Equations of circular motion

Circular motion of rigid body Uniform circular motion Accelerated circular motion

Nature is full of rotating objects, Circular motion being periodic, gives stable structure.

Uniform circular motionMagnitude of v is constantDirection is changesF = forces acting toward center.This force changes direction of velocity

Accelerated circular motionVelocity is changing both in magnitude & directionTwo different forces actF = Centripetal forceF = Tangential forceTwo Accn. existsCentripetal Acc.Tangential Acc,

New concepts of angular measurement( Radian , Steradian )Calculations jn calculus become simpleIt is geometry based definition of angleThere is no natural reason why circus should have 3600. ( Why not 400 ?)

What is Radian; (Measure of angle)3600 = (2 x R ) / R = 2 Radian

What is Radian; (Measure of angle)sS = r

Degree & Radians10 = (p/1800) rad = 0.0174533 rad;

1 rad = 1800/p = 57.2960 = 57.30

The surface area of a sphere is 4r2, The surface area of a steradian is just r2. The name steradian is made up from the Greek stereos for "solid" and radian. The SI Unit abbreviation

Example: a sphere with a radius of 1 (called the "unit sphere"):has a surface area of 4, and a steradian would "cut out" an area of 1.

Concept of rigid bodyEvery particle of in the rigid bodyRotates through the same angle Have the same angular speed Have same angular accelerationPartials do not move with respect to each other during motion.

Terms involved in circular motion

Angular positionAngulardisplacement = Very small angle tending to zero

Angular velocitytWhen initial angle =0

Using following quantities we can study rotation of rigid-body Angular position() , linear position (x) Angular speed ( ), linear speed (v), Angular accn ( ) linear acceleration (a) Period (T) Frequency Angle is always expressed in Radians

Motion with Constant AccelerationEquations of motions: = 2 1 tCircular motion First equations.

Distance moved = average velocity x tSecond equations of liner motion.---(as v = u + at) Angle rotated= Angular velocity x t---(as 2 = 1 + t)

Equations of motions:Third equation.Distance moved = average velocity x t---(as v = u + at) Angle rotated= Angular velocity x t---(as 2 = 1 + t) X

Three main equations for constant accelerated motion are are

Other terms / equations of circular motion

T =Period , time taken for one evolutionTime periodT sec

FrequencyFrequency f = Number of revolution in unit time

Relation between tangential velocity & angular velocity

Tangential velocity

Example: A Rotating shaftA 4.0 cm diameter shaft turns at 2400 rpm. What is the speed of a point on the surface of shaft?Frequency =1f=

Example: A Rotating shaftA 4.0 cm diameter shaft turns at 2400 rpm. What is the speed of a point on the surface of shaft? covered In one revolution = 2 per covered in a min = 2400 = /sec =( 2400 )/60 rad /sec

Angular acceleration

Angular (centripital accelarationIn next slide we will prove that centripetal accelaration always acts towards center of trotation

Direction of Angular acceleration V = V2 V1ABObject move from A to B in time t secV1= Initial velocityV2= Final velocityV= change in velocityThis is centripetal acceleration

Magnitude of Angular acceleration is givenMagnitude of centripetal accelerationby formula acLet us prove this formula

AECBAD CBD = CAD =900 rAs V2 & V1 are tangent to circle + ADB + 90 + 90 =360(interior angles)Consider quadrilateral ADBC + ADB = 180Consider line ADE

Direction of Angular acceleration ABV2CCABAs CB = CA = r, CB = BA= V,& is common Consider triangle ABC & ABC V1Object move from A to B in time t sec

Direction of Angular acceleration ABV-VCCABrVV = V2 = V1 in magnitude of tangential velocityObject move from A to B in time t secV

Consider difference between Arc AB & lenght AB As reduces Diff. between arc AB & length AB reduced

Diff. between arc AB & length AB reducedHence as 0Arc AB = Length AB

Direction of Angular acceleration ABV2CCABrV1Object move from A to B in time t secSAs 0, Length AB = arc ABNow arc AB = s = V x t (V= tangential velocity),

Direction of Angular acceleration ABV2CCABrV1Object move from A to B in time t secS

Direction of Angular acceleration ABV2-VCCABrV1Object move from A to B in time t secSHence ac = v2/ r gives basic relation ship between tangential velocity & centripetal acceleration

Other form of same equation

Ball in circular motion rope provides centripetal force to keep ball in circle. If rope is cut and ball continues in straight line with velocity at the time of cutting the rope,

T =Period , time taken for one evolutionTime period

Frequency f = Number of revolution in unit timeFrequencyor = 2 / T

Centripetal ForceNow Centripetal acceleration acp

Hence centripetal force fcp, required to keep an object of mass m moving in a circle of radius r can be calculated.

Equations of circular motionRPM= rotation per minute

Example: Decelerating Windmill A windmill rotating at w = 2.1 rad/s slows down at a constant angular acceleration of a = -0.45 rad/s2. How long does it take for the windmill to come to a complete stop?

Example: Find the period of a disc rotating at 45 RPM.f = 45/60 rotation per sec.

= 385 106 2.66 10-6 = 1024.1m/sec = 2 / T = ( 2 3.14 ) / ( 27.3 24 60 60) = ( 6.28) / ( 2358720) = 2.66 10-6 rad/secV = r = 2 / T V = r= 385 103 103 metersExample : Calculate tangential velocity of moon

V =1030.3m/secac?= (1030*1030 / 385000*1000) = 0.00273 m/sec sq

Ex :A shaft was initially rotating at 600 rpm. Shaft accelerates at constant rate to 2400 rpm in a time interval of 3.0 s. How many revolutions does the shaft made in this time interval?

90. = 60 + . = 150 radians = 20 rad /sec2Number of revolutions = 150 / 2 = 75 revolutions

Practical applications

. Lesser headCD (Compact disk) or HD (hard disk)VtLesser head details

Reading on landReading on Pit

CD have information is stored (digitally) in a series of pits and flat areas on facePits(1) and land (0) represents binary ones and zeroLesser moves relative to disk at constant liner speedCD or HD surfaceDisk surfaceLesser beam

CD/DVD player Speed. Lesser head

The lesser travel liner speed must be same over full disc surfaceHence disk turns with a variable angular speed to have constant tangential speed vt is at all radiusPitLandVtReading CDWriting CD

CD/DVD player SpeedExampleFind RPM of CD to have vt = 2.0 m/s when the laser beam shines on the disk at 4.0 cm & 5 cm from its center Lesser headVt

Centripetal ForceNow Centripetal acceleration acp

Hence centripetal force fcp, required to keep an object of mass m moving in a circle of radius r can be calculated.

Problem 1

Find (take g=10 m/sec sq.)What is minimum height H from which the ball should be released so that it will not leave the track

Track160cmW = 5kg

=mgForces acting on the ballBall will not fall when of both forces are same & net down ward force will be zero

Centrifugal forceBall will not fall down when= mg= g

V160cmV=4m/secNote that V do not depend on weightdrinkRoller coasterRotating objectSmall loopBig loop

Track160cmPE =KEmgH = m v210*H = 4*4H = 0.8 m

1.8mExample2 : What reading a spring balance will record at bottom most point during rotation? (take g=10 m/sec sq.)m = 5kgr =10m

m = 5kgWhat weight a spring balance will record at bottom most point during rotation?v

1.8mm = 5kgABPE at A = KE at B1.8x10x m = m x V2Spring balance reading = Net down ward force = mg + mv2/rV = 6 m/secSpring balance reading = mg + mv2/r = 5 x 10 + (5 x 62 /10) = 50 + 18 = 68 N10m

Problem 3

Angular movement & Kinetic energy of rotation

Rotation as a VectorRotation and other angular motion quantities are vectors & are defined using the right-hand rule:If the fingers of your right hand follow the rotation direction, then your thumb points along the rotation axis in the direction of the angular velocity w.An alternative definition is that if a right-hand threaded screw is rotated, then w is in the direction that the screw advances.

Rotation as a Vector

The Vector Nature of Rotational MotionThe direction of the angular velocity vector is along the axis of rotation. A right-hand rule gives the sign.

The Vector Nature of Rotational MotionA similar right-hand rule gives the direction of the torque.

Direction of Angular acceleration Change in velocity V = V2 V1ABObject move from A to B in time t secV1= Initial velocityV2= Initial velocityV= Initial velocityThis is centripetal acceleration

A car is moving on circular track at velocity of 14m/sec.Find out coefficient of friction between road & car tiers such that car will remain on track

N = m*g

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