rotational motion is important. The power for a linear motion is denoted as $P=\vec{F}$ where the force is constant. In the section below the torque is considered to be constant. From the work and kinetic energy that has been derived above the power in rotational motion is defined as the rate of doing work.

$P=\dfrac{dW}{dt}$

If the net torque is constant, then work done become $W=\tau\theta$, the power then is calculated as $P=\dfrac{dW}{dt}$

On substituting the value of W we get,

$\begin{align} &\mathrm{P}=\dfrac{d(\tau \theta)}{d t}\\ &\mathrm{P} =\tau \dfrac{d \theta}{d t}\\ &\text { or } P=\tau \omega \end{align}$

Rotational Dynamics

Rotational Dynamics are completely analogous to linear motion. Depicted below is the comparison between the quantities in translation or linear and rotational dynamics.

From the table above it is clear that force in a linear motion is the torque for the rotational motion.

Conclusion

The article traverses through the different concepts of rotational motion. Torque, work done, power and the kinetic energy carried by a rigid body in motion are calculated. A comparison between linear and rotation motion distinguishes between the quantities that are analogous to one another. It is also seen how a rigid body experiences two kinds of motion i.e. translation and rotational motion.

Translational motion is majorly linear while rotational motion is when a rigid body starts rotating along some axis. Examples of a body rotating at some fixed point and an axis of rotation are also looked at.