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Circular Motion 6.1 – Circular Motion A-Level Physics: 0. Introduction 1. Measurements and their Errors 2. Particles and Radiation 3. Waves and Optics 4. Mechanics and Materials 5. Electricity 6. Further Mechanics and Thermal Physics 6.1 Circular Motion 6.2 Simple Harmonic Motion 6.3 Thermal Physics 7. Fields and their Consequences 8. Nuclear Physics 12. Turning Points in Physics Objects moving in a curved path is just a special case of motion along curved path. The main reasons that we focus on circular motion is that the maths works out nicely and that there are countless examples of objects moving in paths which, if not exactly circular, can be approximated as such. The planets orbiting the Sun and the electron orbiting the atom are two examples. There are plenty of examples of circular motion in our everyday life: a CD spinning in a CD player, food in a microwave, a spinning football, a loop-the-loop at the fairground, a car going around a roundabout – these are all examples of circular motion. There are three main parts to this booklet. In the first five sections, we develop the necessary tools to understand circular motion. Sections 6 to 8 then look at applying these tools to three different types of situation. There is only a small amount of new Physics here, but it can be applied to a vast array of different circumstances. My aim is that it will be easier to see the links between all the different types of problems if we break it down like this. Section 10 onwards develops our ideas of circular motion beyond the A-Level course and is intended for anyone looking for more to quench thirst for more circular motion. There are 7 sets of Mastery Problems embedded within the main text. The problems focus on one specific skill that has been discussed in the previous section. The aim here is to isolate different types of problems to ensure we understand the key points before proceeding. The mixed problems in Section 9 are intended to push you and require a greater degree of problem solving. As ever, I am always looking to improve my resources so please let me know if there is anything that needs improving or any errors. Mr Hynes 31/05/2020 Several Corrections Made 18/06/2020 Specification Reference 6.1 Circular Motion = AQA 3.6.1.1 Circular Motion 6.1 Circular Motion Contents 1. Angles & Radians ......................................................................................................... 4 2. Frequency & Time Period ......................................................................................... 5 3. Angular Speed, ω.......................................................................................................... 5 3.1 Angular Speed and Frequency ........................................................................ 6 3.2 Angular Speed and Linear Speed ................................................................... 7 4. Centripetal Force ...................................................................................................... 10 4.1 A Centrifugal Aside ........................................................................................... 11 5. Centripetal Acceleration ........................................................................................ 12 5.1 g-Forces ................................................................................................................ 12 5.2 *Proof of a = v2/r ............................................................................................... 13 6. Motion in Horizontal Circles................................................................................. 14 6.1 The Swinging Mass ........................................................................................... 14 6.2 A Note on Energy .............................................................................................. 16 6.3 Rounding the Bend ........................................................................................... 16 6.4 Take the A Train ................................................................................................ 17 6.5 Using Weight ....................................................................................................... 18 6.6 Other Examples of Centripetal Force ........................................................ 19 6.7 More Worked Examples ................................................................................. 19 7. Conical Pendula and Banked Turns ................................................................... 22 7.1 More Mass on a String ..................................................................................... 22 7.2 The Conical Pendulum .................................................................................... 23 7.3 Banked Turns ..................................................................................................... 24 8. Motion in a Vertical Circle ..................................................................................... 28 8.1 Mass and String ................................................................................................. 28 8.2 Loop the Loop ..................................................................................................... 28 8.3 Bucket of Water ................................................................................................. 30 9. Mixed Problems ......................................................................................................... 32 10. *Friction ..................................................................................................................... 37 10.1 Investigating Friction ................................................................................... 37 10.2 Modelling Friction .......................................................................................... 39 10.3 Friction Problems ........................................................................................... 40 10.4 Bends Reloaded .............................................................................................. 43 10.5 Motorbike Turns ............................................................................................. 44 10.6 The Spinning Terror ...................................................................................... 45 11. *Non-Inertial Frames & Fictious Forces ........................................................ 47 12. *Moment of Inertia ................................................................................................ 47 13. * Angular Momentum ........................................................................................... 47 Important Note: Sections marked with a * and/or indented with a blue bar like this go beyond the course specification and are non-examinable. They build on the core content to help you build your understanding of circular motion. There are many parallels between circular and linear motion and these topics are designed to give the more advanced student a wider circular experience! 1. Angles & Radians If an object is moving along a circular path, it makes more sense to refer to its position on the circle using a radius r and an angle θ, as opposed to coordinates x and y. For circular motion, r will remain constant and θ will change, as shown in the diagram opposite. We saw in the waves topic that it is sometimes more convenient to use radians as a unit to measure an angle. But where does the radian come from? In Figure 2, we see the angle θ in radians is defined as the ratio of the arc length s to the radius r, in other words 푠 휃 = 푟 If s = r then it follows that θ = 1 rad. Therefore, 1 radian is the angle Figure 1. subtended at the centre of a circle by and arc equal in length to the radius. When s = 2πr (the circumference of a circle of radius r), then θ = 2π radians = 360°. Hence 360° = 2휋 radians So why bother using radians then? Degrees can be useful – the number 360 has 24 divisors and is divisible by every number between 1 and 10, other than 7. This is a very useful property when discussing angles. However, radians can prove very useful when simplifying trigonometric expressions for small angles. The details are not necessary here, but it can be shown that sin 휃 ≈ 휃 if 휃 is small and measured in radians. We will not be using that here, but it is always good to know why we do Figure 2 things. It is called the small angle approximation, should you want to go and find out more. Example 1.1: A toy car moves around a circular track of radius 0.60 m. If the car travels 1.8 m along the track, what angles has the car moved through? 푠 1.8 360 휃 = = = 3.0 rad = 3.0 × = 172° 푟 0.6 2π Mastery Problems A – Angles & Radians 1. Without the use of a calculator, convert the following angles into radians in terms of π: 360°; 180°; 90°; 60°; 30°; 45°; 20°; 270°; 150°; 22.5°; 80°; 300°. 2. Without the use of a calculator, convert the following angles into degrees: 3휋 5휋 휋 3휋 11휋 휋 4휋 휋 휋 4휋 ; ; ; ; ; ; ; ; ; 4 6 10 2 6 3 3 12 8 9 3. Express the following angles in radians: 35°; 82°; 53°; 71°; 44°. 4. Express the following angles in degrees: 2.86 rad; 4.21 rad; 1 rad; 3.47 rad; 3 rad. 2. Frequency & Time Period Circular motion is a type of periodic motion. This means the motion repeats itself over a certain time interval. We can therefore use time period and frequency to describe circular motion: Time period, T = time taken for one complete revolution [s]. Frequency, f = number of revolutions per second [s-1 = Hz] where, 1 푓 = 푇 Note: a more commonly used unit for frequency for everyday
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