WJEC Physics for A2: Student Bk

11 3.1 Circular motion Example A vinyl record disc spins at 33 revolutions per minute. Calculate the angular speed in rad s –1 . Answer In 60 s , the angle swept out = 33 × 2 π = 66 π rad ∴ ω = Δ θ = 66 π rad Δ t 60 s = 3.46 rad s − 1 . (b) Relationships between angular and linear speed The linear quantities in circular motion are indicated in red in Fig. 3.1.3; we can use this diagram to relate them to the rotational quantities. By definition If P is moving with speed v , the increase in distance, Δ d in time Δ t is given by Δ d = v Δ t So the angular position increase, ∴ Dividing by Δ t and using gives Example Calculate the speed of the Earth’s motion around the Sun [ 1 AU = 1.50 × 10 11 m ] Answer The angular velocity ω = Δ θ = 2 π Δ t (86400 × 365.25) s = 1.99 × 10 −7 rad s −1 . ∴ The speed, v = r ω = 1.50 × 10 11 m × 1.99 × 10 −7 rad s −1 = 29.9 km s −1 . Angular speed or angular velocity? Motion in a circle is directional so it has vector properties. However, the direction of the linear velocity is constantly changing. The one constant aspect is the direction of the axis of rotation, so this is chosen as the direction of the angular velocity vector (using the right hand grip rule, see Fig. 3.1.5). We shall not be treating the vector aspects of angular motion in this course, though it is touched upon in Option C, The physics of sports. In spite of treating only the scalar aspects of rotational motion, it is common to refer to angular speed as angular velocity and we shall use the terms interchangeably in this section of the book, in the same way that people often talk about the velocity of light. θ = d r Δ θ = v Δ t r ω = v r ω = Δ θ Δ t Study point It is often convenient to leave π explicitly in an angular speed, e.g. in the example ω = 1.1π rad s −1 . 3.1.1 Self-test The magnetic tape in an old reel- to-reel tape player is fed through the play-back heads at a constant 9.5 cm s −1 . Calculate the angular velocity of the reel when the diameter of the tape spool is 15 cm. 3.1.2 Self-test The sound waves on a 33 rpm vinyl record are produced using wavy grooves. (a) Calculate the wavelength of the grooves for a 1 kHz note at the outside of the 30 cm diameter disc. (b) How would a 1 kHz groove be different near the centre of the disc? Fig. 3.1.5 Direction of angular velocity vector