Gravitational Potential Energy - UNSOLVED PRACTICE SET

Q7. Define gravitational potential energy. Why is it negative when the zero is taken at infinity?

Q8. Derive the expression for gravitational potential energy of a body of mass m at height h above Earth's surface (for h << R).

Q9. Calculate the gravitational potential energy of a 1 kg mass on Earth's surface. (M = 6 × 10²⁴ kg, R = 6.4 × 10⁶ m, G = 6.67 × 10⁻¹¹ Nm²/kg²)

Q10. In your school, a student lifts a 5 kg book from the floor to a shelf 2 m high. Calculate the increase in gravitational potential energy using both mgh and the exact formula. Compare the results.

Q11. What is binding energy? Calculate the binding energy of a 1000 kg satellite on Earth's surface.

Q12. Explain why the formula U = mgh is only an approximation and when should the exact formula U = −GMm/r be used?

Q13. Derive the expression for gravitational potential energy of a body of mass m at distance r from Earth's centre, with zero potential at infinity. Show that near Earth's surface, this reduces to U = mgh. Discuss why the potential energy is negative and what this physically signifies.

Q14. A body of mass 1000 kg is taken from Earth's surface to a height of 3R, where R is Earth's radius.

(a) Calculate the change in gravitational potential energy using the exact formula.

(b) Calculate what the change would be if you used mgh with g = 9.8 m/s².

(c) Find the percentage error in using the approximate formula.

(d) At what height does the approximate formula give an error of 10%?

(e) Discuss when each formula is appropriate.

Q15. Three particles of masses m, 2m, and 3m are placed at the vertices of an equilateral triangle of side a.

(a) Calculate the gravitational potential energy of the system.

(b) Calculate the work done in separating the particles to infinity.

(c) If the side of the triangle is doubled, how does the potential energy change?

(d) What is the significance of the negative sign in the potential energy of a bound system?

Q16. A 500 kg satellite orbits Earth at a height of 600 km above the surface.

(a) Calculate the gravitational potential energy of the satellite.

(b) Calculate the work required to move the satellite to a height of 1200 km.

(c) Calculate the minimum energy required to send the satellite out of Earth's gravitational field from its initial orbit.

(d) If the satellite's kinetic energy in orbit is equal to half the magnitude of its potential energy, calculate its total energy.

(e) Discuss the energy transformations if the satellite's orbit slowly decays due to atmospheric drag.

Q17. The gravitational potential energy of a two-body system (Earth + body) is given by U = −GMm/r.

(a) Plot U vs r for r > R.

(b) Show that the slope of this curve gives the gravitational force.

(c) A body falls from rest from height 2R to R. Calculate the change in potential energy.

(d) Using energy conservation, calculate its speed when it hits the surface.

(e) Compare this with the speed obtained using kinematic equations (v² = 2gh).

Q18. In a school experiment, students verify the conservation of energy using a pendulum.

(a) A 200 g bob is raised to a height of 10 cm. Calculate its gravitational potential energy.

(b) Calculate its speed at the lowest point using energy conservation.

(c) If the measured speed is 5% less than the calculated value, calculate the energy lost.

(d) Where does this "lost" energy go?

(e) Design an experiment to minimise this energy loss and improve accuracy.