System of Particles and Rotational Motion - Topics, Characteristics, Notes, Books, FAQs

Edited By Vishal kumar | Updated on Sep 26, 2024 01:45 PM IST

Rotational motion is an integral part of mechanics and the questions asked from this chapter are most difficult. So before starting to solve questions from this chapter, you should have a good grasp of concepts from the topics of laws of motion, center of mass, and linear momentum. The questions that are asked from this topic involve a conceptual approach rather than a formula-based approach. You must read the theory thoroughly before solving the questions. This article will help you understand the system of particles and rotational motion topics and how these concepts make complex questions when mixed with Newton's basic laws.

Rotational Motion Topics Class 11

Important topics of rotational motion class 11 are given below:

Equilibrium of rigid bodies
Rigid body rotation and equations of motion
Comparison of linear and rotational motion
Moment of inertia
Radius of gyration
Values of moments of inertia for simple geometrical objects (no derivation).

Rotational Motion Important Topics

Moment of Inertia, Angular Velocity, Torque, Rotation of a rigid body about a fixed axis, Angular Momentum, Conservation of Angular Momentum, Combined Translational and Rotational Motion of a Rigid Body, Uniform Pure Rolling, Instantaneous, Axis of Rotation, Accelerated Pure Rolling, Angular Impulse, Toppling.

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Overview of Rotational Motion

If a body is pivoted at a point and the force is applied on the body at a suitable point, it rotates the body about the axis passing through the pivoted point. This is the turning effect of the force and the motion of the body is called the Rotational motion. In our day-to-day life, we come across many objects that show rotational motion, the spinning of car wheels, the rotation of washing machine agitators, the rotation of the earth, etc.

Before you proceed further in this chapter, you should know what is a rigid body, a rigid body is made of too many particles but the distance between any two particles is always constant. Suppose you remember the concepts of Translatory motion. In that case, we consider the rigid body as the point object in which we take the same linear displacement, linear velocity as well as same linear acceleration of all the particles on the rigid body. But in the case of rotational motion or translatory plus rotational motion, the particles of the rigid body have different linear displacement, velocity, and acceleration.

For most of the students, rotational motion is often the pain in the neck, because it introduces many new terminologies like Moment of Inertia, Torque, and angular momentum, etc. But it is not difficult, because you can relate these terminologies with your previous knowledge of translatory motion. For example just like the “center of mass”, the moment of inertia is also the property of the object that is related to its mass distribution. The same goes with Torque, you can relate it with force.

System Of Particles And Rotational Motion Important Topics

The Motion Of A Rigid Body

When the particle is rotating or rotating plus translating the different particles P1, P2, P3, and P4 have different linear displacement, velocity, and acceleration.

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Characteristics of Rotational Motion

In rotational motion, the particles of the rigid body follow a circular path around the rotational axis, the rotational axis could be fixed or it could be unfixed. An example of fixed-axis rotational motion is the rotation of a fan, in which each particle on the blade follows a circular path around the axle of the motor of the fan. An example of an unfixed axis of rotational motion is the spinning top, in the spinning top, the tip of the top is an unfixed axis around which all the particles are following a circular path.

Centre Of Mass

Centre of mass of a body or system is a point where the whole mass of the body or system is supposed to be concentrated, and forces are directly applied to this point of translational motion.

Consider a system of particles of masses m₁, m₂,.....m_n whose position vectors are given by r₁,r₂,....r_n respectively. but $\sum_{i=1}^n m_i = M$, mass of system then

$R_{c m}=\frac{\sum_{i=1}^n m_i r_i}{M}$

where m_ir_iis the moment of the particle with mass m_i.

The center of masses of some regular rigid bodies are given in the below table:

• Rigid Body	• Centre Of Mass
• Rod lying along the x-axis	• (L/2, 0, 0)
• Semicircular ring	• $\frac{2R}{\pi}$
• Semicircular disc	• $\frac{4R}{3\pi}$
• Solid hemisphere	• $\frac{3R}{8}$
• Hollow hemisphere	• $\frac{R}{2}$
• Solid sphere	• At its centre
• Square or rectangle	• At its centre
• Solid cone	• $\frac{H}{4} $ from base of the cone
• Hollow cone	• $\frac{H}{3}$ from base of the cone

Comparison Between Translational And Rotational Motion

Feature	Translational Motion	Rotational Motion
Centre of mass	• The motion of a rigid body includes the motion of its center of mass.	• A rigid body can also move while its center of mass is fixed
Defnition	• Movement of an object along a straight line or curved path	• Movement of an object around an axis
Parameters	• Displacement, velocity, acceleration	• Angular displacement, angular velocity, angular acceleration
Force	• Linear force is applied to change motion	• Torque is applied to cause motion
Inertia	• Mass ($m$) resists changes	• Moment of inertia ($I$) resists changes
Kinetic energy	• $K E$ $=\frac{1}{2} m v^2$	• $K E$$=\frac{1}{2} \omega^2$
Example	• A ball rolling in a straight line	• A spinning top

Equilibrium Of Rigid Bodies

The equilibrium of a rigid body is a state where a rigid body is not changing its linear momentum or angular momentum, meaning no net force or net torque is acting on it. It is one of the important topics in the system of particles and rotational motion.

Types Of Equilibrium

Static Equilibrium: A rigid body is in static equilibrium when it is at rest and remains at rest.

For example, a book lying on a table

Dynamic Equilibrium: A rigid body is in dynamic equilibrium when it moves with a constant velocity(not accelerating).

For example, a satellite in orbit around the Earth

Rigid Body Rotation

When a rigid body rotates, all points in the body move in circular paths around a fixed axis.

1. Axis of rotation

It is the line about which the rigid body rotates.

2. Angular Displacement ($\theta$)

The angle through which a point or line has been rotated in a specified sense about a specified axis

3. Angular Velocity

It is the rate of change of angular displacement.

$$\omega \equiv \frac{d \theta}{d t}$$

4. Angular Acceleration

The rate of change of angular velocity is called angular acceleration

$$\alpha=\frac{\mathrm{d} \omega}{\mathrm{dt}}$$

Equations Of Rotational Motion

The three equations of rotational motion :

First Equation

$$\omega=\omega_0+\alpha t$$

2. Second Equation

$$\Delta \theta=\omega_0 t+\frac{1}{2} \alpha t^2$$

3. Third Equation

$$\omega^2=\omega_0^2+2 \alpha \Delta \theta$$

where,

$\omega_0$ = initial angular velocity
$t$ = time
$\theta$ = angular displacement
$\alpha$= angular acceleration
$\omega$ = final angular velocity

Relation between linear and angular velocities, $v=r \omega$
Relation between linear and angular acceleration, $a=\alpha \mathrm{r}$

Moment Of Inertia

It is the measure of of body's resistance to angular acceleration about a given axis.

For a system of point masses, the moment of inertia is:

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

where,

$m_i$ = mass of each point masses
$r_i$ = distance of each mass from the axis of rotation

For a continuous body,

$$I=\int r^2 d m$$

where,

$dm$ = infinitesimal mass element
$r$ = distance from the axis of rotation

Theorem Related To Moment Of Inertia

1. Parallel Axis Theorem

The parallel axis theorem states that the moment of inertia of a body about an axis parallel to an axis passing through the center of mass is equal to the sum of the moment of inertia of a body about an axis passing through the center of mass and the product of mass and square of the distance between the two axes.

$$I_{\|}=I_{c m}+m d^2$$

where,

d is the distance between the two axes

2. Perpendicular Axes Theorem

The moment of inertia about an axis perpendicular to the plane is equal to the sum of moments of inertia about two perpendicular axes in the plane.

$$I_z=I_x+I_y$$

Radius Of Gyration

The radius of gyration is the distance from an axis of rotation where the entire mass of a body is assumed to be concentrated, such that the moment of inertia about that axis remains unchanged.

$$k=\sqrt{\frac{I}{m}}$$

where,

$k$= radius of gyration
$I$ = moment of inertia about the given axis
$m$= total mass the of body

Values Of $I$ For Simple Geometrical Objects

The values of some important geometrical objects are given in the table:

Geometrical Objects	Value of moment of inertia
• Hollow Cylinder Thin-walled	• $I=M r^2$
• Thin Ring	• $I=\frac{1}{2} \mathrm{Mr}^2$
• Hollow Cylinder	• $I=\frac{1}{2}M\left(r_2^2+r_1^2\right)$
• Solid Cylinder	• $\mathrm{I}=\frac{1}{2} \mathrm{Mr}^2$
• Uniform Disc	• $\mathrm{I}=\frac{1}{4} \mathrm{Mr}^2$
• Hollow Sphere	• $\mathrm{I}=\frac{2}{3}\mathrm{Mr}^2$
• Solid Sphere	• $\mathrm{I}=\frac{2}{5} \mathrm{Mr}^2$
• Spherical Shell	• $I=\frac{2}{3} {Mr}^2$
• Thin rod (at the center)	• $\mathrm{I}=\frac{1}{12} \mathrm{Mr}^2$
• Thin rod ( at the end of the rod)	• $\mathrm{I}=\frac{1}{3}\mathrm{Mr}^2$

How to start preparing for rotational motion?

First of all, learn how to calculate the moment of inertia (MOI) of different objects around different axes. You should also get familiar with using two theorems on MOI, i.e. Theorem of Parallel axes and the Theorem of Perpendicular axes. Then only you should go towards calculating other parameters required in the problem. Rotational motion involves a mixture of concepts, hence You need to practice a lot more on this topic. For learning concepts through question-solving students, students can get help from our Entrance360 experts. This platform is best for learning because learning concepts through problem-solving helps in building concepts rather than solving all questions after reading a full chapter. This can create doubts while solving hence it will result in more confusion and the students might get trapped.

Rotational Motion Formulas

- $I=\int r^2 d m$

- $\sum=r \times F$

- $K \cdot E \cdot=\frac{1}{2} I w^2$

- $L=r \times P$

Notes for Rotational Motion

Make a plan to prepare for the chapter and Stick to a Timetable. You can make a timetable according to the available time left for preparation and try to prepare according to it.
Don’t try to memorize MOI of different objects, rather first calculate it yourself and then memorize it.
Read carefully the examples given in the NCERT book. Start solving the questions only after you understand all the examples.
Give mock tests chapter-wise from time to time, this will ensure you have a good hold on concepts.
Look at the solution/answer only after giving a good number of attempts.

NCERT Notes Subject Wise Link:

Books for Rotational Motion

For understanding concepts, students can consider NCERT books. But for question-solving students should consider Understanding Physics by D. C. Pandey (Arihant Publications).

You don't have to study the whole book to understand the concept from this chapter because we will provide you the exact page number and line number of these books where you will get these concepts to read.

NCERT Solutions Subject-wise link:

Summary

The study of systems of particles and rotational motion focuses on how groups of particles interact and move. We discussed the system of particles and rotational motion important topics, rotational motion topics class 11 like torque, the moment of inertia, angular momentum, the radius of gyration, and the value of the moment of inertia of simple geometrical objects in this article.

NCERT Exemplar Solutions Subject-wise link:

Frequently Asked Questions (FAQs)

1. What is a rotational motion?

Rotational motion is the motion of an object that revolves around a fixed axis, characterized by the rotation of its mass at various distances from that axis.

2. What is called a rigid body?

A rigid body is a body that can rotate with all the parts locked together and without any change in its shape.

3. What is a rotational motion example?

A top spinning is an example of rotational motion.

4. What is the application of rotational motion?

Ocean currents, cyclones, and tornadoes are examples of rotational motion.

5. What is the unit of inertia?

$\mathrm{kg} \cdot \mathrm{m}^2$ is the unit of inertia.