RI Chap 7 Circular Motion Lecture Notes

7 CIRCULAR MOTION H2 Physics 9478 Content Page 7.1 Introduction 2 7.2 Kinematics of Uniform Circular Motion 3 7.3 Dynamics of Uniform Circular Motion 7 7.4 Problem Solving Strategy for Uniform Circular Motion 11 7.5 Uniform Horizontal Circular Motion 11 7.6 Vertical Circular Motion 16 Learning Outcomes Candidates should be able to: (a) express angular displacement in radians. (b) show an understanding of and use the concept of angular velocity. (c) recall and use vr ω= to solve problems. (d) show an understanding of centripetal acceleration in the case of uniform motion in a circle, and qualitatively describe motion in a curved path (arc) as due to a resultant force that is both perpendicular to the motion and centripetal in direction. (e) recall and use centripetal acceleration 2ar ω= and 2avr= to solve problems. (f) recall and use 2F mr ω= and 2F mv r= to solve problems.

Page | 2 7.1 Introduction Circular motion is a common occurrence in our daily lives. The second hand of a clock goes round in a circle in 60 seconds while the Earth orbits the Sun once in 365 days. You probably have also experienced circular motion when you sat on a carousel or in the Singapore Flyer. As physics students, you want to go beyond the physical experience. You want to understand what causes an object to move in a circular motion and the physical quantities associated with it. For a body moving in a circular path, it must already be travelling with a certain linear speed and at the same time, there must be a force applied on it that is directed towards the centre of the circular path. As the body tries to move off in a straight line, the applied force towards the centre pulls the body inwards, causing it to move in a curve. The continuous application of this centre- pointing force allows the body to move in a circular path. This force is aptly given the name centripetal force wher e “centri” means centre and “petal” means pointing in Latin. The H2 Physics syllabus classifies circular motion into two categories: uniform and non-uniform circular motion. Uniform circular motion can be described as the motion of a body in a circle at a constant speed. As the body moves in a circle, it is continuously changing its direction. At all instances, the body ’s velocity is tangent to the circle. For the magnitude of the velocity (speed) to remain constant, the resultant force on the body must continuously act perpendicular to it s velocity . Hence the resultant force only has a centripetal component which points radially towards the centre of the circular path, with no component in the tangential direction. In a non-uniform circular motion, the speed of the body is varying. This occurs when the resultant force on the body has both centripetal (radial) and tangential component s. The tangential component of the resultant force causes the speed to vary. The centripetal component continuously changes the direction of the body. In this topic, you will solve problems on uniform circular motion in a horizontal plane and uniform as well as non-uniform circular motion in a vertical plane.

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT 3 | Page 7.2 Kinematics of Uniform Circular Motion It is not possible to apply the kinematics equations for constant acceleration to uniform circular motio n. This is because for a body moving in circular motion, its path is not straight and the direction of its acceleration is constantly changing implying that acceleration varies . Hence, we need a new set of equations with an understanding of certain important quantities. Consider a body P moving in a uniform circle of radius r about point O as shown in Fig. 7.1. Body P has an angular velocity ω and a linear velocity v at any point in its path. Fig. 7.1 Points a, b, c, d and e are points on its circular path. At time 0t = , body P is at point a. After a time interval of t∆ , body P has rotated through an angle of θ to point b. Body P makes one complete revolution when it rotates from points a to c to d to e and back to a.  Period and Frequency S.I. unit: s The period of body P is the total time it takes to movefrom positions a to c to d to e and back to a once. Period The period of a body in circular motion is the time taken for it to make one complete revolution. x y • O P v r θ ω a b c d e

Page | 4 S.I. unit: Hz Period T and frequency f are related by the equation 1f T=  Angular Displacement S.I. unit: radian. The angular displacement of body P in the time interval of t∆ is θ. In general, any angle θ measured in radians is defined by the relation: s rθ = where s is the arc length and r is the radius of the circle. When the body moves one complete circle, the arc length s would be the circumference of the circle, which is equal to 2 rπ . Therefore, 2 2r r= =πθπ Angular displacement Geogebra animation Frequency The frequency of a body in circular motion is the number of revolutions made per unit time. Radian The radian is the angle subtended by an arc length equal to the radius of the arc. Angular Displacement Angular displacement is the angle a body rotates through with respect to its initial position.

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT 5 | Page Since the body moves through 360o in one complete circle, 2 rad 360 rad 180 π π = ° = ° In general, 180 radxx π °= × Conversely, rad180yy π°= × Example 1 Fill in the blanks. Angle in degrees Angle in radians (in terms of π) 360 90 180 45 30 1 Example 2 The laser in a CD player is 5.0 cm from the central of the disc. What length of the disc is scanned by the laser when the disc turns through an angle of 0.45 radians?  Angular Velocity Angular velocity Geogebra animation Angular Velocity Angular velocity of a body is defined as the rate of change of its angular displacement with respect to time.

Page | 6 Angular velocity, d dt θω = where θ is the angular displacement and t is the elapsed time. S.I. unit: rad s−1 Angular velocity is a vector – it has both magnitude and direction. The direction is described as “clockwise” or “anti-clockwise”. For a body in uniform circular motion, its angular velocity is constant.  Relationship between Angular Velocity and Linear Velocity Differentiating s = r θ with respect to t and since r is a constant, ds d rdt dt θ= vr ω∴= where dsv dt= and d dt θω = . Linear velocity v is also known as the tangential velocity. Linear or tangential velocity is a vector quantity. Linear speed is the magnitude of linear velocity. For a body in uniform circular motion, its linear speed is constant.  Relationship between Angular Velocity, Period and Frequency Time taken for one complete revolution, period = T Angular displacement for one complete revolution = 2π From the definition, d dt θω = Therefore, 2 T πω = Since 1T f= , 2 fωπ=

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT 7 | Page  Relationship between Period and Linear Speed Linear speed v of a body moving in a uniform circular motion, circumference of the circlev T= Therefore, 2 rT v π= Example 3 Alice and Bob are riding on a merry -go-round, which is rotating at a constant angular velocity. Alice stands on a point twice as far from the centre of the platform as Bob. Which of the following statements is correct? A. Alice's linear velocity is twice of Bob's. B. Alice's linear velocity is the same as Bob's. C. Alice's linear velocity is half of Bob's. D. Alice's linear velocity is a quarter of Bob's.