Rotational Motion Chapter - Physics JEE Main

JEE Main Physics Chapters 

1

Units and Measurement

11

Electrostatics

2

Kinematics

12

Current Electricity

3

Laws of Motion

13

Magnetic Effects of Current and Magnetism

4

Work, Energy and Power

14

Electromagnetic Induction and Alternating Currents

5

Rotational Motion

15

Electromagnetic Waves

6

Gravitation

16

Optics

7

Properties of Solids and Liquids

17

Dual Nature of Matter

8

Thermodynamics

18

Atoms and Nuclei

9

Kinetic Theory of Gases

19

Electronic Devices

10

Oscillations and Waves

20

Communication Systems

Sl. No

Name of Concept

Key Points to Remember

1.

Rigid Bodies

A rigid body is basically a solid body in which the shape and size of the body remain unchanged under motion and interaction with other bodies.

2.

Motion of a Rigid Body

Rigid bodies undergo mainly three types of motion:

1. Pure Rotational: 

The rotation of the body about a fixed axis is known as rotational motion. Pure rotational motion means that all the points at the same radius from the centre of rotation will have the same velocity.

Pure rotational motion examples or rotary motion examples: Rotation of the ceiling fans

2. Pure Translational: 

A rigid body is said to be in pure translational motion if any straight line fixed to it remains parallel to its initial orientation all the time.

Ex: A bike/car moving along a straight long road

3. Translational+Rotational: 

When a rigid body executes pure rotation around an axis while also translational motion with respect to a reference frame, it is said to be in combined translational and rotational motion.

Ex: A rolling wheel

3.

Rotational Kinematics

Just like in linear kinematics, in rotational kinematics we study the parameters related to rotational motion. Such as angular velocity, angular displacement, angular acceleration, etc..

4.

Moment of Inertia

Moment of Inertia plays an important role in the study of rotational dynamics. It is defined as:

$I=\displaystyle\sum\limits_{i=1}^n {m_i}{r_i}^2$

Where, 

ri- The perpendicular distance of ith particle of mass mi from the axis of rotation

For a continuous body moment of inertia is defined as:

$I=\int {r^2} dm$ 

Where,

dm- Mass of an infinitesimal element of the body at a perpendicular distance r from the axis of rotation

Note: Moment of inertia does not depend on the mass of the body

5.

Centre of the Mass of a Rigid Body

Centre of the mass of a rigid body is an imaginary point where the mass of the entire system is concentrated with external forces focusing on this balancing point.

6.

Parallel Axis Theorem

Statement: 

The Moment of Inertia of a body about any axis is equal to the sum of the Moment of Inertia of the body about a parallel axis passing through its centre of mass and the product of its mass and the square of the distance between the two parallel axes.

$I={I_C}+M {d^2}$

Where,

d- The perpendicular distance between the two axes.

7.

Perpendicular Axis Theorem

Statement:

The moment of Inertia of a planar body about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes concurrent with a perpendicular axis and lying in the plane of the body.

${I_Z}={I_X}+{I_Y}$

8.

Angular Momentum

Angular momentum is the rotational analog of linear momentum. It is a vector quantity that describes the amount of rotation of an object.

9.

Angular Displacement

• When a rigid body rotates about a fixed axis, the angle displaced by a line passing through a point on that body and intersecting the axis of rotation perpendicularly is the angular displacement.

• It can be counterclockwise (positive) or clockwise (negative).

• It is analogous to a component of the displacement vector.

• Its SI unit is the radian. Other units include degree and revolution.

10.

Conservation of Angular Momentum

Statement:

When the resultant torque acting on a system of particles is zero, then the total angular momentum vector of the system remains constant.

I.e., $\vec\tau_{net}=0$

11.

Instantaneous Axis of Rotation

The instantaneous axis of rotation (IAOR) of a rigid body is the axis fixed to the body about which the body appears to be rotating at a particular instant of time. It is also known as the instantaneous center of rotation (ICOR).

To locate the IAOR of a rigid body, we can use the following methods:

Method 1: Draw perpendicular lines from the velocity vectors of two points on the body. The point where the two lines intersect is the IAOR.

Method 2: If the body is in pure rolling, the IAOR is the point of contact of the body with the surface on which it is rolling.

12.

Rolling motion

Rolling motion is a combination of translational and rotational motion. It is a type of motion in which a rigid body moves in such a way that one point on the body remains in contact with a surface at all times. This point is called the point of contact.

13.

Toppling

Toppling is a phenomenon in which a rigid body rotates about a fixed point until it falls over. It is a common occurrence in everyday life, such as when a child stacks blocks or when a person leans too far back in their chair.

14.

Angular Velocity

• It is the average angular velocity and can be negative or positive.

• It is a vector quantity and the direction is perpendicular to the plane of rotation.

• The angular velocity of a particle varies at different points.

• The angular velocity of all the particles of a rigid body is the same as a point.

• It represents the speed (how fast) of an object rotating or revolving relative to another point or the speed at which the angular position or orientation of a body changes with time.

15

Torque

“Torque is a unit of measurement for the force that can cause an item to revolve about an axis”. Torque is what causes an angular acceleration.

Formula: 

• $\tau =F\times r$ 

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

The SI unit of torque is the Newton-metre. Conventionally, the right hand grip rule is used to determine the torque vector's direction. The direction of the resulting vector is shown by the thumb if we point our index finger along vector A and middle finger along vector B

16

Types of Torque

Torque can either be static or dynamic.

• Static Torque: Any torque that does not result in an angular acceleration is static. For example, someone pushing on a closed door is applying a static torque to the door because the door is not rotating about its hinges, despite the force applied.

• Dynamic Torque: A racing car's drive shaft transfers the dynamic torque because it must cause the wheels' angular acceleration if the vehicle is to accelerate along the track.

17

Rotational Equilibrium

• A body is in translational equilibrium if the sum of all external forces acting on it is zero. This means that the body is not accelerating, and it may be either at rest or moving with constant velocity.

• A body is in rotational equilibrium if the sum of all external torques acting on it is zero. This means that the body is not angularly accelerating, and it may be either at rest or rotating with constant angular velocity.

• If a body is in translational equilibrium and rotational equilibrium, then it is in static equilibrium. This means that the body is not accelerating in either direction.

Sr.No

Quantity

Linear Motion

Rotational Motion

1

Force

F =ma

$\tau=r\cdot f=r F \sin \theta$

2

Acceleration

$a=\dfrac{d v}{d t}$

$\alpha=\dfrac{d \omega}{d t}$

3

Velocity

$v=\dfrac{d r}{d t}$

$\omega=\dfrac{d \theta}{d t}$

4

Work

$W=\int F \cdot d r$

$W=\int \tau \cdot d\theta$

5

Power

P=Fv

$P=\tau \omega$

6

Mass

M

I (Moment of Inertia)

7

Linear Momentum

p=Mv

$L=I \omega$

8

Kinetic Energy

$\dfrac{1}{2}Mv^2$

$\dfrac{1}{2}l\omega^2$

9

Displacement

dr

dθ

Sl. No.

Name of Concept

Formula

1.

Moment of Inertia of circular-shaped bodies

• Ring: $I= M {R^2}$

• Disc : $I=\dfrac { M{R^2}}{2}$

• Solid sphere: $I=\dfrac {2}{5} M {R^2}$

• Spherical shell: $I=\dfrac {2}{3} M{R^2}$

2.

Moment of inertia of cylindrical bodies

• Hallow cylinder: $I= M {R^2}$

• Solid cylinder: $I=\dfrac { M{R^2}}{2}$

• Thick-walled cylinder: $I=\dfrac {M \left ({R_1}^2+{R_2}^2\right )}{2}$

• Solid cylinder of length l: 

$I=\dfrac {M{R^2}}{4} +\dfrac {M {l^2}}{12}$

3.

Moment of inertia of a thin rod

• Thin rod of length l: 

$I=\dfrac {M{l^2}}{12}$

• Special case:

If axis of rotation is Perpendicular to rod at one end.

$I=\dfrac {M{l^2}}{3}$

4.

Moment of inertia of a rectangular sheet

$I=\dfrac {M \left ({l^2}+{b^2}\right )}{2}$

5.

Torque

$\tau =\vec r \times \vec F$

6.

Radius of gyration

$K=\sqrt {\dfrac {I}{M}}$

7.

Rotational work and energy

Rotational work: $W_{rot}=\int \tau d\theta$

Kinetic energy: $k_{rot}=\dfrac {1}{2} I {\omega^2}$

8.

Angular momentum

$L=\vec r \times \vec P$

9.

Angular momentum of a rigid body in combined translational and rotational motion

$\vec L=\vec {L_{cm}}+M\left (  \vec {r_o}+\vec {v_o}\right )$

10.

Angular impulse

$\vec J=\Delta \vec L$

Rotational Kinematics

Linear Kinematics

Linear position is denoted by S

Angular position is denoted by $\theta$

Displacement: $\Delta S= {S_2}- {S_1}$

Angular Displacement: $\Delta {\theta}={\theta_2}-{\theta_1}$

Average velocity: $v_{avg}=\dfrac {\Delta S}{\Delta t}$

Angular average velocity: $\omega_{avg}=\dfrac {\Delta \theta}{\Delta t}$

Instantaneous velocity: $ v_{ins}= \lim_{{\Delta t} \rightarrow 0} \dfrac {\Delta S}{\Delta t}$

Instantaneous angular velocity: $ {\omega_{ins}}=\lim_{{\Delta t} \rightarrow 0} \dfrac {\Delta \theta}{\Delta t}$

Average acceleration: $a_{avg}=\dfrac {\Delta v}{\Delta t}$

Average angular acceleration: ${\alpha_{avg}}=\dfrac {\Delta \omega}{\Delta t}$

Instantaneous acceleration:

$a_{ins}=\lim_{{\Delta t} \rightarrow 0} \dfrac {\Delta v}{\Delta t}$

Instantaneous angular acceleration:

${\alpha_{ins}}=\lim_{{\Delta t} \rightarrow 0} \dfrac {\Delta \omega}{\Delta t}$

Linear equations of motion:

$v={v_o}+at$

${v^2}={{v_o}^2}+2as$

$s={v_o}t+\dfrac {1}{2} a{t^2}$

Rotational equations of motion:

$\omega = {\omega_o}+\alpha t$

${\omega^2}={\omega_o}^2+2 \alpha \theta $

$\theta={\omega_o}t+\dfrac {1}{2} \alpha {t^2}$

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