Conservation of Mechanical Energy - UNSOLVED PRACTICE SET

Q7. State the law of conservation of mechanical energy. Under what conditions is it valid?

Q8. Prove that for a freely falling body, the sum of kinetic and potential energies remains constant at all points during the fall.

Q9. A 1 kg stone is thrown upward with a speed of 20 m/s. Calculate its total mechanical energy at:

(a) The point of projection

(b) The highest point

(c) When it returns to the ground

Q10. In your school, a student on a swing is pulled back to a height of 1 m and released. Explain the energy transformations as the swing moves back and forth, assuming negligible air resistance.

Q11. A block slides down a frictionless incline from height h. Show that its speed at the bottom is √(2gh), independent of the angle of inclination.

Q12. Why is mechanical energy not conserved in real-life situations like a bouncing ball or a sliding block? Where does the "lost" energy go?

Q13. State and prove the law of conservation of mechanical energy for a body falling freely under gravity. Show that at any point during the fall, the sum of kinetic and potential energies equals the initial potential energy. Draw energy-position and energy-time graphs.

Q14. A 2 kg block is released from rest at the top of a frictionless curved track of height 4 m. It slides down and encounters a loop of radius 1 m.

(a) Calculate the speed of the block at the bottom of the track.

(b) Calculate the speed at the top of the loop.

(c) Calculate the normal reaction on the block at the top of the loop.

(d) What is the minimum height from which the block must be released to complete the loop?

(e) Discuss what happens if the block is released from a lower height.

Q15. A simple pendulum of length 2 m with a bob of mass 500 g is displaced by 60° and released.

(a) Calculate the maximum potential energy.

(b) Calculate the maximum kinetic energy and maximum speed.

(c) Calculate the speed when the string makes 30° with the vertical.

(d) Calculate the tension in the string at the lowest point.

(e) If the pendulum eventually stops due to air resistance, explain what happens to the mechanical energy.

Q16. A 500 g ball is dropped from a height of 10 m. It bounces back to a height of 6 m.

(a) Calculate the mechanical energy at the initial height.

(b) Calculate the mechanical energy just before the first bounce.

(c) Calculate the mechanical energy just after the first bounce.

(d) Calculate the percentage loss of mechanical energy during the bounce.

(e) To what height will it rise after the second bounce if the same percentage of energy is lost?

Q17. A mass-spring system consists of a 0.5 kg block attached to a spring (k = 200 N/m) on a frictionless horizontal surface. The block is displaced 10 cm and released.

(a) Calculate the total mechanical energy of the system.

(b) Calculate the maximum speed of the block.

(c) At what displacement is the speed half of the maximum?

(d) Calculate the acceleration at the maximum displacement.

(e) Sketch the graphs of kinetic energy, potential energy, and total energy vs displacement.

Q18. In a school physics demonstration, a 5 kg cart is released from rest at the top of a frictionless roller coaster track. The track has the following profile (describe verbally): starts at height 10 m, descends to 2 m, rises to 6 m, descends to 0 m, and ends at height 4 m.

(a) Calculate the speed of the cart at each point.

(b) Calculate the maximum speed during the journey.

(c) At which point is the kinetic energy minimum?

(d) If a 500 N braking force is applied at the lowest point, calculate the stopping distance.

(e) Discuss whether mechanical energy is conserved throughout the motion.