HCI 06 Circular Motion Lecture Notes

1 Chapter 6 Circular Motion Image credits to https://en.wikipedia.org/wiki/Singapore_Flyer#/media/File:Singapore_Flyer.JPG The Singapore Flyer is currently one of the tallest Ferris wheel in the world. Described by its operators as an observation wheel, it reaches 42 stories high. At a total height of 165 m (541 feet) and a diameter of 150 m (492.1 feet), it was the tallest Ferris wheel at its inception, 5 m (16 feet) taller than the previous record holder, the Star of Nanchang. The Flyer was itself surpassed in 2014 by the High Roller in Las Vegas, which stands at 167.6 m (550 feet) with a diameter of 158.5 m (520 feet). The Ain Dubai in Dubai which was launched in 2021 stood at a height of 250 m (820 feet). D B C A

2 TOPIC 6: CIRCULAR MOTION Circular Motion Learning Outcomes Students should be able to: Kinematics of uniform circular motion (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) derive, from the definitions of velocity and acceleration, equations which represent uniformly accelerated motion in a straight line. Centripetal acceleration (e) 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. (f) recall and use centripetal acceleration a = rω2, a = v2/r to solve problems. Centripetal force (g) recall and use centripetal force F = mrω2, F = mv2/r to solve problems. Playlist of lecture examples: https://youtube.com/playlist?list=PL_b5cjrUKDlaAYoOvqyWUXVITtqdkKOSO Further Readings / References 1) University Physics with Modern Physics, 14th Ed., Young & Freedman, Chapter 9, Pg 298. 2) Advanced Level Physics, 6th Ed., Nelkon & Parker, Chapter 2, Pg 48. 3) College Physics, 8th Ed., Young & Geller, Chapter 6, Pg 161. 4) Physics for Scientists and Engineers with Modern Physics, 6th Ed., Serway Jewett, Chapter 6, Pg 150.

3 Contents TOPIC 6: CIRCULAR MOTION ................................ .. Error! Bookmark not defined. 6.1 Introduction …………………….…………………………………………………………4 6.2 Kinematics of Circular Motion …………………………………………………………5 6.2.1 How angle is measured ................................ ................................ ................ 5 6.2.2 Angular Displacement (θ) ................................ ................................ .............. 5 6.2.3 Angular Velocity (ω) ................................ ................................ ...................... 6 6.2.4 Relationship between Tangential Speed v and Angular Speed  .................. 6 6.2.5 Period (T) and Frequency (f) ................................ ................................ ......... 7 6.3 Uniform Circular Motion ………………………………………………………………….8 6.3.1 Derivation of the equation of centripetal acceleration ................................ .... .9 6.3.2 Centripetal Force (Fc) ................................ ................................ ................ ..10 6.4 Vertical Circular Motion …………………………………………………………………122 6.5 Problem solving for Circular Motion …………………………………………………...14 6.5.1 Further examples ………………………………………………………………….15 Tutorial 6 Self review Questions …………………………………………………………………………….17 Discussion Questions …………………………………………………………………………….19 Answers ……………………………………………………………………………………………24

4 6.1 Introduction Circular motion is a type of motion we often encounter in our everyday lives, - the motion of a car and its passengers when navigating a bend, the motion of the capsules of a Ferris wheel when in rotation, children on a spinning merry-go-round, and clothes being spun in a spin dryer. On a larger scale, in astronomy, we know that the moon circles the Earth, which circles the Sun, which circles the centre of the Milky Way. On the atomic scale, for early models of the hydrogen atom , we picture an electron orbiting around the nucleus at the centre of the atom (a single proton in hydrogen’s case). Have you ever wondered? • Why must an airplane tilt when executing a turn? • Why the reading of your weight would be different at the bottom of a Ferris wheel as compared to the top? • Why do people in a roller coaster not fall out at the top of the loop? (No, it is not because of the belts!) The exploration of the fascinating world of circular motion begins here. A Short Revision on basic ideas of Newton's Laws of Motion Newton's 1st Law: Describes the motion of any object not subjected to a net external force, specifically that • an object at rest remains at rest, or • an object continues with constant speed in a straight line. In the presence of a net external force, a constant mass experiences an acceleration governed by Newton's 2nd Law: = netFa m Note: • Forces which are parallel to the motion of an object result in its motion either speeding up or slowing down, with the direction of motion remaining along the original axis. • Forces which are perpendicular to the motion of the object cannot affect its speed, but instead will change its direction of motion. • Forces which are neither wholly parallel nor perpendicular to the direction of motion will affect both the speed and direction of the object upon which the force acts. (Consider projectile motion.) In this chapter you will learn that objects may move in circular paths when they experience a force or forces acting perpendicularly to their motion.

5 6.2 Kinematics of Circular Motion When an extended object such as the wheel of a bicycle is rotating , different parts of the wheel will travel different distances in the same time. The outer parts of the wheel will travel larger distances and hence will have higher speeds than the inner parts of the wheel. However, all parts of the wheel will pass through the same angle in a given time. Hence the speed of rotation of a wheel is better expressed as an angular speed, as this is the same for all parts of the wheel. 6.2.1 How angle is measured The S.I. unit for angles is not the degree but the radian. An angle measured in radians is actually the ratio of two lengths: arc length and radius. The arc length s is the distance travelled along the circular path, and the angle θ is said to subtend the arc length at the circle centre (Figure 1). Hence, note that θ is actually a dimensionless quantity since it is a ratio of 2 lengths. θ (in rad) = arc length radius s r= s = r θ One radian is the angle subtended at the centre of the circle by an arc equal in length to the radius of the circle. If the circumference of a circle s = 2πr , θ = s/r = 2π rad . Since for a complete circle, the angle subtended is 360, 360  2π rad We can deduce that 1 radian = 360 2π   = 57.3 Conversion between degrees and radians: 180  π rad Z (degrees)  Z x 180  (radians) 6.2.2 Angular Displacement (θ) The angular displacement θ is the angle an object has turned about a fixed point. In circular motion, the fixed point is taken to be the centre of the circle (Figure 2). 0Δθ θ θ=− If 0θ = 0, then Δθθ= . S.I. unit of θ: radian (rad) r s  r Figure 1 r  r Figure 2 0θ  O O

6 6.2.3 Angular Velocity (ω) The description of circular motion in angular form is analogous to the description of linear motion. Angular velocity ω (pronounced as omega) is defined as the rate of change of angular displacement (Recall: In linear motion, linear velocity is defined as the rate of change of linear displacement). Angular velocity is the rate of change of angular displacement. Average angular velocity, ω = change in angular displacement Elapsed time ω = 0 0θθ tt − − = Δ Δ θ t The instantaneous angular velocity ω is the angu