WJEC Physics for A2: Student Bk

17 3.1 Circular motion At the bottom of the loop (Fig. 3.1.17) the contact force, C' , and mg now oppose each other so the resultant towards the centre is C' – mg . Hence mv 2 r = C' – mg , ∴ C' = mv 2 r + mg The minimum value of C' is mg as expected. Example Find an expression for the speed with which a car enters a loop-the-loop or radius r at the bottom, if it just remains in contact with the loop at the top. Answer The speed at the top is given by v 2 = rg . So its kinetic energy at the top is 1 2 mrg . The gain in potential energy as the car climbs the loop = 2 mgr ∴ The loss in kinetic energy as the car climbs the loop = 2 mgr ∴ Initial kinetic energy = 1 2 mrg + 2 mgr = 5 2 mrg ∴ 1 2 mv 2 = 5 2 mrg , ∴ The initial velocity v = √ 5 rg (b) Newton’s law of gravitation Until the 16th century it was a widely held belief that the scientific laws of the sublunary sphere (the geocentric universe below the sphere of the Moon, i.e. the Earth and its atmosphere 3 ) were different from those of the rest of creation, which was made of quintessence . Newton showed that the inverse square law related the acceleration due to gravity on the surface of the Earth (i.e. 9.81 m s − 2 ) with the centripetal acceleration of the Moon in orbit around the Earth. Self-tests 3.1.10 and 3.1.11 take you through this calculation. Note that the calculation is not 100% accurate; because the Moon doesn’t orbit the centre of the Earth but the centre of mass of the Earth-Moon system (see Section 4.3). (c) The shape of a rotating planet What is the shape of the Earth? Apart from a few wrinkles (mountains. continents, oceanic basins…) it is spherical. Well not quite. If you examine the Hubble Space Telescope image of Jupiter in Fig. 3.1.19 and compare its outline with the red circle, you’ll notice that it is slightly flattened at the poles. The Earth is similarly shaped, but the deviation from a sphere is less. Why is this? We’re not going to allow ourselves to say, ‘Centrifugal force,’ and swiftly pass on. So, what’s the story? A non-rotating planet would be spherical. This is the minimum potential energy shape. Even dwarf planets – Ceres and Pluto – have sufficient mass to make solid rocks flow into this form. So we’ll start by looking at the forces on a hypothetical spherical rotating planet – it might as well be the Earth. Study point The same principles hold for the roller coaster (Fig. 3.1.18). This time, however, the questions is, ‘What is the maximum speed to stay in contact with the track?’ Fig. 3.1.18 Roller coaster 3.1.10 Self-test The mean radius of the Earth is 6371 km. g at its surface in 9.81 m s −2 . Use the inverse square law to estimate the acceleration due to the Earth’s gravity at 385 000 km, the orbital radius of the Moon. 3.1.11 Self-test The orbital period of the Moon is 27.3 days. Using the orbital radius from Self-test 3.1.10, calculate the Moon’s centripetal acceleration and compare it with the answer to Self-test 3.1.10. r Fig. 3.1.19 The shape of Jupiter 3 … also comets, which were long held to be atmospheric phenomena.