System of Particles and Rotational Motion

Chapter 7: System of Particles and Rotational Motion

This chapter extends mechanics from single particles to systems of particles and rigid bodies undergoing rotation. Rotational motion has analogues for every quantity in linear motion.

Centre of Mass (COM)

The centre of mass is the point where all the mass of a system can be considered to be concentrated for translational analysis.

For two particles:
x_COM = (m1 x1 + m2 x2) / (m1 + m2)

For an extended body, COM is found by integrating over all mass elements.

Key result: The net external force on a system equals (total mass) × (acceleration of COM).
F_ext = M a_COM

This means the COM moves as if all mass were concentrated there and all external forces acted on it.

Rotational Kinematics

The analogues of linear quantities in rotation:
Angular displacement: theta (radians)
Angular velocity: omega = d(theta)/dt (rad/s)
Angular acceleration: alpha = d(omega)/dt (rad/s²)

1.Equations of rotational motion (constant alpha):
2.omega = omega₀ + alpha t
3.theta = omega₀ t + (1/2) alpha t²
4.omega² = omega₀² + 2 alpha theta

Relation between linear and angular quantities: v = r omega, a_t = r alpha, a_c = omega² r

Torque

Torque (tau) is the rotational equivalent of force:
tau = r × F = r F sin(theta)

where r is the perpendicular distance from the axis to the line of action of force.

SI unit: N m (same dimensions as energy, but a different physical quantity).

Condition for rotational equilibrium: net torque = 0

Moment of Inertia

Moment of inertia (I) is the rotational analogue of mass — it measures resistance to angular acceleration.

I = Sigma m_i r_i² (sum over all mass elements)

Common values:
Solid disc: I = (1/2) M R²
Ring/hollow cylinder: I = M R²
Solid sphere: I = (2/5) M R²
Thin rod (about centre): I = (1/12) M L²
Thin rod (about end): I = (1/3) M L²

Parallel Axis Theorem: I = I_COM + M d² (d = distance between the two parallel axes)
Perpendicular Axis Theorem (planar bodies only): I_z = I_x + I_y

Rotational Dynamics

Newton's second law for rotation: tau_net = I alpha

Angular momentum: L = I omega (vector, along axis of rotation)
Conservation of angular momentum: If net external torque is zero, L = I omega = constant.
(Example: a spinning skater pulls in arms → I decreases → omega increases to keep L constant)

Rotational Kinetic Energy

KE_rot = (1/2) I omega²

For rolling without slipping: KE_total = (1/2) M v² + (1/2) I omega²

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Example 1

Find the COM of a system: 2 kg at x = 0 and 3 kg at x = 5 m.
x_COM = (2×0 + 3×5)/(2+3) = 15/5 = 3 m from the 2 kg mass.

Example 2

A wheel starts from rest with angular acceleration 2 rad/s². Find angular velocity after 5 s.
omega = omega₀ + alpha t = 0 + 2 × 5 = 10 rad/s

Example 3

A force of 10 N acts at distance 0.5 m from the axis at 90°. Find torque.
tau = r F sin(90°) = 0.5 × 10 × 1 = 5 N m

Example 4

Find moment of inertia of a solid disc (M = 4 kg, R = 0.5 m) about its axis.
I = (1/2) M R² = (1/2)(4)(0.25) = 0.5 kg m²

Example 5

A skater with I = 5 kg m² spins at 2 rad/s. She pulls in arms so I becomes 2 kg m². New angular velocity?
L = I1 omega1 = I2 omega2 → 5 × 2 = 2 × omega2 → omega2 = 5 rad/s

Example 6

Find angular acceleration of a disc (I = 0.5 kg m²) when net torque = 3 N m.
alpha = tau/I = 3/0.5 = 6 rad/s²

Example 7

A solid cylinder rolls down a slope. Fraction of total KE in rotation?
KE_rot = (1/2)I omega² = (1/2)(1/2 MR²)(v/R)² = (1/4)Mv²
KE_total = (1/2)Mv² + (1/4)Mv² = (3/4)Mv²
Fraction = (1/4)/(3/4) = 1/3

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Common mistakes

Confusing torque (N m) with work/energy (also J = N m) — they have same units but are very different quantities.
Using I = MR² for a solid disc — that is the formula for a ring. Solid disc: I = (1/2)MR².
Forgetting that conservation of angular momentum applies only when net external torque is zero.

Summary

The COM of a system moves as if all external forces act on a single particle at that point. Rotational motion has perfect analogues to linear motion: omega ↔ v, alpha ↔ a, I ↔ m, tau ↔ F, L ↔ p. The parallel axis theorem extends moment of inertia calculations. Angular momentum is conserved when net external torque is zero.