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Rotational Dynamics

When an object moves in a straight line then the motion is considered as translational but when the same object moves along an axis in a circular path then it is known as Rotational Motion. Consider a wheel rolling without slipping in a straight line. The forward displacement of the wheel is equal to the linear displacement of a point fixed on the rim. 

As we know that in linear motion, an object is moved till it feels any external force. This force changes the motion of objects. But when an object is moving about a fixed axis in a circular or curved path then, it can also feel the force better known as Torque. It is easier to open a door by pushing on the edge farthest from the hinges than by pushing in the middle. Thus, the torque is a force which is studied under rotational dynamic. 

In rotational motion, the object is not treated as a particle but is treated in translational motion. 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.

To understand the Rotational dynamics, we need to know about many terms like torque, angular acceleration etc. Angular displacement is denoted by $\theta$. For a particle moving in a circle of radius r and assuming that it has moved an arc length of s, the angular position theta is given by,

$\theta$ = $\frac{s}{r}$

 Angular Acceleration
Unit: radians
1 radian = $\frac{180}{\pi}$  

Angular Displacement:
 The angular displacement can be defined as the change in the angular position of the particle or object.
Angular Displacement
$\Delta$ $\theta$ = $\theta_2$ - $\theta_1$
$\theta_2$ = Final angular position
$\theta_1$ = Initial angular position

Angular Velocity and Angular Acceleration
The rate of change of angular displacement is called Angular velocity
Unit: radian per second
$\overline{\omega}$ = $\frac{\Delta \Theta}{\Delta t}$ 
$\omega$ = 2$\pi$f
$\omega$ = angular velocity

Angular Acceleration: The rate at which angular velocity changes with time is called Angular acceleration.
$\overline{A}$ = $\frac{\Delta\omega}{\Delta t}$
Here A is pronounced as alpha
Also a = r$\alpha$ = translational acceleration
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Rotational Dynamics Equations

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There are 3 Rotational Dynamics equations. They are,
$\omega$(t) = $\omega$0+ at
q(t) = q0 +  $\omega$0t + $\frac{1}{2}$at2
$\omega_0$ = $\omega_0^{2}$ + 2a (q - q0)  
$\omega_0$ = magnitude of the initial angular velocity
$\omega_t$ = angular velocity’s magnitude after time t
q0 = Initial angular position
q(t) = Angular position after time t

Conservation of Rotational Energy

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The law of conservation of rotational energy states that the energy of a system remains constant. Energy can neither be destroyed nor be created. It changes from one state to another.
For rotational energy, E = K.E + P.E
m g h = $\frac{1}{2}$m v2 + $\frac{1}{2}$m$\omega$2

Rotational Kinetic Energy

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Rotational kinetic energy is the energy which is gained by an object or body on virtue of its rotation.
It is given by,
Ek = $\frac{1}{2}$ I ω2

Rotational kinetic Energy Formula

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As we know for translation
K E =>  K.E = $\frac{1}{2}$mv2

Similarly, if we consider the rotational component then
K.E = $\frac{1}{2}$ I$\omega$2

Thus, the mathematical formula for rotational K E is,
 Ek = $\frac{1}{2}$ I$\omega$2
Here the momentum of inertia takes the place of mass and translational velocity is replaced by angular velocity ω.
For example:
Earth time period = 23.93 hours
Angular velocity = 7.29 × 10-5 rad /s
Moment of inertia I = 8.04 × 1037 kgm2
Hence rotational kinetic energy = 2.138 x 1029 J

The rotational kinetic energy is basically the energy associated with every part of the object. Also the kinetic energy of any object at any instance can be the sum of both rotational as well as translational kinetic energy.

Rotational kinetic Energy Units

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The units of rotational kinetic energy = JOULES for S.I. system
Let us derive the units by using the formula, We know that,
Ek (rotational) = $\frac{1}{2}$I$\omega$2
Unit of Inertia, I = m r2
Hence I can be expressed as Kg m2
Also for angular momentum unit is $\frac{rad}{second}$
So, K.E (rotational) = Kg m2
rad / second = watt sec = JOULES
Torque can be defined in various ways:

1. Torque is the turning effect of the force about the axis of rotation.

2. The moment of force is called Torque.

3. Torque is positive for a force that causes or tends to cause counterclockwise rotation and vice-versa.

4. The rate at which there is a change in angular momentum is also called Torque.

5. Any force whose line of action is not directed towards the axis of rotation or the center of mass for an object will provide a Torque on that object.
Torque = perpendicular distance from the axis of rotation x Force
$\tau$ = r x F
          = r F sin$\theta$
r = a distance between the point from which torque is measured to the point where force is applied
$\tau$ is the torque and
F = force applied
Its dimensional formula is [M L2 T -2], which is same as that of work.
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The Center of Mass is an imaginary point where one can assume the entire mass of the given system or object to be located.
Center of Mass
Characteristics of Center of Mass:
1) The point can be real or imaginary, e.g., in case of a hollow or empty box the mass is physically not located at the center of mass point.
2) This mass is supposed to be located at the center of mass in order to simplify calculations.
3) The motion of the center of mass characterizes the motion of the entire object.
4) The center of mass may or may not be the same to the geometric center if a rigid body is considered.
5) It is considered as a reference point for many other calculations of mechanics.
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This concept of Moment of Inertia was introduced by Leonhard Euler. It is defined as the property of matter that resists change in rotational motion or acceleration. The rate of rotation can vary because the product of the moment of inertia and the rate of rotation are equal to the angular momentum, which is fixed.

The symbols “I” and sometimes “J” are used to refer Moment Of Inertia.
Moment of Inertia
I = $\sum_{i=1}^{n}$ mr2where,
m = mass
r = Distance from the axis of rotation

The difficulty faced to change the angular motion of any object about an axis is shown or given or measured by calculating moment of inertia about that particular axis. It includes how far each bit of mass constituting the object from the axis.

Greater the distance of the mass, greater would be the rotational force  required to change its rate of rotation.
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Rotational Dynamics Problems

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Some numericals are given below:

Solved Examples

Question 1: A disc is rotating with a constant angular acceleration of p radian/s2 about a fixed perpendicular axis which is passing through its centre.
a) Find the angular velocity of the disc after 4 seconds?
b) Find the angular displacement of the disc after 4 sec?
c) Find the number of turns accomplished by the disc in 4 sec?
$\omega$(4 sec) = $\omega_{0}$ + at
          = 0 + (p rad/sec2) 4 sec
          = 4p rad/sec.
 Angular velocity, q(4 sec) = q0+ $\omega_{0}t$ + $\frac{1}{2}$at2
                                      = 0 + $\frac{1}{2}$ (p rad/sec2) (16 sec2) = 8 p radian.
 Let the number of turns be n,
 then n x 2 p rad = 8 p rad and n = 4

Question 2: A boy rotates a disc and the disc starts rotating with a constant angular acceleration of p radian/s2 about a fixed perpendicular axis which is passing through its centre.
a) Calculate the angular velocity of the disc if the time given is 8 seconds?
b) Calculate the angular displacement of the disc if the time is 8 sec?
c) Calculate the number of turns accomplished by the disc if the time is 8 sec?
$\omega$(8 sec) = $\omega_{0}$ + at
             = 0 + (p rad/sec2) 8 sec
             = 8p rad/sec.
Angular velocity, q(8 sec) = q0+ $\omega_{0}$t + $\frac{1}{2}$ at2
                                     = 0 + $\frac{1}{2}$ (p rad/sec2) (64 sec2)
                                     = 32 p radian.
Let the number of turns be n, then n x 2 p rad = 16 p rad and n = 8

More topics in Rotational Dynamics
Moment of Inertia Center of Mass
Torque Circular and Rotational Motion
Related Topics
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