2Rotational MotionThe body is rigid. (i.e. It does not suffer deformation by external forces.)The forces on the body may act at different points.
3The Kinematics of Rotation Axis of rotation – the body is rotating about a fixed axis.side viewaxis of rotation
4The Kinematics of Rotation Axis of rotation – the body is rotating about a fixed axis.ωaxis ofrotationtop view
5The Kinematics of Rotation Angular displacement – The reference line moves an angle Δθ about the axis of rotation.ωΔθaxis ofrotationtop view
6The Kinematics of Rotation Average angular speed -ωΔθaxis ofrotationtop view
7The Kinematics of Rotation Instantaneous angular speed -ωΔθaxis ofrotationtop view
8The Kinematics of Rotation Example 1 – Find the angular speed.
9The Kinematics of Rotation Average angular acceleration -
10The Kinematics of Rotation Instantaneous angular acceleration -
11The Kinematics of Rotation Constant angular acceleration αωo = initial angular velocityω = final angular velocityθ = angular displacementt = time taken
12The Kinematics of Rotation Constant angular acceleration αωo = initial angular velocityω = final angular velocityθ = angular displacementt = time taken
13The Kinematics of Rotation Constant angular acceleration αωo = initial angular velocityω = final angular velocityθ = angular displacementt = time taken
14The Kinematics of Rotation Constant angular acceleration α
15The Kinematics of Rotation Note that the following quantities, except time t, are vectors.
16The Kinematics of Rotation We may use + and – signs to indicate the direction of the vectors.
17The Kinematics of Rotation Example 2 – to find the angular acceleration.The negative sign of α indicates that it is in opposite direction to the positive angular velocity.ωαO
18Linear AccelerationWhen the object is rotating, it has two components of linear acceleration.Tangential acceleration atIt is the linear acceleration along the tangent.Radial acceleration arIt is the centripetal acceleration pointing radially inwards.
19Tangential acceleration at = r. αIt changes the angular velocity.atrO
34Experiment to determine the moment of inertia of a flywheel Supplement Ch.6The gravitational potentialenergy of the weight is convertedinto the rotational kinetic energyof the flywheel and the kineticenergy of the weight.However there is loss of energydue to friction.
35Table for Moment of Inertia Hoop about cylindrical axisI = MR2Hoop about any diameterI = MR2M = mass of the hoopR = radius of the hoop
36Table for Moment of Inertia Solid Cylinder about cylindrical axisI = MR2Solid Cylinder about central diameterI = MR2 + ML2M = mass of the cylinderR = radius of the cylinderL = length of the cylinder
37Table for Moment of Inertia Thin Rod about axis through centre perpendicular to its lengthI = ML2Thin Rod about axis through one end and perpendicular to its lengthI = ML2M = mass of the rodL = length of the rod
38Table for Moment of Inertia Solid sphere about any diameterI = MR2Hollow sphere about any diameterI = MR2M = mass of the sphereR = radius of the sphere
39Parallel Axes Theorem m is the mass of the object G is the centre of gravityof the objectGIG is the moment of inertiaabout the axis through thecentre of gravityhIP is the moment of inertiaabout the axis through thepoint P.GPNew axis of rotation
40Parallel Axes Theorem G Note that the two axes are parallel. h G P New axis of rotation
41Parallel Axes TheoremExample 7IGGIPhGPNew axis of rotation
42Perpendicular Axes Theorem For a lamina lying in the x-y plane, the momentof inertia IX , IY and IZ, about three mutuallyperpendicular axes which meets at the samepoint are related byIZ = IX + IY
43Perpendicular Axes Theorem IZ = IX + IYExample 8
44Moment of force ΓMoment of force (torque) It is the product of a force and its perpendicular distance from a point about which an object rotates.Unit: NmFOaxis of rotation
45Moment of force ΓFaxis of rotationOTop viewOFΓ = F d
46Moment of force Γ The force F acts at point P of the object. The distance vector from O, the point of rotation, to P is r.θis the angle between the force F and the distance vector. Γ = F.r.sinθTop viewOFθrPΓ = F d
47Moment of force Γ Moment of force Γis a vector. In the following diagram, the moment of force is an anticlockwise moment.It produces an angular acceleration α in clockwise direction.Top viewOFrPθαΓ = F d
48Moment of force on a flywheel A force F acts tangentially on therim of a flywheel.FrΓ = F × r
49Work done by a torque Suppose a force F acts at right angle to the distance vector r.FrO
50Work done by a torque What is the moment of force about O? Γ= F × r O
51Work done by a torque The moment of force turns the object through an angle θ with a displacement s.FFrθOΓ= F × rs
52Work done by a torque What is the work done by the force? W= F × s F O θOsW= F × s
53Work done by a torque Express the work done by the force in terms of Γ and θ.UseF =ands = r. θFrθOsW = F × s= Γ× θ
54Example 9Work done against the moment of friction is equal to the loss of rotational kinetic energyof the flywheel.
55Torque and Angular acceleration = . is the torqueI is the moment of inertia is the angular accelerationCompare to F = m.a in linear motion.
56Torque and Angular acceleration In an angular motion with uniform angular acceleration :
58Conditions for equilibrium A body will be in static equilibrium, if1. net force is zero2. net moment of force about any point is zero
59Angular momentum LThe angular momentum L of an object about an axis is the product of the angular velocity and its moment of inertia.L = I.Unit of L: kg m2 s-1 or Nms.L is a vector. Its direction is determined by the direction of the angular velocity .
60Angular momentum of a rotating point mass A point mass m is rotating tangentially at speed v at a distance r from an axis.From I = mr2 , L = I and v = r vrmaxis ofrotation
61Example 11Find the angular momentum of a solid sphere.
62Newton’s 2nd law for rotation The torque acting on a rotating body is equal to the time rate of change of the angular momentum.
63Newton’s 2nd law for rotation If the net torque is zero, the angular momentum is a constant. The angular acceleration is zero.
68Typical examples of second law Flywheel with moment of inertia I.rmass maxisαFind the angular acceleration αin terms of I, m and r.
69Typical examples of second law Flywheel with moment of inertia I.mass mTmgaraxisαTT.r = I. αa = r. αmg – T = ma
70Typical examples of second law Smooth pulley with moment of inertia I and radius r.αFind the linear accelerationa of the two masses in termsof m1, m2, I and r.rm1m2aa
71Typical examples of second law Smooth pulley with moment of inertia I and radius r.m2aT2m2gαm1aT1m1grT2T1T2.r-T1.r = I.αT1-m1g = m1am2g-T2 = m2aa = rα
72The law of conservation of angular momentum If external net torque = 0, the sum of angular momentum of the system is zero.If Γ=0,
73The law of conservation of angular momentum For a system with initial moment of inertia I1 and initial angular velocity ω1, its initial angular momentum is I1ω1.If the system changes its moment of inertia to I2 and angular velocity ω2, its final angular momentum is I2ω2.If there is not any external net torque, thenI1ω1 = I2ω2