System of Particles and Rotational Motion

Topic-by-topic explanations

1. Introduction

A system of particles can be treated using its centre of mass for overall translation and separate coordinates for internal motion.

2. Centre of mass

The centre of mass is the mass-weighted average position: r_cm=Σmᵢrᵢ/Σmᵢ. It can lie outside the material of an object.

3. Motion of centre of mass

Total momentum P=Σmᵢvᵢ=Mv_cm. The centre of mass obeys M a_cm=F_ext; internal forces cancel in pairs for the system.

4. Linear momentum of a system of particles

Total linear momentum is the vector sum of the momentum of all particles. If net external force is zero, momentum is constant.

5. Vector product of two vectors

The vector product A×B has magnitude AB sinθ and direction given by the right-hand rule. It appears in torque and angular momentum.

6. Angular velocity and its relation with linear velocity

Angular velocity is rate of angular displacement; for circular motion v=rω. Tangential acceleration is rα, and centripetal acceleration is rω².

7. Torque and angular momentum

Torque τ=r×F has magnitude rF sinθ. For fixed-axis rotation of constant I, τ=Iα. Angular momentum of a rigid body about the axis is L=Iω.

8. Equilibrium of a rigid body

Static rigid-body equilibrium requires both ΣF=0 and Στ=0. Choose a pivot that eliminates unknown forces from the torque equation when possible.

9. Moment of inertia

Moment of inertia I=Σmᵢrᵢ² measures rotational inertia about a specified axis. It depends on the axis and mass distribution. Parallel-axis theorem: I=I_cm+Md².

10. Kinematics of rotational motion about a fixed axis

For constant angular acceleration: ω=ω₀+αt, θ=ω₀t+½αt² and ω²=ω₀²+2αθ, with angular displacement in radians.

11. Dynamics of rotational motion about a fixed axis

Rotational dynamics links net torque to angular acceleration; rotational kinetic energy is ½Iω². For rolling without slipping, v_cm=Rω and total kinetic energy includes translation and rotation.

12. Angular momentum in case of rotations about a fixed axis

If net external torque about an axis is zero, angular momentum is conserved. Pulling mass closer to the axis decreases I and increases angular speed if L stays fixed.

Formula sheet

Use consistent SI units, identify the system and assumptions, and show the steps of each calculation. Check units and whether the result is physically reasonable.

Worked example

Common errors and revision checklist

Moment of inertia depends on axis. Torque uses the perpendicular component of force. Rigid-body equilibrium needs both force and torque balance.

• Write the relevant law or relation before substituting values.
• Show units, signs and directions.
• Check assumptions and round only the final result.