Recall from last lecture: Simple Harmonic Motion

Kinematic of Simple Harmonic MotionThe most general solution is given byCompare:Units: is a pure number is a frequency with units s -1 or Hz (hertz)must have the unit of lengthThe motion of a simple harmonic oscillator is sinusoidal

Simple Harmonic MotionA is the amplitude of the motionφ is the phase of the motionω is the frequency of the motionAΤ is the period of the motionTPlot of amplitude of oscillations of springversus time

Uniform circular motionDefinition: Uniform circular motion is the motion of an object travelling at aconstant (uniform) speed on a circular path.Motion of a model airplane, travelling at constant speed on a horizontalcircular path is an example of uniform circular motion.Period, T, is time required (in seconds) to travel once around thecircle, R is the radius of the circle, and v is the speed.T2πR=v

Motion on a Circular PathTside viewThe force that maintains the plane on itscircular path is the tension in the stringThis is a centripetal force, it always actstowards the centre(Newton's 2 nd law)When a car moves around a curve,static friction between the tyres androad surface prevent the car fromskidding. This static friction providesthe centripetal force.

Kinematics of Circular Motion0v'time= t+∆t∆sθRvtime=tConsider a body moving along the segment ofa circle, ∆s (delta s) and the circle hasradius R. At time t, it has velocity v, andsome later time t+∆t it has velocity v’.Assume magnitude of velocity constant: | v |= constantBy definition of the radianfor a segment of a circle:Note: We will generally use radians to measure aunit of plane angle: 360° is equal to 2π radian.Define the angular speed, ω (omega):Since velocity is displacement/time= ω, angular speedSI unit: rad s -1

Kinematics of Circular MotionBut:magnitude of the accelerationDefine centripetal force:Fc2mv= ma ⇒ Fc= Always acting towards centreRDirection of centripetal acceleration parallel to change in velocity:asvai.e. towards the centreCentripetal acceleration

Kinematics of Circular MotionExample: determine the centripetal force for the following:(i.e. one revolution each second)radius:F cmass:We have thatcentripetalacceleration:2va = and vR2⇒ a = ω R⇒ a =≈ 80 ms(-12πrads )−2= ωR2× 2 mCentripetal Force:F c= ma==3 kg × 80 ms240 kgms−2-2or 240 NN = Newtons

Circular Motion in Cartesian CoordinatesyRrθxGeneral phase, θ 0is the starting angle and ωt is the timedependent changing angle going around the circle.Circle of radius RThe acceleration points along the direction of the radius vector rbut towards the centre

Kinematics of Circular MotionExample: Conical PendulumA metal bob at the end of a string is given a push so that it rotatesin a circular path in the horizontal planeFree body diagram to represent the forces:αl = length of stringm= mass of bobR = radius of circle followed+xTφmgαα+yThe centripetal force is given by sum of forces acting on the bob:T is tension, mg is weightacting towards the centre (i.e. in the +x-direction)

Kinematics of Circular MotionLet's write : F c = F , T = Tx-components:y-components:c( F )cx⇒Fc⇒F( F )c= Tx+ mgx= Tsinα+ 0= Tsinα= T+ mg⇒ 0 = T cosα− mgmg⇒ T =cosαsubstituting value for T abovecyyyNewton's 2 nd lawbut:

Kinematics of Circular MotionTime for one complete revolution:

SummaryUniform circular motionPeriod is time required (in seconds) to travel once around the circle:R is the radius of the circle, and v is the speed.T=2πRvCentripetal forceCritical velocityFcma⇒mvRv is the critical velocity to keep the bodymoving on a circular path.=Fc=2v =rg tanθ