WJEC Physics for A2: Student Bk

10 Every six seconds or so, the excited riders on the swing carousel in Barcelona travel along a horizontal circle about 20 m in diameter. Nor are they alone in this behaviour as the wall of death rider shows. We all, along with other denizens of the Earth, execute a multiplicity of motions that, if not exactly, are close to circular: • Daily rotation around the axis of the Earth • Annual path of the Earth around the Sun • 240 million-year orbit of the Solar System around the 8 kiloparsec distant galactic centre. As this section is being written, astronomers are celebrating the detection of the gravitational waves which were emitted when two black holes, previously in mutual orbit, coalesced into a single body. This one observation simultaneously confirmed the outstanding prediction of General Relativity theory and provided the first direct observational evidence for black holes themselves (rather than the matter which spirals into them). To get back to Earth, this section introduces the mathematics needed to describe and understand the behaviour of circling objects – their motion and the forces which produce it. The mathematics is not difficult but the physics concepts contain some traps for the unwary. You have been warned! 3.1.1 The kinematics of ‘uniform’ circular motion 1 For the most part we shall limit our study to objects which are moving at a constant speed in circles. This is not unduly restrictive – we shall still allow the swing carousel to spin up and down but only consider the forces and motion when it is at a steady speed. Option C, The physics of sport, considers angular accelerations and decelerations. (a) Angular position and speed Consider an object, P , which is moving at a constant speed, v , in a circular path of radius r about a central point, O . We measure its linear motion along the circumference from the point X. The angular position, θ , as shown is measured from the line OX . We define the angular speed, ω , as the rate of change of θ , i.e. the angle swept out by the radius per unit time. Mathematically: Units Scientists often express angles in degrees (°) for reasons of convenience and the familiarity of the audience with these units. Astronomers often subdivide degrees into minutes (') and seconds (") of arc. However, for rotational and vibrational purposes we shall always use radians. The definition of radian measure is covered in the Year 1 / AS textbook, Section 4.2.4. Hence: Unit of angular position, θ : rad . Unit of angular speed, ω : rad s − 1 . ω = Δ θ Δ t 3.1 Circular motion 1 The quotation marks around ‘uniform’ are because an object which is travelling in a circle has not got a uniform velocity because the direction is changing – something to be aware of. Fig. 3.1.1 Circular motion in Barcelona Fig. 3.1.2 Wall of death rider O X r v P d Fig. 3.1.3 Angular position Fig. 3.1.4 Multiple orbits in a star / black hole system