System of Particles and Rotational Motion - Topics, Characteristics, Notes, Books, FAQs
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.
This Story also Contains
- Rotational Motion Topics Class 11
- Overview of Rotational Motion
- System Of Particles And Rotational Motion Important Topics
- Comparison Between Translational And Rotational Motion
- Equilibrium Of Rigid Bodies
- Rigid Body Rotation
- Equations Of Rotational Motion
- Moment Of Inertia
- Theorem Related To Moment Of Inertia
- Values Of $I$ For Simple Geometrical Objects
- How to start preparing for rotational motion?
- System of Particles and Rotational Motion Important Formulas
- Books for Rotational Motion
- Summary
Rotational Motion Topics Class 11
Topics of rotational motion class 11 are given below:
- Centre of Mass
- Motion of Centre of Mass
- Linear Momentum of a System of Particles
- Vector Product of Two Vectors
- Angular Velocity and Its Relation with Linear Velocity
- Torque and Angular Momentum
- Equilibrium of a Rigid Body
- Moment of Inertia
- Kinematics of Rotational Motion About a Fixed Axis
- Dynamics of Rotational Motion About a Fixed Axis
- Angular Momentum in Case of Rotation About a Fixed Axis
Overview of Rotational Motion
Rotational motion refers to the movement of a body around a fixed axis. Unlike linear motion where an object moves in a straight line rotational motion involves circular paths of particles around an axis. Every point in the body follows a circular trajectory, and all points complete one rotation in the same time. This type of motion is seen in many real-life objects such as wheels, fans, planets, and rotating machinery.
To describe rotational motion clearly, physical quantities like angular displacement, angular velocity, angular acceleration, moment of inertia, torque, and angular momentum are used. Understanding rotational motion helps students relate linear concepts to rotational systems and forms the foundation for studying rigid body dynamics in physics.
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,.....mn whose position vectors are given by r1,r2,....rn 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 miri is the moment of the particle with mass mi.
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 :
1. 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.
System of Particles and Rotational Motion Important Formulas
Centre of Mass
The centre of mass (CM) of a system is the point where the entire mass of the system appears to be concentrated.
For two particles:
$\vec{R}=\frac{m_1 \overrightarrow{r_1}+m_2 \overrightarrow{r_2}}{m_1+m_2}$
Motion of CM represents the overall motion of the system.
Motion of Centre of Mass
- The CM of a system moves as if all external forces acted at that point.
- Total external force $=$ total mass × acceleration of CM.
- Internal forces do not affect the motion of CM.
Linear Momentum of a System of Particles
Total linear momentum = vector sum of momenta of all particles.
For a system:
$\vec{P}_{\text {total }}=M \vec{v}_{C M}$
Conserved when net external force is zero.
Vector Product (Cross Product)
Defined as:
$\vec{A} \times \vec{B}=A B \sin \theta \hat{n}$
- Used to define torque and angular momentum.
- Direction given by Right-Hand Rule.
Angular Velocity and Its Relation with Linear Velocity
- Angular velocity $(\omega)$ is the rate of change of angular displacement.
- Linear velocity (v) of a rotating body:
$v=\omega r$ - Every point on a rigid rotating body has same $\omega$ but different v.
Torque and Angular Momentum
Torque -Turning effect of a force.
$\vec{\tau}=\vec{r} \times \vec{F}$
Angular Momentum
For a particle:
$\vec{L}=\vec{r} \times \vec{p}$
Related to torque:
$\vec{\tau}=\frac{d \vec{L}}{d t}$
Equilibrium of a Rigid Body
A rigid body is in equilibrium when:
1. Net external force $=0$
2. Net external torque $=0$
Two types:
Translational equilibrium
Rotational equilibrium
Moment of Inertia
Rotational inertia of a body.
$I=\sum m_i r_i^2$
- Depends on mass distribution and axis of rotation.
- Greater distance from axis → larger I.
Parallel Axis Theorem:
$I=I_{C M}+M d^2$
Perpendicular Axis Theorem:
$I_z=I_x+I_y$
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)
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.
A rigid body is a body that can rotate with all the parts locked together and without any change in its shape.
A top spinning is an example of rotational motion.
Ocean currents, cyclones, and tornadoes are examples of rotational motion.
kg⋅m2 is the unit of inertia.