Rigid Body

Particle Object without extent Point in space Solid body with small dimensions

Rigid Body An object which does not change its shape Considered as an aggregation of particles Distance between two points is a constant Suffer negligible deformation when subjected to external forces Motion made up of translation and rotation

Motion of a Rigid Body Translational – Every particle has the same instantaneous velocity Rotational – Every particle has a common axis of rotation

Centre of Mass Centre of mass of a system of discrete particles:

Centre of mass for a body of continuous distribution: It is the point as if all its mass is concentrated there Located at the point of symmetry

Conditions of Equilibrium For particle – Resultant force = 0 For rigid body – Resultant force = 0 and – Total moments = 0

Toppling An object will not topple over if its centre of mass lies vertically over some point within the area of the base Figure

Stability Stable Equilibrium – The body tends to return to its original equilibrium position after being slightly displaced – Disturbance gives greater gravitational potential energy – Figure Figure

Unstable Equilibrium The body does not tend to return to its original position after a small displacement Disturbance reduces the gravitational potential energy

Neutral Equilibrium The body remains in its new position after being displaced No change in gravitational potential energy

Rotational Motion about an Axis The farther is the point from the axis, the greater is the speed of rotation (v  r) Angular speed, , is the same for all particles

Rotational K.E. The term is known as the moment of inertia

Moment of Inertia (1) Unit: kg m 2 A measure of the reluctance of the body to its rotational motion Depends on the mass, shape and size of the body. Depends on the choice of axis For a continuous distribution of matter:

Experimental Demonstration of the Energy Stored in a Rotating Object

Moment of Inertia (2) A body composed of discrete point masses A body composed of a continuous distribution of masses

Moment of Inertia (3) A body composed of several components: – Algebraic sum of the moment of inertia of all its components A scalar quantity Depends on – mass – the way the mass is distributed – the axis of rotation

Radius of Gyration If the moment of inertia I = Mk 2, where M is the total mass of the body, then k is called the radius of gyration about the axis

Moment of Inertia of Common Bodies (1) Thin uniform rod of mass m and length l – M.I. about an axis through its centre perpendicular to its length – M.I. about an axis through one end perpendicular to its length

Moment of Inertia of Common Bodies (2) Uniform rectangular laminar of mass m, breadth a and length b – About an axis through its centre parallel to its breadth – About an axis through its centre parallel to its length

Moment of Inertia of Common Bodies (3) – About an axis through its centre perpendicular to its plane

Uniform circular ring of mass m and radius R – About an axis through its centre perpendicular to its plane Moment of Inertia of Common Bodies (4)

Uniform circular disc of mass m and radius R – About an axis through its centre perpendicular to its plane Moment of Inertia of Common Bodies (5) – The same expression can be applied to a cylinder of mass m and radius R

Uniform solid sphere of mass m and radius R Moment of Inertia of Common Bodies (6)

Theorems on Moment of Inertia (1) Parallel Axes Theorem

Perpendicular Axes Theorem Theorems on Moment of Inertia (2)

Torque (1) A measure of the moment of a force acting on a rigid body – T = F · r Also known as a couple A vector quantity: direction given by the right hand cork- screw rule Depends on – Magnitude of force – Axis of rotation

Work done by a torque – Constant torque: W = T  – Variable torque: Torque (2)

Kinetic Energies of a rigid body (1) Translational K.E. Rotational K.E. – It is the sum of the k.e. of all particles comprising the body – For a particle of mass m rotating with angular velocity  :

If a body of mass M and moment of inertia I G about the centre of mass possesses both translational and rotational k.e., then Kinetic Energies of a rigid body (2) =

Moment of inertia of a flywheel (1) Determination of I of a flywheel – Mount a flywheel – Make a chalk mark – Measure the axle diameter by using slide calipers – Hang some weights to the axle through a cord – Wind up the weights to a height h above the ground – Release the weights and start a stop watch at the same time

Moment of inertia of a flywheel (2) – Measure: – the number of revolutions n of the flywheel before the weights reach the floor – the number of revolutions N of the flywheel after the weights have reached the floor and before the flywheel comes to rest

Theory Moment of inertia of a flywheel (3) where f = work done against friction per revolution ….. (1)

– When the flywheel comes to rest: Loss in k.e. = work done against friction Moment of inertia of a flywheel (4) ….. (2) (2) In (1) …. (3)

Moment of inertia of a flywheel (5) – The hanging weights take time t to fall from rest through a vertical height h Total vertical displacement = average vertical velocity  time Knowing v, I can be calculated from (3)(3)

Applications of flywheels In motor vehicle engines In toy cars

Angular momentum The angular momentum of a particle rotating about an axis is the moment of its linear momentum about that axis.

Conservation of angular momentum (1) The angular momentum about an axis of a given rotating body or system of bodies is constant, if the net torque on the object is zero As – If T = 0, I  = constant

Examples – High diver jumping from a jumping board Conservation of angular momentum (2)

– Dancer on skates Conservation of angular momentum (3) Experimental verification using a bicycle wheel – Mass dropped on to a rotating turntable

Application – Determination of the moment of inertia of a turntable Set the turntable rotating with an angular velocity  Drop a small mass to the platform,  changes to a lower value  ’ If there is no frictional couple, the angular momentum is conserved, I  = I ’  ’ = (I + mr 2 )  ’ ,  ’ can be determined by measuring the time taken for the table to make a given number of revolutions and I can then be solved Conservation of angular momentum (4)

Rotational motion about a fixed axis (1) T = I  d ’ Alembert ’ s Principle – The rate of change of angular momentum of a rigid body rotating about a fixed axis equals the moment about that axis of the external forces acting on the body

Rotational motion about a fixed axis (2) i.e. I  = T

Compound pendulum (1) Applying the d ’ Alembert ’ s Principle to the rigid body But where k is the radius of gyration about its centre of mass G

For small oscillations Compound pendulum (2)  SHM with period

It has the same period of oscillation as the simple pendulum of length Compound pendulum (3) l is called the length of the equivalent simple pendulum

The point O, where OS passes through G and has the length of the equivalent simple pendulum, is called the centre of oscillation S and O are conjugate to each other The period T is a minimum when h = k (see expt. results)expt. results Compound pendulum (4)

Torsional pendulum where c = torsional constant I = moment of inertia  SHM with period

Rolling objects (1)

P has two components: – v parallel to the ground – r  (=v) perpendicular to the radius OP If P coincides with Q, the two velocity components are oppositely directed. Thus Q is instantaneously at rest Rolling objects (2)

Hence, for pure rolling, there is no work done against friction at the point of contact Rolling objects (3)

Kinetic energy of a rolling object Total kinetic energy = translational K.E. + rotational K.E. =

Stable Equilibrium

No toppling

Compound pendulum