Simple Pendulum - UNSOLVED PRACTICE SET

Q7. Derive the expression for the time period of a simple pendulum for small oscillations. State the assumptions made.

Q8. Why does the time period of a simple pendulum not depend on the mass of the bob? Explain using the equation of motion.

Q9. A simple pendulum has a time period of 2 s on Earth. What will be its time period on a planet where g is twice that of Earth?

Q10. Explain why the motion of a simple pendulum is only approximately SHM. What happens if the amplitude is large?

Q11. A pendulum clock keeps correct time at a place where g = 9.8 m/s². If it is taken to a mountain top where g = 9.78 m/s², will it gain or lose time? Explain.

Q12. What is a seconds pendulum? Calculate its length at a place where g = 9.8 m/s².

Q13. Derive the expression for the time period of a simple pendulum. Start with the forces acting on the bob, write the equation of motion, and show that for small angles (sin θ ≈ θ), the motion is simple harmonic. Derive T = 2π√(L/g) and discuss the factors on which the time period depends and does not depend.

Q14. A simple pendulum of length 1 m is taken to different locations:

(i) On Earth's surface (g = 9.8 m/s²)

(ii) On the Moon (g = 1.6 m/s²)

(iii) Inside a freely falling elevator

(iv) On an orbiting space station

For each location, calculate the time period if possible, or explain why the pendulum does not oscillate. Discuss the physical reasoning in each case.

Q15. A student conducts an experiment with a simple pendulum in the school lab. She measures the time for 20 oscillations with different lengths and records the following data:

L = 25 cm, T₂₀ = 20 s

L = 50 cm, T₂₀ = 28 s

L = 100 cm, T₂₀ = 40 s

(i) Calculate the time period T for each length.

(ii) Plot a graph of T² versus L.

(iii) From the graph, determine the value of g.

(iv) Discuss possible sources of error in this experiment.

Q16. A simple pendulum of length 2 m is suspended from the ceiling of a room. The bob has mass 0.5 kg and is displaced by a small angle of 5° and released.

(i) Calculate the time period of the pendulum.

(ii) Calculate the maximum speed of the bob.

(iii) Calculate the tension in the string when the bob passes through the mean position.

(iv) Calculate the tension in the string at the extreme position.

(v) If the length is reduced to 0.5 m, how does the time period change?

(Given: g = 9.8 m/s²)

Q17. A pendulum clock loses 10 seconds per day at a place where g = 9.8 m/s². The owner wants to adjust it to keep correct time.

(i) Calculate the percentage change in time period required.

(ii) Should the length of the pendulum be increased or decreased?

(iii) Calculate the required change in length if the original length is 1 m.

(iv) If instead of adjusting the length, the clock is taken to a different location, what should be the value of g at the new location?

Q18. In a school science exhibition, a student builds a "pendulum wave" demonstration using 15 simple pendulums of different lengths suspended from a common support. The longest pendulum has length 1 m and makes 15 oscillations in 30 seconds.

(i) Calculate the time period of the longest pendulum.

(ii) The student wants each successive pendulum to complete one more oscillation than the previous one in the same time interval. Calculate the lengths of the first five pendulums.

(iii) Explain why the pendulums appear to move in waves and then come back together periodically.

(iv) If the demonstration is taken to a hill station where g = 9.78 m/s², how will the pattern change?

(Given: g = 9.8 m/s²)