Spring Mass System - UNSOLVED PRACTICE SET

Q7. Derive the expression for the time period of a spring-mass system executing SHM on a frictionless horizontal surface.

Q8. Two springs of force constants k₁ and k₂ are connected in parallel. Show that the effective force constant is k₁ + k₂.

Q9. Two springs of force constants k₁ and k₂ are connected in series. Show that the effective force constant is k₁k₂/(k₁ + k₂).

Q10. A spring of force constant k is cut into n equal parts. What is the force constant of each part? Explain your reasoning.

Q11. Explain why the time period of a spring-mass system depends on mass but the time period of a simple pendulum does not.

Q12. A block attached to a spring is displaced and released on a horizontal frictionless surface. At what point during the motion is:

(i) The speed maximum?

(ii) The acceleration maximum?

(iii) The potential energy maximum?

(iv) The kinetic energy maximum?

Q13. A block of mass m is attached to a spring of force constant k on a frictionless horizontal surface. The block is displaced by distance A from equilibrium and released.

(i) Derive the equation of motion and show that it represents SHM.

(ii) Derive expressions for displacement, velocity, and acceleration as functions of time.

(iii) Calculate the total energy of the system and show that it is conserved.

(iv) Discuss how the motion would change if the surface had friction.

Q14. Consider a block of mass m attached to two identical springs each of force constant k, arranged in different configurations:

(i) Springs in parallel, both attached to the block and fixed walls on opposite sides

(ii) Springs in series, one after another, with the block at the free end

(iii) Springs in parallel, both attached to the same wall and the block

For each case, derive the effective force constant and the time period of oscillation.

Q15. A student performs an experiment with a vertical spring-mass system. A spring of force constant k hangs from a support, and a mass m is attached to its lower end. The mass is pulled down slightly and released.

(i) Show that the motion is SHM and derive the time period.

(ii) How does this time period compare with the horizontal spring-mass system?

(iii) What is the new equilibrium position compared to the spring's natural length?

(iv) Explain why the time period does not depend on the acceleration due to gravity, even though gravity acts on the mass.

Q16. A block of mass 2 kg is attached to a spring of force constant 200 N/m on a frictionless horizontal surface. The block is displaced 10 cm from equilibrium and released from rest.

(i) Calculate the time period and frequency of oscillation.

(ii) Calculate the maximum speed of the block.

(iii) Calculate the maximum acceleration of the block.

(iv) Calculate the total energy of the system.

(v) Calculate the speed of the block when it is 6 cm from equilibrium.

Q17. Two springs with force constants k₁ = 100 N/m and k₂ = 200 N/m are connected to a block of mass 0.5 kg in two different arrangements:

Arrangement A: Springs in parallel

Arrangement B: Springs in series

For each arrangement:

(i) Calculate the effective force constant.

(ii) Calculate the time period of oscillation.

(iii) If the block is displaced 5 cm and released, calculate the maximum speed.

(iv) Which arrangement gives a longer time period? Explain physically why.

Q18. In a school physics lab, a student investigates a vertical spring-mass system. A spring of natural length 20 cm stretches to 25 cm when a 0.5 kg mass is attached.

(i) Calculate the force constant of the spring.

(ii) The mass is pulled down an additional 5 cm and released. Calculate the time period of oscillation.

(iii) Calculate the maximum speed of the mass.

(iv) Calculate the total energy of the oscillation.

(v) The student repeats the experiment with a 1 kg mass. How do the equilibrium extension and time period change?

(Given: g = 9.8 m/s²)