Circular motion Motion of a particle along thecircumference of a circle is calledcircular motion.
If the magnitude of this linear velocity(linear speed) remains constant, thebody is said to be in uniform circularmotion. (U.C.M.)
Position Vector or RadiusVector A vector drawn from the centre of thecircular path to the position of theparticle performing circular motion at ainstant, is called the position vectoror radius vector of that particleperforming circular motion at a instant
Angular Displacement Angular displacement θ of a particle ina given time is defined as the angletraced by its radius vector in that timeat the origin of frame of reference. ∴ angular displacement = ∠BOA = θarcBAButrsrIt is expressedin radian sr
Angular Velocity The rate at which the angulardisplacement takes place is calledangular velocity (ω)qvAverage angular velocitytThe instantaneous angular velocity can be obtained at limitingvalue of δtt 0dlimt dtIt is measured in rad/sec.
U.C.M. “ Motion of a body along thecircumference of a circle withuniform linear speed or uniformangular velocity.”
Angular Acceleration The rate of change of angular velocityis called „angular acceleration (α)‟2 1avti 2t 0d radlim It is measured int dt s
Multiple Choice Questions 1. In U.C.M. -------------- remainconstant(a) Velocity (b) linear velocity(c) angular velocity(d) Magnitude of linear velocity. 2. Determine the angular accelerationof arotating body which slows down from500rpm to rest in 10 seconds.(a) 5.233rad/s2 (b) - 5.233 rad/s2(c) 5.233m/s2 (d) -5.233m/s2
22 121 122 : b 5.233 rad / sBut 02252 3.143105.233 rad / stnt tAns
Right Hand Rule “ Imagine the axis of rotation to be held in your righthand with the fingers curled round the axis and thethumb out stretched along the axis. In this situation,if the curled fingers indicate the direction of rotation,the thumb indicates the direction of angulardisplacement, angular velocity and angularacceleration.
If the particle is moving in anticlockwisedirection, the directions of angulardisplacement and velocity are along theaxis in upward direction. The direction ofangular acceleration is along the axis inupward direction if the angular velocity isincreasing and downward if the angularvelocity is decreasing.
Relation between Linearvelocity and Angular velocity
According to the definition, velocity isdistance travelled in unit time.svtarcangleradiussrs rInstantaneous value of velocity can be obtained atthe limiting value of δt ast 0rv limt
As r is not changing with respect totime, its limiting value is same tt00rBut,lim is instantaneov rus angular velocity .tV vectov r limtral y, rl
Relation between linear accelerationand angular accelerationAccording to definition of linearacceleration, time rate of change of velocityis the magnitude of acceleration. dvi.e. adtda rdtda rdtBut v rBut r is constant.dBut angular accelerationdta r Vectorally, a r
Acceleration of a body in U.C.M. 1. By Vector Method
Acceleration is time rate of change ofvelocity.BR BQ QR QR BR BQ QR Velocity at B Velocity at AQR Change in velocityQR v According to the triangle law of vectorsvaccelerationdt
Instantaneous acceleration can beobtained by limiting δt to 0.t 0va limt∠QBR ≅ ∠AOB
∵ angle between the tangent is anglebetween the corresponding radii. As δt approaches O, B approaches Aand δθ becomes so small that we canwrite.
t 0t 0t 0cva limtBut V is constanta v limtBut lim angular velocityta v22c cvSubstituting v r or ,rva r or ar
2. By Calculus MethodThe object moves from A to P in time t subtending angle θ at thecentre Thereforet∴ θ = ωt
r xi yjBut tr r i cos t j sin tTime derivative of this positionvector will give us the linear velocity.d di.e. v r r i cos t j sin tdt dtv rr i r coi sin t j cs j r siosnt
Time derivative of this velocity will giveus acceleration. n 2n 2nn 2 2n 20dv di.e. accel r i sin t j cos tdt dtdvaccel r i cos t j sin tdtaccel i r cos t j r sin tBut(i r cos t j r sin t ) raccel . raccel r r
Home Work 1. The extremity of the hour hand of aclock moves 1/20 th as fast as theminute hand. What is the length of thehour hand if the minute hand if theminute hand is 10 cm long? ( vm = 20vh , v = rw, Ans = 6 cm) 2. Find the speed at which the pointson the equator move as the earthrotates about its axis. Radius of theearth = 6400 km. ( w = 2π/ T, v = rw,Ans = 465 m/s)