FLEXIBLE LEARNING APPROACH TO PHYSICS Module P2.6 Circular motion 1 Opening items 3.4 Centripetal force 1.1 Module introduction 4 Applications of circular motion 1.2 Fast track questions 4.1 Designing safe bends for motorways 1.3 Ready to study? 4.2 Satellites and communications satellites 2 Characteristics of uniform circular motion 5 Closing items 2.1 A satellite in a circular orbit 5.1 Module summary 2.2 A review of Newton’s first and second laws of 5.2 Achievements motion 5.3 Exit test 2.3 A simple experiment Exit module 2.4 Circular motion in general 3 A mathematical description of circular motion 3.1 Speed, velocity and angular speed 3.2 The period of circular motion 3.3 Centripetal acceleration FLAP P2.6 Circular motion COPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1 1 Opening items 1.1 Module introduction Why study circular motion? For the ancient Greeks the circle symbolized perfection. The fact that the Sun, Moon and stars appeared to move around the Earth in circular paths was evidence of their status as ‘heavenly bodies’. Circular motion was supposedly their natural state. Today we know that the Greek picture of how things move was wrong. The ‘natural state’ of any body, heavenly or otherwise, is motion in a straight line at constant speed unless it is subject to a force. Why then do we need to study circular motion? Although we don’t regard movement in a circle with the same reverence as did the ancient Greeks, it is still a very useful concept. If we restrict our discussion to circular motion at constant speed (called uniform circular motion), then its mathematical description turns out to be quite straightforward. There are many situations in nature which approximate to uniform circular motion; the motion of the Earth around the Sun, the orbits of the Moon and artificial satellites around the Earth, fairground rides, cars going around bends and the internal workings of washing machines are just some examples which come to mind. FLAP P2.6 Circular motion COPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1 This module deals with the physics of uniform circular motion and its applications. Some examples of uniform circular motion, e.g. artificial Earth satellites, are described in Section 2 to give you a ‘feel’ for what they have in common. This leads in to a more rigorous, mathematical description of circular motion in Section 3, where the terms angular speed and centripetal acceleration are introduced. Some practical applications of these ideas are discussed in Section 4. Before you start to work through the material, here are a couple of questions you might like to ponder. By the time you reach the end of Section 3 you should be in a position to answer them yourself. Detailed explanations of the answers to these questions are given in Section 4. Suppose you are designing a motorway. How can you ensure that the bends will not cause vehicles travelling at 701mph to skid? Communications satellites are most effective if they are permanently positioned above a particular point on the Earth’s surface. How is this possible? Can you position such a satellite above any point on the Earth’s surface? Could you, for example, have one permanently stationed over Britain? Study comment Having read the introduction you may feel that you are already familiar with the material covered by this module and that you do not need to study it. If so, try the Fast track questions given in Subsection 1.2. If not, proceed directly to Ready to study? in Subsection 1.3. FLAP P2.6 Circular motion COPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1 1.2 Fast track questions Study comment Can you answer the following Fast track questions?. If you answer the questions successfully you need only glance through the module before looking at the Module summary (Subsection 5.1) and the Achievements listed in Subsection 5.2. If you are sure that you can meet each of these achievements, try the Exit test in Subsection 5.3. If you have difficulty with only one or two of the questions you should follow the guidance given in the answers and read the relevant parts of the module. However, if you have difficulty with more than two of the Exit questions you are strongly advised to study the whole module. Question F1 Show, from first principles, that an object undergoing uniform circular motion must be experiencing an acceleration which is directed towards the centre of the circle and has a magnitude v2/r where v is the speed of the object and r is the radius of the circle. FLAP P2.6 Circular motion COPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1 Question F2 A planet orbiting a star experiences a force of magnitude 3.02×2101221N due to the gravitational attraction of the star. If the planet has a speed of 2.02×210151m1s1−11 and it takes 2.62×210171s to complete one orbit, calculate the mass of the planet and the radius of its orbit. Study comment Having seen the Fast track questions you may feel that it would be wiser to follow the normal route through the module and to proceed directly to Ready to study? in Subsection 1.3. Alternatively, you may still be sufficiently comfortable with the material covered by the module to proceed directly to the Closing items. FLAP P2.6 Circular motion COPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1 1.3 Ready to study? Study comment To study this module you will need to understand the following terms: acceleration, arc length, force, gravity, mass, Newton’s laws of motion, radian, radius, speed, tangent, velocity, weight. There are also some mathematical topics with which you will need to be familiar before starting to study: derivative, magnitude of a vector, Pythagoras’s theorem, scalar, triangle rule for addition of vectors, trigonometry, vector. If you are uncertain about any of these ideas then you can review them now by reference to the Glossary, which will also indicate where in FLAP they are developed. The following Ready to study questions (the answers for which are given in Section 6) will allow you to establish whether you need to review some of the topics before embarking on this module. FLAP P2.6 Circular motion COPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1 Question R1 A cyclist moving along a straight road accelerates uniformly from 5.001m1s1−11 to 10.01m1s1−11 in 10.01s. Calculate: (a) the magnitude of the acceleration; (b) the average speed over the interval; (c) the distance travelled over the interval. FLAP P2.6 Circular motion COPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1 Question R2 Figure 1 shows a graph of distance s travelled against time for an aircraft on a runway, accelerating towards 400 take-off. (a) Calculate the average speed between times t2=251s 300 and t2=2151s. (b) Use the graph to estimate the instantaneous speeds at the same two times. 200 (c) Estimate the average acceleration between t2=251s distance/m and t2=2151s. 100 0 246 81012 14 16 time/s Figure 13See Question R2. FLAP P2.6 Circular motion COPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1 Question R3 In Figure 2 if the arc length AB is 101cm, and the radius r is 121cm, calculate the angle θ in radians. B r Question R4 θ The vector a points North and has a magnitude of 4 units; the vector b points East and has a magnitude of 3 units. Describe the vector c where c2=2a2+2b. A Figure 23See Question R3. FLAP P2.6 Circular motion COPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1 2 Characteristics of uniform circular motion Uniform circular motion is simply motion in a circle at constant speed. In this section we will look at some of the situations in which this kind of motion occurs and we will consider some of its characteristics. N 2.1 A satellite in a circular orbit Imagine yourself at the North Pole, throwing a snowball horizontally as hard as you can. It would hit the ground some distance away from you (Figure 3a) and if you knew its initial velocity you could easily work (a) out how far it would travel by considering it as a projectile, moving Figure 3a3Launching a snowball . along a parabolic path. ☞ FLAP P2.6 Circular motion COPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1 N N (b) If you were able to project it faster and faster (say from some kind of gun), it would travel further and further before touching down. At higher speeds you would have to take the curvature of the Earth into account in order to predict where it would land (Figure 3b). Finally, if you gave the snowball exactly the right horizontal velocity, you would find that the Earth’s surface would ‘fall away’ from the snowball, because of the curvature, at just the same rate as the snowball falls towards the Earth. It would then describe a circle (c) around the Earth and arrive back at the starting point with exactly the same speed as when it set out (Figure 3c). For its whole journey, the Figure 3b & 3c3Putting a snowball into snowball would be travelling at constant speed at the same fixed orbit. height above the Earth’s surface. FLAP P2.6 Circular motion COPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1 Could this really happen? I have to admit the answer is ‘No’. The snowball would be slowed down by air resistance, its path would be affected by the slightly non-spherical shape of the Earth and, travelling just above the ground, it would almost certainly collide with something.