Gravitation

Why does an apple fall to the ground, and why does the Moon keep moving around the Earth instead of flying away? Newton realised that the same force explains both. This chapter begins with Kepler's three laws of planetary motion: the law of orbits, the law of areas and the law of periods, T^2 proportional to a^3. It then states Newton's universal law of gravitation, F = Gm1m2/r^2, and describes the gravitational constant G = 6.67 x 10^-11 N m^2 kg^-2, first measured by Cavendish. You will learn that a uniform spherical shell attracts an outside mass as if all its mass were at its centre, and exerts no force on a mass inside it. The chapter derives the acceleration due to gravity g = GM/R^2 and explains how g decreases with height above the surface and with depth below it. You will study gravitational potential energy, -GMm/r, and the escape speed sqrt(2gR), about 11.2 km/s for the Earth. Finally, you will find the orbital speed, period and energy of satellites in circular orbits.

Detailed Worksheet: Gravitation Section A - Definitions (10 marks) 1. State Kepler's law of areas. (2 marks) 2. Write the SI unit and value of the gravitational constant G. (2 marks) 3. What is the force between two point masses of 1 kg each placed 1 m apart? (2 marks) 4. What is the value of g at the centre of the Earth? Give a reason. (2 marks) 5. Why is the total energy of an orbiting satellite negative? (2 marks) Section B - Calculations and Applications (15 marks) 6. Using g = GM/R^2, calculate g at the Earth's surface. Take M = 6 x 10^24 kg, R = 6.4 x 10^6 m and G = 6.67 x 10^-11 N m^2 kg^-2. (3 marks) 7. A body weighs 72 N on the Earth's surface. Find its weight at a height equal to the Earth's radius and at a depth of half the Earth's radius. (3 marks) 8. Calculate the escape speed from the Earth's surface. Take g = 9.8 m/s^2 and R = 6.4 x 10^6 m. (3 marks) 9. A planet has twice the mass and twice the radius of the Earth. Compare its g and escape speed with those of the Earth. (3 marks) 10. Mars is 1.52 times as far from the Sun as the Earth. Use Kepler's third law to find the length of a Martian year in Earth years. (3 marks) Section C - Diagrams (10 marks) 11. Draw an elliptical orbit of a planet around the Sun, marking the perihelion, aphelion and equal areas swept in equal times. (4 marks) 12. Draw a graph showing the variation of g with distance r from the centre of the Earth, both inside and outside it. (3 marks) 13. Draw a diagram of a satellite in a circular orbit showing the gravitational force and velocity. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Derive the expression for the variation of g with height h above the Earth's surface, and show that for small h, g' = g(1 - 2h/R). (5 marks) 15. Derive the expression for escape speed from the surface of a planet. Show that it does not depend on the mass of the escaping body. (5 marks) 16. A 400 kg satellite is in a circular orbit of radius 2R about the Earth. How much energy is needed to transfer it to a circular orbit of radius 4R? Take g = 9.8 m/s^2 and R = 6.4 x 10^6 m. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Take g = 9.8 m/s^2, R = 6.4 x 10^6 m and G = 6.67 x 10^-11 N m^2 kg^-2 unless stated otherwise.