EJC Physics H206 Motion in a Circle 2023 1. Notes (FULL)

9749(202 3) H2 Physics H206 Motion In a Circle – Notes Page 1 of 16 H2 Topic 06 – Motion In a Circle We often experience the effects of centripetal (“centre- seeking”) motion on amusement rides. Thrill-seekers can opt for vertical loops in fast roller coasters, while those looking to chill can opt for carousel rides. Content: • Kinematics of uniform circular motion • Centripetal acceleration • Centripetal force Learning Objectives: Candidates should be able to: (a) express angular displacement in radians (b) show an understanding of and use the concept of angular velocity to solve problems. (c) recall and use v = rω to solve problems. (d) describe qualitatively motion in a curved path due to a perpendicular force, and understand the centripetal acceleration in the case of uniform motion in a circle. (e) recall and use centripetal acceleration a = rω2 and a = v2 r to solve problems. (f) recall and use centripetal force F = mrω2 and F = mv2 r to solve problems.

9749(202 3) H2 Physics H206 Motion In a Circle – Notes Page 2 of 16 6.0 Introduction So far, our analysis has been limited to that for linear motion. Here we will work with a simplified view of rotational motion. Nonetheless, we will be able to explain the workings of a centrifuge, the need for inclination of the road on high speed race tracks, the reason behind the thrill -factor of a roller coaster ride, and even the possibility of simulating gravity in space. 6.1 Kinematics of Uniform Circular Motion 6.1.1 Angular Displacement θ Consider an object moving in a typical anti - clockwise circular path of radius r from A to B. Since s rθ = is a ratio, angular displacement θ is technically dimensionless, though we refer to values of angular displacement as having a “unit” of radian: 6.1.2 Angular Velocity ω In an analogous sense, the angular speed of an object is the rate of change of angle (no mention of “displacement”) swept out by a radius.s θ r s A B O Legend s = arc length r = radius (or )s s r r θ θ = = One radian is the angle subtended at the centre of a circle by an arc length that is equal to the radius 1 rad s = r r Angular displacement θ is the angle swept out by a radius. θ : angular displacement (rad) s : arc length (m) r : radius (m) Angular velocity ω is the rate of change of angular displacement swept out by a radius d dt θω = A screen for the read-out of a radar. Quite often seen in popular culture such as movies, the radar readout screen, where a radius sweeps out and reveals “enemies”, is nonetheless a useful image to base your understanding of angular velocity on. ω : angular velocity (rad s−1) d dt θ : rate of angular displacement swept out by radius (rad s−1)

9749(202 3) H2 Physics H206 Motion In a Circle – Notes Page 3 of 16 6.1.3 Uniform Circular Motion Uniform circular motion occurs when an object moves round a circle with the same speed. The time it takes to complete one revolution is called the period T within which the angular velocity is d2 2d T ft θπωπ= = = . f is the frequency of the rotation: the number of rotations per unit time, 1T f= . 6.1.4 Relationship between Angular Velocity ω and Linear Velocity v quantity linear angular relation rate of change with respect to time↓ displacement s θ sr θ= velocity v ω vr ω= An object in uniform circular motion travels a full circumference in a period: circumference per 22 iodv TT r rrππ ω= = == v : linear velocity or tangential velocity (m s -1) r : radius of the circular path (m) ω : angular velocity (rad s−1) Example 1 Points A, B and C lie on the second hand of an analogue watch. (a) State and explain the point which has the largest (i) angular velocity ω, (ii) linear speed v. (b) Point C is 2.0 cm from the centre of the face, calculate the (i) angular velocity of point C, (ii) linear velocity of point C. Solution (a)(i) all 3 points have same angular velocity as they lie on the same radius and is swept through the same rate of change of angular displacement (a)(ii) so linear velocity is directly proportional to distance from centre of circular motion. C is furthest away so has largest linear velocity A • B • C • The second hand takes 60 s to complete one cycle. vr ω= 2 2 f T πω π = =

9749(202 3) H2 Physics H206 Motion In a Circle – Notes Page 4 of 16 Example 2 Singapore lies approximately on the Earth’s equator. The radius of the Earth is about 6370 km. Calculate the (i) angular velocity and (ii) linear speed of Singapore. Solution ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) 51 2 31 2 i 7 27 10 rad s west to east ii 6370 10 463 m s faster than speed of sound in air! 2 2 24 60 2 24 60 . v T r πω π ω π −− − = = = × =  = ×=  6.2 Centripetal Acceleration and Centripetal Force Centripetal means “centre-seeking”. In circular motion, the force (also acceleration) is directed towards the centre of the circular path. By Newton’s 1st Law, if there is no net force on an object, it will continue in its state of rest or constant velocity. Inertia will cause the object to continue moving with initialv along a straight line ( ). For an object in uniform circular motion, speed remains constant but direction changes so velocity changes. Therefore, acceleration must exist. ( )final initial final initialvv v v v = +∆= − − By Newton’s 2nd Law, the object has to experience a net force that is given by its rate of change of momentum. The net force (also the acceleration) is directed towards the centre of the circular path. θ θ The Earth takes 24 h to rotate one cycle about its axis.

9749(202 3) H2 Physics H206 Motion In a Circle – Notes Page 5 of 16 centripetal acceleration ca centripetal force cF 2 c va r= 2 c mvF r= 2 ca rω= 2 cFm rω= • Centripetal force cF is a resultant force cF is the “leftover” net force resulting from real forces such as contact, tension, weight etc. Therefore do not draw centripetal force in a free body diagram. • For uniform circular motion, constant linear speed implies zero tangential force. The centripetal force only changes the direction of motion but not the speed. Tension in the string provides the centripetal force to keep the ball in circulation motion. If the string were to snap suddenly, by Newton’s 1st Law, the ball will tend to continue in its state of constant velocity just before the string snaps: Ball will continue moving off tangential to the original circular motion. A toy tower has a curved track for a marble to roll down along a circular path onto the floor smoothly. Ball will continue on the floor along path C, which is tangential to the curved track just before the track ends. Example 3 Explain why is there no work done during circular motion by a centripetal force. Solution [direction] Force* that provides centripetal force for the object to undergo uniform circular motion is always perpendicular to the direction of motion of the object [w.d. definition] Hence no displacement in the direction of the force and therefore no work done. Note: need to specify the type of real force when answering such questions to demonstrate the knowledge that centripetal force is a resultant force and not a new type of force. e.g. “tension provides the centripetal force” / “gravitational force provides the centripetal force “ etc. A B C D

9749(202 3) H2 Physics H206 Motion In a Circle – Notes Page 6 of 16 Example 4 6.3 General Approach to Solving Problems relating to Circular Motion 1. Draw a free body diagram showing all the forces acting on the body. 2. Determine the centre of the circular path.