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# Rotational Dynamics

Top
 Sub Topics Dynamics talks about how the force and torques acts for a body in motion. What's peculiar about Rotational dynamics? It’s a known fact that a rigid body rotates in two dimensions in a fixed axis. The force and torque even acts in these. Then how is that rotational dynamics show there peculiar nature in dynamics! Let’s learn more about it.

## What is Rotational Dynamics?

Rotational dynamics is a part of dynamics which conveys rotational motion. It analyzes how the body rotates and the force acting on it. The rotational dynamics starts with the study of Torque that causes angular accelerations of objects. Let’s discuss of the Rotational Dynamics and its basic terms. Force changes the motion of objects but when an object is moving in a fixed axis in a circular or curved path then, it can also feel the force known as Torque. In rotational motion we go for a body rather than particles. Each particle behaves in a different manner like the body rotates more if rotations are given at edges rather than at center so we consider body as a whole. Since the force acts on the edges away from the center we consider the concept of Torque. Torque is a force which is studied under rotational dynamic. It defines how force acts in a rotating body. Let’s take point P at a distance r from the axis of rotation if force acts in angle $\theta$.

The torque at point p is given by
Torque defines how force acts when a body when it rotates.

## Angular Acceleration

Angular acceleration tells more about rotational motion. At what rate the angular velocity changes with time defines it! Any rotating body has its own angular velocity. The rate at which the velocity varies gives its angular acceleration.
Let’s consider a body rotating having initial angular velocity ?1 and attains some angular velocity of ?2 after some time t. The angular acceleration is

Lets consider a body rotating having initial angular velocity $\omega_1$ and attains some angular velocity of $\omega_2$ after some time t.The angular acceleration is

The unit for $\alpha$ is radian second-2 or rad s-2.

## Center of Mass

Center of mass is a point where the body mass is concentrated. For a rotating body it defines a point where a body is free to rotate if not compelled by force. It can be given for a single particle or a system of particles.

If x1, x2 .... xn are the positions of the particles. The position of center of mass is

If v1,v2.....vn are the velocities of the particles. The velocity of center of mass is

If a1,a2,....an are the acceleration of the particles. The acceleration of center of mass is

## Moment of Inertia

This concept was given by Leonhard Euler. It measures the resistance of a body to change its rotation. It is denoted by I or J given as

Where m is the mass of the particle and
r is the radius from the axis of rotation.

## Rotational Dynamics Equations

Let us imagine a body rotating in a fixed axis due to a force applied with initial angular velocity $\omega$o and attains angular velocity $\omega$ moving with angular acceleration a after time t. If the twist in the angle is , and the angular acceleration gained is a then the rotational dynamics equations related to angular velocity are

These define the rotational motion of a particle.

## Rotational Dynamics Problems

Here are some problems on rotational dynamics you can go through it:

### Solved Examples

Question 1: A body of mass 5 kg acquires an acceleration of 10 rad s-2 by an applied torque of 2 Nm. Calculate its Moment of inertia.
Solution:

Given: $\tau$ = 2 Nm, Angular acceleration $\alpha$ = 10 rad s-2

Moment of inertia I = $\frac{\tau}{\alpha}$

= $\frac{2}{10}$

= 0.2 kgm-2.

Question 2: Calculate the kinetic rotational energy if moment of inertia of 0.5 kgm-2 and angular velocity is 2 $\pi$ rad s-1.
Solution:

Given: Moment of inertia I = 0.5 kgm-2, Angular velocity $\omega$ = 2 $\pi$ rads-1

The rotational kinetic energy is

KErot = $\frac{1}{2}$ I $\omega^{2}$

= $\frac{1}{2}$ $\times$ 0.5 kg m-2 $\times$ (2 $\pi$ rad s-1)2

= 9.8696 kg m-2 rad2 s-2.

Question 3: The speed of rotating body increases from 120 rpm to 240 rpm in 20s. Calculate its angular acceleration.
Solution:

Initial angular velocity $\omega_0$ = 2 $\pi$ fo = $\frac{2 \pi \times 120}{60}$ = 4 $\pi$ rad s-1

Final angular velocity $\omega$ = 2 $\pi$ f = $\frac{2 \pi \times 240}{60}$ = 8 $\pi$ rad s-1

Angular acceleration $\alpha$ = $\frac{\omega - \omega_o}{t}$

= $\frac{ 8 \pi - 4 \pi}{20}$

= $\frac{\pi}{5}$