What is a Rigid Body?

A rigid body is an assemblage of a large number of material particles which do not change their mutual distance under any circumstances.

What is axis of rotation?

An imaginary line perpendicular to the plane of circular paths of particles of a rigid body in rotation and containing the centres of all these circular paths is known as the axis of rotation.

Rotational Motion: Key Concepts & Applications

1.

If angular acceleration $(α = 0)$ is zero, $ω$ =constant and $θ = ω t$

2.

If angular acceleration $(α = co n s t an t)$ , then

(a) = $θ = \frac{( ω _{0} + ω ) t}{2}$

(b) = $α = \frac{ω - ω _{0}}{t}$

(c) = $ω = ω_{0} + α t$

(d) = $θ = ω_{0} t + \frac{1}{2} α t^{2}$

(e) = $ω^{2} = ω_{0}^{2} + 2 α θ$

(f) = $θ_{n}^{th} = ω_{0} + \frac{α}{2} (2 n - 1)$

3.

If angular acceleration is not constant the above equation will not be applicable, in this case

(a) = $ω = \frac{d θ}{d t}$

(b) = $α = \frac{d ω}{d t} = \frac{d ^{2} θ}{d t ^{2}} = ω \frac{d ω}{d θ}$

S.No

Topic

Diagram

Formula

1

Moment of Inertia For Discrete System of Particles

$I = m r^{2}$

m=mass,

r=Perpendicular distance from AOR

$\Rightarrow$ I depends on mass, mass distribution, position of AOR.

$\Rightarrow$ I does not depend on angular velocity, angular acceleration, torque, angular momentum.

$\Rightarrow$ As the distance of mass increases from the rotational axis, MOI increases

$\Rightarrow$ SI Unit- $K g - m^{2}$

2

Moment Of Inertia(MOI) for Continuous Mass Distribution

$I_{A B} = \int r^{2} d m$

S.No.	MOI of Body	Geometrical Figure	Axis
1.	Rod about an axis passing through the centre		$I = \frac{M L ^{2}}{12}$
2.	Rod about an axis passing through its end and perpendicular to its Length		$I = \frac{M L ^{2}}{3}$
3.	Rod Inclined at some angle		$I = \frac{M L ^{2}}{3} S i n^{2} θ$
4.	Ring		$I = M R^{2}$
5.	Disc		$I = \frac{1}{2} M R^{2}$
6.	Hollow Cylinder		$I = M R^{2}$
7.	Solid Cylinder		$I = \frac{M R ^{2}}{2}$
8.	Rectangular Plate about an axis that lies in the plane and parallel to shorter sides		$I = \frac{M L ^{2}}{3}$
9.	Rectangular Plate about the axis AA’		$I_{A A^{'}} = \frac{M L ^{2}}{12}$
10.	Rectangular Plate about the axis AA’		$I_{A A^{'}} = \frac{M b ^{2}}{3}$
11.	Rectangular Plate about the axis AA’		$I_{A A^{'}} = \frac{M b ^{2}}{12}$
12.	Solid Sphere about its diametric axis		$I = \frac{2}{5} M R^{2}$
13.	Hollow Sphere about its diametric axis		$I = \frac{2}{3} M R^{2}$

S.No

Theorem

Diagram

1.

Perpendicular Axis Theorem-The moment of inertia (MI) of a plane lamina about an axis perpendicular to its plane is equal to the sum of its moments of inertia about any two mutually perpendicular axes in its own plane intersecting each other at the point through which the perpendicular axis passes.

$I_{Z} = I_{X} + I_{Y}$

Applicable only for two dimensional bodies or plane lamina

2.

Parallel Axis Theorem-The moment of inertia of a body about any axis is the sum of its moment of inertia about a parallel axis through the center of mass and the product of the body's mass and the square of the distance between the two axes.

$I = I_{CM} + M d^{2}$

S.no	Conceptual Points	Diagram
1.	If the line of action of a force passes through point O, then the Torque of this force about point O will be zero.
2.	If a force is acting parallel to the axis of rotation, then its rotational effect about that axis will be zero.
3.	If the line of action of a force passes through the axis of rotation, then its rotational effect (Torque) about that axis of rotation will be zero.

S.No	Topic	Diagram	Formula
1.	Angular Momentum of a Rigid Body Rotating about Fixed Axis		$L = I ω$
2.	Angular Momentum of Combined Rotation and Translation		$L_{P} = I_{CM} ω + r \times m v_{CM}$
3.	Total Kinetic Energy in Combined Translation and Rotation		$K_{Total} = \frac{1}{2} M V_{C}^{2} + \frac{1}{2} I_{C} ω^{2}$

S.No	General Conditions	Diagrams
1.	No Friction Pure Rolling, $V_{A} = V_{B}$ $V_{C} - ω R = V_{S u r f a ce}$
2.	$V_{C} > ω R$ , Contact point slips forward so kinetic friction will act in backward direction, so $V_{C}$ decreases and $ω$ Increases.
3.	$V_{C} < ω R$ , Contact point slips in backward direction, so kinetic friction will act in forward direction, so $V_{C}$ increases and decreases.
4.	$V = ω R$ (Initial), Static Friction whose values can lies between zero and $μ N$ will act in backward direction.

S.no

Conditions

Diagram

1.

Accelerated Pure Rolling,

$\Rightarrow a_{COM} = α R$

2.

If Surface is also Moving,

$\Rightarrow a_{COM} - α R = a_{Pl ank}$

S.No

Velocity

Acceleration

Time taken to reach bottom

1.

$V e l oc i t y_{R o ll in g} = \frac{2 g h}{1 + \frac{K ^{2}}{R ^{2}}} = \frac{2 g ss in θ}{1 + \frac{K ^{2}}{R ^{2}}}$

$V e l oc i t y_{s l i d in g} = 2 g h = 2 g s S in θ$

$\Rightarrow V_{Sl i d in g} > V_{R o ll in g}$

$a_{ro ll in g} = \frac{g ss in θ}{1 + \frac{K ^{2}}{R ^{2}}}$

$a_{s l i d in g} = g S in θ$

$\Rightarrow a_{s l i d in g} > a_{ro ll in g}$

$t_{ro ll in g} = \frac{1}{s in θ} \frac{2 h}{g} (1 + \frac{K ^{2}}{R ^{2}})$

$t_{s l i d in g} = \frac{1}{S in θ} \frac{2 h}{g}$

$\Rightarrow t_{ro ll in g} > t_{s l i d in g}$

S.No	General Conditions
1.	Least $\frac{K ^{2}}{R ^{2}}$ will reach first.
2.	Maximum $\frac{K ^{2}}{R ^{2}}$ will reach last.
3.	Equal $\frac{K ^{2}}{R ^{2}}$ will reach together
4.	Change in Kinetic Energy Due to Rolling $v_{2} > v_{1}$ $= \frac{1}{2} m (1 + \frac{K ^{2}}{R ^{2}}) (v_{2}^{2} - v_{1}^{2})$
5.	Rolling v/s Sliding $v_{ro ll in g} = \frac{v _{s l i d in g}}{1 + \frac{K ^{2}}{R ^{2}}}$ , $a_{ro ll in g} = \frac{a _{s l i d in g}}{1 + \frac{K ^{2}}{R ^{2}}}$ , $t_{ro ll in g} = t_{s l i d in g} 1 + \frac{K ^{2}}{R ^{2}}$ vRolling<vSliding , aRolling<aSliding , tRolling>tSliding

Centre of Mass And Collision	Centre of Gravity	Kinetic Energy
Centre of Mass	Escape Velocity	Elastic Collision
moment of inertia of a hollow sphere	Kepler's Law of Planetary Motion	Conservation of Linear Momentum

Frequently Asked Questions

What is a Rigid Body?

What is axis of rotation?

Join ALLEN!

Frequently Asked Questions

What is a Rigid Body?

What is axis of rotation?

Join ALLEN!

Rotational Motion

1.0Definition

2.0Axis of Rotation

3.0Kinematics of Rotational Motion

4.0Moments of inertia of some regular shaped bodies about specific axes

Rectangular Plate about the axis AA’

Rectangular Plate about the axis AA’

Rectangular Plate about the axis AA’

5.0Perpendicular Axis Theorem and Parallel Axis Theorem

6.0Radius of Gyration

7.0Moment of Force or Torque

Couple of Forces

8.0Equilibrium of Rigid Bodies

9.0Rotational Equilibrium

10.0Relation Between Torque and Angular Acceleration

11.0Rotating Pulley

12.0Toppling

13.0Angular Momentum of a Particle

14.0Conservation of Angular Momentum(COAM)

15.0Rotational Kinetic Energy

16.0Rotational Power

17.0Work Energy Theorem in Rotational Motion

18.0Rolling Motion

Pure Rolling

19.0Rolling With Slipping

20.0Accelerated Pure Rolling

21.0Rolling Motion on an Inclined Plane

22.0Sample Questions on Rotational Motion

Table of Contents