System of Particles and Rotational Motion: common doubts, answered
The questions students ask most often about System of Particles and Rotational Motion, each with a short answer. For the full chapter, read the System of Particles and Rotational Motion notes.
Rigid bodies and their motion
Read this section in the notes →What is a rigid body and does it really exist?
A rigid body is an ideal object in which the distance between every pair of particles stays fixed whatever forces act. No real body is perfectly rigid, since everything deforms slightly under load. The idea is still useful because, for wheels, rods and tops, the deformation is too small to affect the motion, so the body can be treated as rigid.
What is the difference between translation and rotation of a rigid body?
In pure translation every particle has the same velocity at a given instant, so the body slides without turning. In rotation about a fixed axis every particle moves in a circle centred on that axis, and particles farther from the axis move faster. A cylinder rolling down an incline combines both: its centre moves forward while the cylinder turns about it.
Centre of mass
Read this section in the notes →Can the centre of mass lie outside the body?
Yes. The centre of mass is a calculated point, not a piece of material, so it can lie in empty space. For a ring it is at the centre, where there is no mass at all, and for a boomerang it lies between the arms. It must lie inside the body only for shapes that are filled and convex, such as a solid sphere.
How do you find the centre of mass of two particles?
Take a mass-weighted average of their positions: X = (m₁x₁ + m₂x₂)/(m₁ + m₂). The centre of mass lies on the line joining them, closer to the heavier particle, and divides that line in the inverse ratio of the masses. For equal masses it is exactly at the midpoint.
Motion of the centre of mass
Read this section in the notes →Does an explosion change the path of the centre of mass?
No. An explosion involves only internal forces, which cancel in pairs and cannot change the motion of the centre of mass. A shell that bursts in mid-air has its centre of mass continue along the same parabola it was following, at least until a fragment hits the ground and an external force acts.
Why do internal forces not affect the motion of the centre of mass?
Because internal forces between particles come in action-reaction pairs that are equal and opposite, so they add to zero when summed over the whole system. Only external forces remain, giving M A = F_ext. The centre of mass therefore moves like a single particle carrying all the mass, acted on by the external force alone.
Vector product of two vectors
Read this section in the notes →Is the cross product commutative?
No. Swapping the order reverses the direction: a × b = −b × a, though the magnitude ab sin θ stays the same. The direction is set by the right-hand rule, turning from the first vector to the second. This is why the order in τ = r × F and l = r × p must be kept exactly as written.
What is the cross product of two parallel vectors?
It is zero, because the magnitude is ab sin θ and sin 0° = 0. In particular any vector crossed with itself gives zero, so î × î = 0. This is why a force along the line through the pivot produces no torque, since r and F are then parallel or antiparallel.
Angular velocity and angular acceleration
Read this section in the notes →Do all points on a rotating rigid body have the same angular velocity?
Yes. In a rigid body turning about a fixed axis, every particle sweeps the same angle in any given interval, so one angular velocity ω describes the whole body. Their linear speeds differ, v = ωr, growing with distance from the axis. Particles lying on the axis itself have zero linear speed.
Torque and angular momentum
Read this section in the notes →Is torque simply force times distance?
Only when the force is perpendicular to the line joining the pivot to the point where it acts. In general τ = rF sin θ = r⊥F, where r⊥ is the perpendicular distance from the axis to the line of the force. A force pointed straight at the pivot gives no torque however far away it acts.
What is the relation between torque and angular momentum?
Torque is the rate of change of angular momentum: dL/dt = τ_ext. This is the rotational version of F = dp/dt. If no external torque acts, the angular momentum of the system stays constant, which is the basis of many rotation problems involving skaters, divers and spinning platforms.
Equilibrium of a rigid body
Read this section in the notes →Can a body be in equilibrium with zero net force but non-zero torque?
No. Mechanical equilibrium of a rigid body needs both zero net force and zero net torque. Two equal and opposite forces acting along different lines, called a couple, give zero net force but a non-zero torque, so the body starts rotating. Zero net force only removes linear acceleration.
Principle of moments and centre of gravity
Read this section in the notes →What is the difference between the centre of gravity and the centre of mass?
The centre of mass depends only on how mass is distributed, while the centre of gravity is the point where the total gravitational torque is zero. They coincide when gravity is uniform over the body, which is true for everyday objects. For a body so tall that g varies across it, the two points differ slightly.
How does a lever give mechanical advantage?
By the principle of moments, a load F₁ at distance d₁ from the fulcrum and an effort F₂ at distance d₂ balance when d₁F₁ = d₂F₂. The mechanical advantage is F₁/F₂ = d₂/d₁, so placing the effort much farther from the fulcrum than the load lets a small effort balance a large load. The trade-off is that the effort must move through a larger distance.
Moment of inertia
Read this section in the notes →Is moment of inertia a fixed property of a body like mass?
No. Moment of inertia depends on the mass, how that mass is spread out, and the chosen axis of rotation. The same rod has a smaller moment of inertia about its centre than about one end. Mass is fixed, but moment of inertia must always be stated together with its axis.
What is the radius of gyration?
It is the distance k from the axis at which the entire mass could be placed to give the same moment of inertia, so I = Mk². It summarises how far, on average, the mass lies from the axis. Like moment of inertia, it depends on the choice of axis, not just on the body.
Kinematics of rotation
Read this section in the notes →How do the equations of rotational motion compare with linear ones?
For constant angular acceleration they have the same form, with angle in place of displacement: ω = ω₀ + αt, θ = ω₀t + ½αt² and ω² = ω₀² + 2αθ. Angles must be in radians and angular speeds in rad/s, so convert rpm by multiplying by 2π/60 before substituting.
Dynamics of rotation
Read this section in the notes →What plays the role of mass and force in rotation?
Moment of inertia plays the role of mass, and torque plays the role of force, giving τ = Iα in place of F = ma. Similarly, rotational kinetic energy is ½Iω², angular momentum is L = Iω, and power is P = τω. Recognising these pairs makes rotational problems much easier to set up.
Angular momentum and its conservation
Read this section in the notes →Why does a skater spin faster when she pulls her arms in?
With no external torque about the vertical axis, her angular momentum Iω stays constant. Pulling her arms in brings mass closer to the axis and reduces I, so ω must increase. Her rotational kinetic energy actually rises, because she does work with her muscles to pull her arms inward, so kinetic energy is not conserved here.
When is angular momentum conserved?
Angular momentum about an axis is conserved when the net external torque about that axis is zero. Internal forces cannot change it. External forces may still act, provided their total torque is zero, such as forces passing through the axis. This lets you relate I₁ω₁ = I₂ω₂ without knowing the details of the internal forces.
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