Master Systems of Particles and Rotational Motion Class 11 with Formula Sheet

A rigid body is an idealisation or an approximation in physics that meets the following conditions. 

• All particles maintain fixed distances from each other
• The shape never changes during motion
• Internal forces don't cause deformation

Before moving to rotational motion, it’s better to be clear with the concept of what makes bodies rigid.
Through earlier chapters, you've learned how to treat objects as point particles. These are massless dots with all mass concentrated at a single point. 

This simplification works for projectile motion, circular motion, and basic dynamics. 

But real objects have size, shape, and mass distribution.

That’s the main conceptual jump you have to consider while preparing for exams with rotational motion class 11 notes. 

Types of Motion for Rigid Bodies

Type of Motion	Description	Example
Pure Translation	Every particle of the rigid body has identical velocity	A book sliding across a table
Pure Rotation (Fixed Axis)	A body rotates about a stationary line. Also, know that different particles have different linear velocities	A wheel rotating about its axle
General Rotation (Moving Axis)	Axis moves, but one point remains fixed	A spinning top
Combined Motion	There is simultaneous translation and rotation	A ball rolling down an incline

 

The centre of mass is the balance point of a rigid body system. You can observe this when you try to balance an irregular object on your finger. The point where it balances perfectly is its centre of mass. The common abbreviations for centre of mass are COM and CM. 

Centre of Mass and the Weighted Average Concept

For two particles, 

• If masses are equal, the centre of mass is exactly midway
• If one mass is larger than the other, the centre of mass shifts toward the heavier mass
• Physical intuition: Heavier objects "pull" the CM toward themselves

Centre of Mass Formula

For discrete particles, the centre of mass formula is

R_cm = (m₁r₁ + m₂r₂ + ... + mₙrₙ) / (m₁ + m₂ + ... + mₙ)

For continuous bodies, the centre of mass formula

R_cm = ∫ r dm / ∫ dm

Common Misconceptions Clarified

Misconception	Correct Concept
The centre of mass must be inside the object	The centre of mass can lie outside the object. One example is CM in an L-shaped object.
The centre of mass is the point with the most mass	The centre of mass is the weighted average position of all mass in the object

To understand why there are two formulas for centre of mass, we have to begin to learn what systems of particles are. We discuss in the next section.

In this section, we will see why we should consider systems of particles. 

First, it would be impossible to track the motion of every particle of a real-world object, even if it’s a tiny object like a coin. Every particle of the coin will have unpredictable motions.

It’s ideal, and let’s say, more convenient to treat the entire system as a single entity. That would also make calculations for complex problems solvable. Because we can predict better how the whole object behaves while in motion. 

Here is a bigger example. 

When you see a crowded stadium, the individual people move chaotically. But you observe the crowd as a whole. They would seem to move in predictable patterns. While the individual particles in a system may have complex motions, the system's centre of mass follows simple rules.

Now, which rules do you think apply for motion? 

You guessed it right, Newtonian mechanics!

Newton's Laws for Systems of Particles

We know from Newton’s Second Law that for a single particle, force is mass x acceleration, that is, F = ma 

For a system of particles

F_external = M_total × A_cm

The core insight

Internal forces between particles cancel out completely due to Newton's third law. 

Only external forces affect the system's overall motion.

Apart from this, we also know that a rigid body or a system of particles will have a centre of mass. That means the centre of mass will also have motion. 

Motion of Centre of Mass 

If you read motion of centre of mass, you might have come to the conclusion that the centre of mass of a system moves as if:

1. All the system's mass were concentrated at the centre of mass
2. All external forces were applied directly at the CM

Mathematical expression

M_total × A_cm = Σ F_external

Using this, we can easily analyse a flying hammer by tracking just one point (its CM) instead of every atom in the hammer.

Linear Momentum of Systems of Particles

So, why does system momentum matter?

Individual particles in a system might have complex motions. But the system's total momentum follows simple and logical rules. This is the power of thinking in terms of systems rather than individual particles. This is the crux of linear momentum. 

Rule for Linear Momentum

P_system = M_total × V_cm

This means:

• Total momentum = (total mass) × (CM velocity)
• You don't need to track every particle individually
• The motion of a centre of mass tells you everything about the system's momentum

Internal vs External Forces in Systems of Particles

 

In a system of particles, there are two types of forces to consider. 

 

Aspect	Internal Forces	External Forces
Definition	Act between particles within the system	Act on the system from outside
Newton's Third Law	Always appear in equal and opposite pairs within the system	Reaction forces lie outside the system
Effect on System Motion	Cancel out and do not affect the system's overall motion	Directly influence the system's overall motion
Examples	Gravitational forces between planets, collision forces between balls	Gravity from Earth, friction from ground, applied forces like a push or pull

Conservation of Momentum

While you learn linear momentum, you should also consider the condition when momentum is conserved. 

• External force = 0 (or negligible)
• Even if individual forces are large, if they're internal, they cancel out

Practical examples of momentum conservation

• Explosion problems (internal forces, external forces negligible)
• Collision problems (short time, external forces negligible). Refer to the Collision section from your previous chapter. 
• Rocket propulsion (internal combustion, external forces negligible)

Real-World Applications for Systems of Particles

Projectile explosion: When a projectile explodes midair, the fragments scatter in all directions, but the centre of mass continues on the original parabolic path.

Recoil problems: When a gun fires, the bullet goes forward and the gun recoils backwards. Total momentum of the gun-bullet system remains zero.

Human walking: When you walk, you push backwards on the ground (internal force on Earth-you system), and the ground pushes forward on you. For the Earth-you system, these are internal forces.

Exam Problem-Solving Framework for Systems of Particles

• Step 1: Identify the system clearly 
• Step 2: Classify forces as internal or external 
• Step 3: Check if external forces are negligible 
• Step 4: Apply conservation: P_initial = P_final 
• Step 5: Use P = MV_cm to find the motion of a centre of mass

Advanced System of Particles Concepts

A few things to remember when you are planning to prepare for JEE Mains. Find some of the modified formulas, when there are external forces at play or when subsystems have different centres of mass. 

Variable mass systems: Rockets, raindrops, etc.

• Mass of system changes with time
• Requires modified equation: F_external = d(Mv)/dt

Multi-body systems: Solar system, atomic structure

• Each subsystem has its own CM
• Total CM depends on all subsystems

Earlier in this overview on the system of particles, we came across rigid bodies. We also learnt about linear momentum of the system of particles. Most of the knowledge till here has been about linear motion, where the system of particles moves as a whole. 

What we should consider here is that the same system of particles can also rotate. Here we need to learn about angular velocity, torque, and angular momentum. 

How is rotational motion connected with systems of particles?

Everything we learned about systems (CM motion, momentum conservation) still applies. But now we need additional tools to describe rotational aspects.

To recap on our current notion of a rigid body. It is a system of particles where:

• Inter-particle distances remain constant
• The shape never changes
• We can treat it as a single entity for rotational analysis

Angular velocity (ω) tells you how fast something rotates. But unlike linear velocity, it's the same for all particles in a rigid body. Linear velocities, on the other hand, vary with distance from the axis. Until this chapter 6 Physics Class 11, you have learned about instantaneous velocity, where the motion is mostly linear and it’s a single point particle. 

Formula for Angular Velocity 

v = ωr

• Particles farther from axis move faster linearly
• All particles complete one rotation in the same time

Vector Nature of Angular Velocity

Angular velocity is a vector quantity pointing along the rotation axis. The key mathematical relation to remember here is the cross product of two vectors. The cross product of two vectors a and b produces a new vector c. This new vector c is perpendicular to both original vectors, with magnitude |a||b|sin(θ). Here θ is the angle between them.

The direction follows the right-hand rule.

• Curl fingers in the direction of rotation
• Thumb points in direction of ω vector

Considering the vector product of two vectors when using the right-hand rule

• Point fingers along a
• Curl toward b, and 
• Your thumb points in the direction of a × b.

Angular Acceleration: The Rate of Change

α = dω/dt

Just like linear kinematics, rotational kinematics has similar equations:

• ω = ω₀ + αt
• θ = θ₀ + ω₀t + ½αt²
• ω² = ω₀² + 2α(θ - θ₀)

Problem-Solving Strategy for Exams

For uniform angular acceleration problems:

1. Identify given quantities
2. Choose an appropriate and equivalent kinematic equation for rotational dynamics
3. Convert units carefully (rpm to rad/s is common)
4. Solve systematically

Just as force changes linear momentum, torque changes angular momentum. 

For your exams that ask you questions on torque, just remember

For a system of particles, we need to consider torques about the same point for all particles.

Considering angular momentum for a system of particles

• Each particle has its angular momentum: L_i = r_i × p_i
• Total angular momentum: L_total = Σ L_i
• Rate of change of total angular momentum = Total external torque

Why Internal Torques Cancel

Internal torques between particles cancel out (just like internal forces) when we consider the system as a whole.

Mathematical proof: If particle i exerts force F_ij on particle j, then:

• Torque on j due to i: τ_ji = r_j × F_ij
• Torque on i due to j: τ_ij = r_i × F_ji = r_i × (-F_ij)
• If forces act along the line joining particles, these torques cancel

The System Equation for Angular Momentum

dL_total/dt = τ_external

This is the rotational analogue of Newton's second law for systems, just as F_external = dP/dt was the translational analogue.

Torque (τ) is the rotational equivalent of force. It's what makes objects start rotating, stop rotating, or change their rate of rotation.

Torque in physics measures the "turning effect" of a force about a point.

Torque Formula

τ = r × F

Key components in Torque

• r: Position vector from axis to point of force application
• F: Applied force
• ×: Cross product (perpendicular components matter)

Maximising Torque: Practical Insights

Torque is maximised when

1. Force is large
2. Distance from axis is large
3. Force is perpendicular to position vectors