Dynamics Rotational Motion - Definition, FAQs

What is Rotational Motion?

Rotational motion definition: Rotational motion is the motion of an object in a circular path about a fixed axis. It is also called rotary motion. In rotational dynamics class 11, the particle is moving in a circular or curved path having:

• Torque
• Angular acceleration
• Moment of Inertia
• Angular displacement

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Dynamics Of Rotational Motion About a Fixed Axis

It is a circular path in which an object moves around a fixed common axis. Every point of the object undergoing rotational motion about a fixed axis has the same angular velocity and angular acceleration about the axis. The axis of rotation remains fixed. It does not change.

• Angular Acceleration

It is the rate of change of angular velocity over time.

$$
\alpha=\frac{\Delta \omega}{\Delta t}=\frac{\omega_2-\omega_1}{t_2-t_1}
$$

where,

• $\alpha$ is the angular acceleration
• $\Delta \omega$ is the change in angular velocity
• $\Delta t$ is the time interval

• Moment of Inertia(I):

It is a physical property of any body, by which it resists any rotational mechanics change exerted by external torque. It is denoted by (I). The moment of Inertia is given by

$$
I=\sum m_i r_i^2
$$

where,

• $I$ is the moment of inertia
• $m_i$ is the mass of each particle in a system
• $r_i$ is the perpendicular distance of each particle from the axis of rotation

It depends on mass: the higher the mass higher the moment of inertia.

• Torque:

It is a vector quantity. It is the product of the perpendicular distance from the axis of rotation and the force applied to it. Torque has a twisting effect on the body.

Numerically, it is given by

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

where,

• $\tau$ is the torque
• $r$ is the distance from the axis of rotation to the point where the force is applied
• $F$ is the magnitude of the applied force
• $\theta$ is the angle

• Angular Momentum:

It is a measure of difficulty possessed by a rotatory body to come to rest.

Numerically it is given by:

$$
L=I \omega
$$

where,

• $I$ is the moment of inertia of the object about the axis of rotation
• $\omega$ is the angular velocity of the object

Angular motion examples: Orbit of the earth around the sun, rotation of the tire

• Rotational Kinetic Energy

The kinetic energy in a rotating body is due to the rotational motion. The formula for rotational kinetic energy is,

$$
K_{\mathrm{rot}}=\frac{1}{2} I \omega^2
$$

where,

• $K_{\text {rot }}$ is the rotational kinetic energy
• $I$ is the moment of inertia of the object about the axis of rotation
• $\omega$ is the angular velocity of the object

• Power in Rotational Mechanics

Power is the rate of work done by the torque in rotating an object.

$$
P=\tau \omega
$$

where,

• $P$ is the power
• $\tau$ is the torque
• $\omega$ is the angular velocity

Rotational Motion Examples

1. Rotating blades in electric fans provide air
2. Applying rotational motion to clean the clothes in the washing machine
3. The bits in drills undergo rotational motion to bore holes
4. Generators use rotational motion to convert mechanical energy to electrical energy
5. Gears in clocks use rotational motion to keep time

Work done In Rotational motion

Work done in rotary motion is defined by the product of torque applied and change in angular displacement.

Work-Energy Principle

Let consider small angle $\Delta \theta$ be the angular displacement under the effect of torque . Then linear displacement will be

$\Delta r=r \Delta \theta$

Therefore the work done is given as,

$W=\tau \Delta \theta$

Let's say the number of force acting, so net torque will be

$($ total $)=\left(\tau_1+\tau_2+\ldots \ldots\right) \Delta \theta$

As we know $\Delta \theta$ is very small for all the torque thus net work done is zero.

Equations of Motion in Rotational Motion Class 11

First Equation:

$$
\omega=\omega_0+\alpha t
$$
Second Equation:

$$
\theta=\omega_0 t+\frac{1}{2} \alpha t^2
$$
Third Equation:

$$
\omega^2=\omega_0^2+2 \alpha \theta
$$
where,

$\omega$ is the final angular velocity
$\omega_0$ is the initial angular velocity
$\alpha$ is the angular acceleration
$t$ is the time taken
$\theta$ is the angular displacement

Applications Of Rotational Motion Class 11

1. Wind turbines
2. Amusement park rides
3. Flywheels
4. Electron spin in quantum physics
5. Tractors and harvesters
6. Wheels of vehicles