System of Particles and Rotational Motion

Physics · Class 11

Lesson 13 of 13 · 16 min

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Must-know facts

18 facts

  1. 1Rigid body: all interparticle distances fixed. Pure translation: same velocity for every particle; fixed-axis rotation: same ω for every particle.
  2. 2X = Σmᵢxᵢ / Σmᵢ; for two equal masses the centre of mass is midway, for three equal masses at the centroid.
  3. 3L-shaped 3 kg plate of three 1 m squares: centre of mass at (5/6 m, 5/6 m).
  4. 4MA = F_ext; internal forces cannot move the centre of mass.
  5. 5An exploding projectile's centre of mass continues on the original parabola.
  6. 6P = MV; if F_ext = 0, P and V_cm are constant.
  7. 7|a × b| = ab sin θ; a × b = −b × a; i × j = k, j × k = i, k × i = j.
  8. 8v = ωr, or v = ω × r; ω points along the axis by the right-hand rule.
  9. 9τ = r × F, τ = rF sin θ = r⊥F; unit N m, dimensions M L² T⁻², a vector unlike work.
  10. 10l = r × p; dL/dt = τ_ext; L constant if τ_ext = 0.
  11. 11Equilibrium: ΣF = 0 and Στ = 0; a couple gives zero net force but non-zero torque, the same about every point.
  12. 12Lever: d₁F₁ = d₂F₂; M.A. = d₂/d₁.
  13. 13Centre of gravity = centre of mass only when g is uniform over the body.
  14. 14I = Σmr², K = ½Iω², I = Mk².
  15. 15Ring MR²; disc MR²/2; rod about its middle ML²/12; solid sphere 2MR²/5; hollow cylinder MR²; disc about a diameter MR²/4.
  16. 16ω = ω₀ + αt, θ = ω₀t + ½αt², ω² = ω₀² + 2αθ; rpm × 2π/60 = rad/s.
  17. 17τ = Iα, W = τθ for constant torque, P = τω.
  18. 18L = Iω for a symmetric body about its axis; Iω constant with no external torque; K = L²/2I rises when I falls.

Common traps

Where marks are lost

Multiplying the force by the distance r from the pivot to the point of application, whatever the angle.

Only the perpendicular distance counts: τ = rF sin θ = r⊥F. A push along the door towards the hinge gives zero torque.

Assuming the centre of mass must lie inside the body.

For a ring, a hollow sphere or an L-shaped plate it can lie in empty space.

Thinking an explosion in mid-air changes the path of the centre of mass.

The explosion forces are internal; with only gravity acting outside, the centre of mass stays on the same parabola.

Putting rpm straight into ω = ω₀ + αt.

Convert first: rad/s = rpm × 2π/60, so 1200 rpm is 40π rad/s.

Treating the moment of inertia as a fixed property of a body, like mass.

I depends on the axis: a disc has MR²/2 about its central perpendicular axis but MR²/4 about a diameter.

Saying rotational kinetic energy is conserved when a skater pulls her arms in.

L = Iω is conserved; K = L²/2I increases as I falls, supplied by the work of her muscles.

Believing a body with zero net force must be in equilibrium.

A couple has zero net force but a net torque, and it sets the body spinning.

Writing a × b = b × a.

The cross product changes sign when the order is reversed: b × a = −a × b.

Equating the centre of gravity with the centre of mass for any body.

They coincide only when g is the same across the body; for a very tall or very large body they differ.

Formulas

11 to know

Centre of mass

R = Σmᵢrᵢ / M; X = Σmᵢxᵢ / M

Continuous body: R = (1/M)∫r dm.

Motion of the centre of mass

M A = F_ext; P = M V; dP/dt = F_ext

Internal forces cancel.

Vector product

|a × b| = ab sin θ; a × b = −b × a

Direction by the right-hand rule; i × j = k.

Linear and angular velocity

v = ω × r; v = ωr

r is the perpendicular distance from the axis in v = ωr.

Torque

τ = r × F; τ = rF sin θ = r⊥F

SI unit N m.

Angular momentum

l = r × p; dL/dt = τ_ext

L constant when τ_ext = 0.

Principle of moments

d₁F₁ = d₂F₂; M.A. = F₁/F₂ = d₂/d₁

Reaction at the fulcrum R = F₁ + F₂.

Moment of inertia

I = Σmᵢrᵢ² = Mk²; K = ½Iω²

Unit kg m².

Rotational kinematics

ω = ω₀ + αt; θ = ω₀t + ½αt²; ω² = ω₀² + 2αθ

Constant α only.

Rotational dynamics

τ = Iα; dW = τ dθ; P = τω

Fixed axis.

Angular momentum about a fixed axis

L = Iω; I₁ω₁ = I₂ω₂ when τ_ext = 0

Symmetric body about its axis.

Key terms

14 terms

Rigid body
A body whose particles all keep fixed distances from one another.
Pure translation
Motion in which every particle of the body has the same velocity at each instant.
Axis of rotation
The line, held fixed, around which each particle of a turning body traces a circle.
Precession
The slow turning of a spinning body's axis about another direction, as a top's axis sweeps a cone.
Centre of mass
The mass-weighted mean position of a system, which moves as if all the mass and external force were there.
Vector product
A vector of size ab sin θ perpendicular to a and b, directed by the right-hand rule.
Torque
The turning effect of a force about a point, r × F.
Angular momentum
The moment of linear momentum about a point, r × p.
Couple
Two equal and opposite forces along different lines, which turn a body without moving it along.
Mechanical advantage
The ratio of load to effort for a lever, equal to effort arm over load arm.
Centre of gravity
The point about which the gravitational torques on a body add to zero.
Moment of inertia
Σmr² about an axis: the rotational counterpart of mass.
Radius of gyration
The distance k at which the whole mass, as a point, would have the same moment of inertia.
Flywheel
A heavy wheel with a large moment of inertia that smooths out changes in an engine's speed.
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