Circular mtion

.Circular motionUniform circular mtionAn object is said is said to be in uniform circular motion if the position vector joining the centre of the circle and the given object sweeps out equal angles in every unit time.Angular displacementThe angle swept by the radius vector in a given time interval is called the angular displacement ().Unit is radians.Angular velocityThe change in angular displacement per second or the rate of change of angular displacement is called the angular velocity. Unit is radians per second.Dimension of angular velocity is Time periodThe time taken by the to complete one revolution in the circular path is called the time periodAngular velocity π/T rad/sFrequencyFor a body going uniformly in a circular path the of revolutions completed in one second is called the frequencyFrequency υ=1/TAlso we have ω = 2π/T =2πυRelation between linear velocity v and angular velocity ωConsider an object moving in a circular path of radius r with uniform angular velocity ω .At any instant let the object at point A and OA is the radius vector.After time t seconds the object is at B and OB is the new radius vector.Intime t seconds the radius vector sweeps out an angle Э radiansBut angle Э = =Angular velocity ω = = = = Hence we get ω =Or v = rω Angular accelerationIt is the change in angular velocity per second or the rate of change of angular velocity.Angular acceleration α= = Relation between linear and angular accelerationLinear acceleration a =αrCentripetal ForceConsider an object moving uniformly in a along a circle of radius r. The speed of the objectremains constant. But its velocity continuouslybecause the direction of velocity changes continuously from point to point in the circularpath. Continuous change in velocity produces a continuous change in momentumof the object. By Newton’s second law a force is required to produce a change in momentum Hence a force must act on the object . This force must be such as to deviate the object continuously towards the centre of the circle. Such a force is called the centripetal force. Hence a force which continuously deviates the object from its linear path to make it move along a circle and is always directed radially inward is called centripetal force. Centripetal accelerationby Force produces change in velocity resulting in acceleration. The acceleration produced the centripetal force is called the centripetal acceleration It is also directed radially inwardEx pression for centripetal accelerationConsider an object moving round a circle with uniform speed v. At any instant t The object is at A .After time δt seconds the object moves to B along AB. At A the velocity of the object is v directed along the tangent at A. At B the velocity f the object is v directed along the tangent at B.If the interval t is very small, A and B will be very close to each other. Angle will be very small. When is very small we have cos = 1 and sin = As the object moves from A to B the change in vAelocity along the tangential direction is vcos - v =vcos0 –v= v- v= 0 So there is no acceleration along the tangential direction . The change in velocity along the radial direction isvsin - 0 = vsin = v Acceleration along the radial direction = =vAcceleration along the radial direction is called centripetal acceleration.Centripetal acceleration = vAlso we know that angular velocity =Hence centripetal acceleration = Expression for centripetal forceIf m is the mass of the object we have force F= mass x acceleration = m×a Therefore centripetal force F = mv = The centripetal force is always directed radially inwards. That is towards the centre of the circular path.Examples :1.Planets revolve round the sun in circular orbits. The centripetal force required for circular motion is provided by the gravitational force of attraction between the and the planets.2.Electrons revolve round the nucleus in circular orbits. The required centripetal force provided by the electrostatic force of attraction between the electron and the nucleus.3. A stone tied to one end of a string is whirled in a horizontal circle by holding the other end in your hand. The centripetal force required for circular motion is provided the tension in the string.Centrifugal forceWhen a body is forced to move in a circular path it has always a tendency to move away from the centre of the circle. This is called centrifugal tendency. By NewtonsFirst law every body has a tendency to move along a straight line path. In circular motion the body continuously deviates from the straight line path by an external force. But the inertia of the body oppose this deviation. The opposing force due to inertia tends thbody to go away from the centre of the circle. This opposing force due to inertia is called the centrifugal force. It is always equal and opposite to the centrifugal force ApplicationsBanking of roads and railsConsider a car going round a curved level road. The centripetal force required for circular motion is provided by the friction between the tyres and the road surface. This will result in the wear and tear of the tyres . If the road surface is Smooth friction will be small So the frictional force will not be sufficient to provide the required centripetal force for circular motion. This will result in skidding of the car.To avoid this curved roads are usually banked.For banking the outer edge of the road surface is raised a little over the inner so that the road surface is inclined to the horizontal. The angle which the road surface makes with the horizontal is called the angle of banking ( ). Now the reaction of the road surface will be inclined to the vertical making an angle( ).The horizontal component of the reaction will supply the centripetal force required for circular motion while the vertical component will balance the Weight of the carThe angle of banking is given by tan =Where v is the speed with which the car is going around a circular track of radius r.A cyclist has to lean inward when moving around a circular trackA cyclist going around a curve leans inwards in order to provide the necessary centripetal force required for motion around a circular path. As the cyclist leans in wards the reaction of the road surface becomes inclined to the vertical. The horizontal component of the reaction provides the centripetal force requiredfor circular motion while the vertical component balances the weight of the cycle and the cyclist.If is the angle through which the cyclist leans inward We have tan =