Systems of Particles and Rotational Motion: Comprehensive NEET Physics Notes

1. Introduction

In previous chapters, we focused primarily on the motion of single particles, which we represented as point masses. However, real-world objects have finite sizes, and their motion often involves complex dynamics that cannot be adequately described by treating them as single particles. In this chapter, we explore the motion of systems of particles and extend our understanding to the rotational motion of rigid bodies, which are idealized as having unchanging shapes.

2. Centre of Mass

2.1 Concept of Centre of Mass

The centre of mass (COM) of a system is the point at which the total mass of the system can be considered to be concentrated for the purpose of analyzing translational motion. For a system of particles, the position of the COM is given by:

$X = \frac{\sum m _{i} x _{i}}{\sum m _{i}}$

$Y = \frac{\sum m _{i} y _{i}}{\sum m _{i}}$

$Z = \frac{\sum m _{i} z _{i}}{\sum m _{i}}$

Where $x_{i}$ , $y_{i}$ , $z_{i}$ are the coordinates of the $i$ th particle, and $m_{i}$ is the mass of the $i$ th particle. The COM is crucial in understanding the motion of extended bodies.

Did You Know? The concept of the centre of mass is used extensively in sports, such as gymnastics and diving, where athletes manipulate their bodies around their COM to achieve precise movements.

3. Motion of Centre of Mass

3.1 Translational Motion of COM

The motion of the centre of mass of a system of particles is determined by the external forces acting on the system. According to Newton's second law, the motion of the COM is given by:

$M \frac{d ^{2} R}{d t ^{2}} = F_{ext}$

Where $M$ is the total mass of the system, $R$ is the position vector of the COM, and $F_{ext}$ is the sum of all external forces.

Real-life Application: In rocket propulsion, engineers calculate the motion of the rocket by analyzing the motion of its COM, which shifts as fuel burns and mass is expelled.

4. Rotational Motion and Angular Variables

4.1 Angular Velocity and Its Relation with Linear Velocity

In rotational motion, every particle of a rigid body moves in a circle centered on the axis of rotation. The angular velocity $ω$ is the rate of change of the angular displacement and is related to the linear velocity $v$ by:

$v = ω r$

Where $r$ is the radius of the circular path of the particle. The angular velocity vector points along the axis of rotation, following the right-hand rule.

4.2 Angular Acceleration

Angular acceleration $α$ is the rate of change of angular velocity:

$α = \frac{d ω}{d t}$

This quantity is crucial in analyzing the dynamics of rotational motion.

Mnemonic: "Angular velocity makes the particles swirl, and acceleration makes them twirl."

5. Torque and Angular Momentum

5.1 Torque

Torque $τ$ is the rotational analog of force and is given by the vector product:

$τ = r \times F$

Where $r$ is the position vector from the axis of rotation to the point where the force $F$ is applied. Torque causes angular acceleration in a rigid body.

5.2 Angular Momentum

Angular momentum $L$ is the rotational counterpart of linear momentum and is defined as:

$L = r \times p$

Where $p = m v$ is the linear momentum of a particle. The time rate of change of angular momentum is equal to the torque:

$\frac{d L}{d t} = τ$

Common Misconception: Some students confuse torque with force. Remember that torque depends not just on the magnitude of force but also on where and how the force is applied relative to the axis of rotation.

6. Equilibrium of Rigid Bodies

6.1 Conditions for Equilibrium

A rigid body is in mechanical equilibrium when both the net force and net torque acting on it are zero:

$\sum F = 0 (translational equilibrium)$

$\sum τ = 0 (rotational equilibrium)$

These conditions ensure that the body has no linear or angular acceleration.

NEET Tip: In NEET exams, problems often involve analyzing the equilibrium conditions for rigid bodies, so mastering these principles is crucial.

Quick Recap

Centre of Mass (COM): The point where the total mass of a system is concentrated.
Motion of COM: Governed by external forces acting on the system.
Angular Velocity: Relates to the linear velocity in rotational motion.
Torque: The rotational analog of force, causing angular acceleration.
Equilibrium Conditions: Both net force and net torque must be zero for a rigid body to be in equilibrium.

Concept Connection

Understanding rotational motion and the centre of mass is critical in studying topics in Chemistry (such as molecular dynamics) and Biology (such as the biomechanics of movement).

Practice Questions

Question: Calculate the angular velocity of a wheel of radius 0.5 m rotating at 600 RPM. Solution: $ω = \frac{2 π \times 600}{60} = 62.8, rad/s$
Question: A uniform rod of length 2 m and mass 5 kg is hinged at one end. Calculate the torque required to hold the rod horizontally. Solution: $τ = Force \times Perpendicular distance = (5 \times 9.8) \times 1 = 49, Nm$
Question: Find the angular momentum of a particle of mass 2 kg moving with a velocity of 3 m/s at a perpendicular distance of 0.5 m from the axis of rotation. Solution: $L = m v r = 2 \times 3 \times 0.5 = 3, kg \cdot m^{2} / s$
Question: Determine the centre of mass of a system of three particles located at coordinates (0,0), (2,0), and (1,1) with masses 1 kg, 2 kg, and 3 kg respectively. Solution: $X_{COM} = \frac{1 ( 0 ) + 2 ( 2 ) + 3 ( 1 )}{1 + 2 + 3} = 1.17, m$ $Y_{COM} = \frac{1 ( 0 ) + 2 ( 0 ) + 3 ( 1 )}{1 + 2 + 3} = 0.5, m$
Question: A disc rotates with a constant angular acceleration of 2 rad/s². If its initial angular velocity is 5 rad/s, calculate its angular velocity after 4 seconds. Solution: $ω = ω_{0} + α t = 5 + (2 \times 4) = 13, rad/s$

Glossary

Torque: A measure of the rotational force acting on a body.
Angular Momentum: The rotational equivalent of linear momentum.
Centre of Mass: The point at which the total mass of a system can be considered to be concentrated.
Angular Velocity: The rate of change of angular displacement.