Its SI unit is rad/s and is an axial vector having its direction given by right hand thumb rule.

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Chester Ferguson
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1 Circular motion An object is said to be having circular motion if it moves along a circular path. For example revolution of moon around earth, the revolution of an artificial satellite in circular orbit around earth etc. are all examples of circular motion. Angular displacement: It is the angle traversed by radius vector of an object having uniform circular motion in given time. It is denoted by Its SI unit is radian (rad) and it is an axial vector having its direction given by right hand thumb rule. Angular velocity: It is the rate of change of angular displacement i.e. Its SI unit is rad/s and is an axial vector having its direction given by right hand thumb rule. Uniform circular motion An object is said to be having uniform circular motion if it moves with constant speed along a circular path. Time Period: It is the time taken by object in uniform circular motion to complete one revolution. It is denoted by T Frequency: It is the no. of revolutions completed by object in one second in uniform circular motion. It is denoted by. Frequency is the reciprocal of time period of object. Angular frequency: It is the product of 2π with the frequency of object in uniform circular motion. RPM: It stands for revolution or rotation per minute. It is related to rad/s by following relation RPS: It stands for revolution or rotation per second. Uniform circular motion is accelerated motion In uniform circular motion direction of velocity of object at any instant is along the direction of tangent to the circular path. As the object moves direction of tangent changes. Velocity is a vector quantity so change in its direction causes change in velocity. This means that in circular motion velocity changes continuously because of change in its direction. Hence circular motion is an example of accelerated motion because velocity changes continuously. Direction of acceleration acting on object in uniform circular motion is toward centre of circular path and is known as centripetal acceleration.

2 Relation between linear velocity and angular velocity Consider and object moving uniform in a circular path of radius r. It covers angle θ in time t. Suppose its speed is v. Suppose arc length AB is l. A B O From (2) and (3) Using equation (1) we get In vector form Angular acceleration

3 It is the rate of change angular velocity of an object in circular motion. It is equal to change in angular velocity per unit time taken. Its SI unit is rad/s 2 and is an axial vector having its direction given by right hand thumb rule. Tangential acceleration It is the rate of change of linear velocity of an object in circular motion. Its SI unit is m/s 2 and its direction is along tangent to the circular path. Considering equation (2) Relation between tangential and angular acceleration Putting Using (1) In vector form Equations of circular motion Centripetal Acceleration It is an acceleration that acts on an object moving along circular path towards centre of circular path. Expression for centripetal acceleration Consider an object moving along circular path of radius r. Let

4 Velocity of object at A Velocity of object at B Displacement from A to B time taken by object to go from A to B A B O Equating (1) and (2) we get Dividing both sides by Putting

5 Resultant acceleration Net resultant acceleration is the resultant of tangential and centripetal acceleration. ASSIGNMENT 1. The wheel of an automobile is rotating with 4 rotations per second. Find its angular velocity. If radius of the fly wheel is 50 cm, find the linear velocity of a point on its circumference. [Ans: 8π rad.s -1, cms -1 ] 2. An insect trapped in a circular groove of a radius 12 cm moves along the groove steadily and completes 7 revolutions in 100 s. (a) what is the angular speed and the linear speed of the motion? (a) Is the acceleration vector a constant vector? What is its magnitude? [Ans: 0.44 rad/s, 2.3 cms -2 ] 3. What is the angular velocity in radian per second of a fly wheel making 3300 rpm? [Ans: 31.4 rad/s] 4. Calculate the angular velocity in rad/s of a particle which makes 300 rev per minute. What is the linear velocity if the radius 4 cm? [Ans: 10 rad/s, 40 cm/s] 5. Blades of an aeroplane propeller are rotating at a rate of 600 revolutions per minute. Calculate its angular velocity. [Ans: 20π rad/s] 6. The radius of the earth s orbit around the sun is m. Calculate the angular and linear velocity of the earth. Through how much angle does the earth revolve in 2 days? [Ans: rad/s, m/s, rad] The blades of an aeroplane propeller are 2 m long and rotate at the rate of 300 rpm. Calculate (i) the frequency (ii) the period of rotation (iii) the angular velocity (iv) the linear velocity of a point 0.5 m from the tip of the blade. [Ans: 5 rps, 0.2 s, rad/s] 9. Calculate the angular velocity of a geostationary satellite, if the unit of time is an hour and that of angle is degree. [Hint: time period of geostationary satellite is 24 hour]. [Ans: π/12 rad/h] Equations of circular motion 10. A wheel starts from rest and attains a rotational speed of 240 rev/s in a time of 2 min. What was its average acceleration? [Ans: 2 rev s -2 ] 11. The angular speed of a motor wheel is increased from 1200 rpm to 3120 rpm in 16 seconds. (i) What is its angular acceleration, assuming the acceleration to be uniform? (ii) How many revolutions does the engine make during this time? [Ans: 40π rad/s, 4π rad/s 2, 576] 12. A washing machine s spin drier rotates at 1200 rpm when the machine is on. After the current is switched off, its rotation rate slows down uniformly to 600 rpm in 50revolutions. Find (i) the angular acceleration and (ii) the time taken by the spin drier to make these 50 revolutions. [Ans: -6π rad s -2, 3.33 s] 13. The angular velocity of a particle moving in a circle of radius 50 cm is increased in 5 minutes from 100 revolutions per minute to 400 revolutions per minute. Find (a) angular acceleration (b) linear acceleration. [Ans: π/30 rad/s 2, 5π/3 cm s -2 ] Centripetal acceleration 14. A body is moving in a circle of radius 100 cm with a time period of 2 second. Find the acceleration.

6 15. If a body moves on the circumference of a circle with a speed equal to that which it would acquire by falling freely through half the radius of the circle, prove that its centripetal acceleration equals the acceleration of free fall (g). 16. Calculate the centripetal acceleration of the moon moving around the earth in a circular orbit in terms of its time period T and radius of the orbit R. [Ans: 4π 2 R/T 2 ] 17. Calculate the centripetal acceleration of a point on the equator of earth due to the rotation of earth about its own axis. Radius of earth = 6400 km. [Ans: 439 km h -2 ] 18. A stone tied to one end of a string 80 cm long is whirled in a horizontal circle with a constant speed. If the stone makes 14 revolutions in 25 seconds, what is the magnitude and direction of acceleration of the stone? [Ans: 9.9 m/s 2 ] 19. An air craft executes a horizontal loop of radius 1 km with a steady speed of 900 km/h. compare its centripetal acceleration with the acceleration due to gravity. [Ans: Ratio 6.38]. 20. A cyclist riding with a speed of 27 km/h. As he approaches a circular turn on the road of radius 80 m, he applies brakes and reduces his speed at the constant rate of 0.5 m/s. What is the magnitude and direction of the net acceleration of the cyclist on the circular turn? [Ans: 0.86 m/s 2, ]

Uniform Circular Motion AP

Uniform Circular Motion AP Uniform circular motion is motion in a circle at the same speed Speed is constant, velocity direction changes the speed of an object moving in a circle is given by v circumference