System of Particles & Rotational Motion

From a single point mass to a spinning rigid body · Class 11 Physics

1The big idea

Until now every object has been a single point — a ball, a stone, a car shrunk to a dot. That works only while the object moves as a whole. But a spinning wheel, a rolling can and a tumbling rod all have parts moving in different directions at the same time. This chapter gives you the two ideas that tame them.

First, any rigid body's motion splits cleanly into two pieces: the translation of one special point — the centre of mass — plus the rotation of everything about that point. Second, rotation obeys equations that mirror the straight-line ones you already know, with each linear quantity swapped for its rotational twin. Master that dictionary and half the chapter is just Newton's laws in new clothing.

2Centre of mass

The centre of mass (CM) is the mass-weighted average position of all the particles in a system — the one point that moves as if the whole mass were concentrated there and all external forces acted there. For particles of mass m₁, m₂, … at positions x₁, x₂, …

The payoff is a theorem you should hold onto: the centre of mass moves exactly as though the total external force were applied to the total mass sitting there. Internal forces — the pushes and pulls between the parts — always cancel in pairs (Newton's third law) and never shift the CM. That is why a wrench flung spinning across a table traces a smooth straight line at its centre of mass while its ends whirl around it, and why an exploding shell's fragments straddle the parabola their unbroken CM would have followed.

3Torque — the turning ability of a force

Where a force changes straight-line motion, a torque changes rotational motion. Torque measures how effectively a force twists a body about an axis. Push a door at the handle and it swings; push equally hard right beside the hinge and nothing happens. Same force, wildly different turning effect — because torque depends on where and in which direction you push.

Here r is the distance from the axis to the point where the force acts, and θ is the angle between r and the force F. The quantity r sin θ is the moment arm — the perpendicular distance from the axis to the line of the force. Only the component of the force perpendicular to r turns the body; a force pointing straight at or away from the axis (θ = 0) gives zero torque, which is why pushing the door toward its hinge does nothing.

Why τ = Iα

Take one particle of mass m at distance r from the axis. A tangential force F gives it a tangential acceleration a_t = r α (from a_t = rα, since the whole body shares one angular acceleration α). Newton's second law for that particle reads F = m a_t = m r α. Multiply both sides by r to turn force into torque:

Add up every particle. The α is common to all of them, so it factors out, leaving τ_net = (Σ m_i r_i²) α. That sum in the brackets is the moment of inertia I — and we have arrived at the rotational Newton's second law:

4Moment of inertia

Moment of inertia I = Σ m_i r_i² is the rotational counterpart of mass: it measures how hard a body resists having its spin changed. But unlike mass it is not a fixed number for a body — it depends on the axis, because r is measured from that axis. The same rod is far easier to spin about its centre than about one end, because about the end its mass sits, on average, farther out. Mass placed far from the axis counts much more heavily — it enters as r².

You are expected to know these standard results cold. Each is written as some fraction of MR²; that fraction, called k²/R² (where k is the radius of gyration), is the single number that decides who wins a rolling race later on.

Notice the pattern: the ring has all its mass at the rim (MR², the largest); the solid sphere has most of its mass tucked in near the centre (⅖MR², the smallest). The hollow sphere (⅔MR²) sits between the ring and the disc because its shell is spread over the surface. Same mass, same radius — but the distribution changes I by more than a factor of two.

5Parallel & perpendicular axis theorems

You only ever memorise moments of inertia about an axis through the centre. Two theorems shift you to any other axis you need.

Parallel-axis theorem

To find I about an axis parallel to one through the centre of mass, and a distance d away, simply add Md²:

For a rod about its end, take the centre value ML²/12 and shift by d = L/2: I = ML²12 + M(L2)² = ML²3. Because Md² is never negative, the CM axis always gives the smallest possible moment of inertia — a body is easiest to spin about its centre.

Perpendicular-axis theorem (flat bodies only)

For a planar (laminar) body lying in the xy-plane, the moment of inertia about the axis sticking out of the plane equals the sum of the two in-plane values:

For a disc about a diameter, symmetry gives I_x = I_y, and the axis through the centre is I_z = ½MR². So 2 I_diameter = ½MR², giving I_diameter = ¼MR² in one line — no integration needed.

6Angular momentum & its conservation

Angular momentum is the rotational version of ordinary momentum. For a rigid body spinning about a fixed axis,

Just as F = dp/dt, torque is the rate of change of angular momentum: τ = dL/dt. The consequence is the chapter's most beautiful result — the law of conservation of angular momentum:

This is the spinning skater. A skater turning with arms outstretched has a large I and spins slowly. When she pulls her arms in, she brings mass close to the axis, so I drops sharply. With no external torque, L = Iω cannot change — so as I falls, ω shoots up and she whirls faster. She did no work to speed her rotation up in the usual sense; she simply redistributed her own mass. A diver curling into a tuck to somersault faster is doing exactly the same thing.

7Rolling without slipping

A wheel that rolls smoothly, without skidding, satisfies the rolling condition v = ω R: the speed of the centre equals the rim speed. The point of contact is instantaneously at rest against the ground — it neither slips forward nor back. Such a body carries energy in two forms at once, translation and rotation, and the split is the key to everything that follows.

The kinetic energy of a rolling body

Add the translational kinetic energy of the centre of mass to the rotational kinetic energy about the centre. Write I = M k² and use ω = v/R:

The bracket (1 + k²/R²) is the extra "tax" that rotation levies on the energy. The larger a body's moment of inertia (larger k²/R²), the more of its energy is locked up in spin rather than in racing forward.

Acceleration down an incline

Release such a body from rest and let it roll a distance s down a slope of angle θ, dropping a height h = s sin θ. Energy is conserved (the friction that makes it roll does no work, since the contact point does not slide), so the lost gravitational energy becomes rolling kinetic energy:

Since v² = 2 a s for constant acceleration, compare the two and the 2s cancels, leaving the result you must know:

Read what this says. The acceleration does not depend on the mass or the radius — only on the shape factor k²/R². A small marble and a boulder of the same material roll down neck-and-neck. And the smaller that factor, the larger the acceleration: the solid sphere (⅖) beats the disc (½), which beats the ring (1). Every solid sphere beats every ring, whatever their sizes.

8Common confusions to clear up

• Moment of inertia is not a fixed property of a body. It changes with the axis you choose. Always ask "about which axis?" before quoting a value.
• The centre of mass need not lie inside the body. For a ring, a horseshoe or a boomerang it sits in empty space — it is a mathematical average, not a material point.
• Torque is not force, and angular momentum is not momentum. A large force close to the axis can give less torque than a small force far out. Location and direction matter as much as size.
• Rolling friction does no work. In rolling without slipping the contact point is momentarily at rest, so static friction — though essential to cause the rolling — transfers no energy. That is why we can use energy conservation on the incline.
• The skater speeds up without any torque. Pulling her arms in changes I, not L. The faster spin comes from conserving angular momentum, not from an outside twist.
• A rolling body is slower than a sliding one. A frictionless sliding block down the same slope accelerates at the full g sin θ, beating every rolling body — because none of its energy is diverted into spin.

9Check yourself

Class 11 Physics · System of Particles & Rotational Motion · aligned to NCERT Chapter 7 · SmartStudy.School