Simple Harmonic Motion SHM Definition - UNSOLVED PRACTICE SET

Q7. Define simple harmonic motion. Write the condition that must be satisfied for a motion to be SHM.

Q8. Show that the equation F = –kx leads to the differential equation d²x/dt² + ω²x = 0, where ω = √(k/m).

Q9. Explain why the negative sign in F = –kx is essential for the motion to be oscillatory.

Q10. A particle is executing SHM. At what positions is:

(i) The speed maximum?

(ii) The acceleration maximum?

(iii) The restoring force maximum?

(iv) The kinetic energy maximum?

Q11. What is the significance of the phase constant in SHM? How does changing the phase constant affect the motion?

Q12. Can a motion be oscillatory but not simple harmonic? Give an example and explain why it is not SHM.

Q13. Define simple harmonic motion and derive the differential equation of SHM starting from the definition of restoring force. Show that the general solution is x = A sin(ωt + φ) or x = A cos(ωt + φ). Explain the physical meaning of each constant (A, ω, φ) and derive expressions for velocity and acceleration as functions of time.

Q14. A particle executes SHM along the x-axis. At t = 0, the particle is at x = A/2 and moving towards the positive x-direction.

(i) Determine the phase constant φ if the equation is written as x = A cos(ωt + φ).

(ii) Write the equations for displacement, velocity, and acceleration.

(iii) At what time does the particle first reach the mean position?

(iv) Sketch the displacement-time graph for one complete cycle.

Q15. Discuss the conditions under which a physical system can execute simple harmonic motion. For each of the following systems, determine whether SHM is possible and explain why or why not:

(i) A ball bouncing perfectly elastically on a hard floor

(ii) A block attached to a spring on a frictionless horizontal surface

(iii) A simple pendulum with large amplitude (greater than 15°)

(iv) A particle moving in a uniform gravitational field near Earth's surface

Q16. A particle executes SHM with an amplitude of 5 cm and a time period of 2 s. At t = 0, the particle is at the mean position and moving in the positive direction.

(i) Write the equation of motion.

(ii) Calculate the displacement at t = 0.5 s.

(iii) Calculate the velocity at t = 0.5 s.

(iv) Calculate the acceleration at t = 0.5 s.

(v) At what time does the particle first reach x = 2.5 cm?

Q17. A body of mass 0.5 kg executes SHM with an amplitude of 10 cm. The maximum restoring force on the body is 2 N.

(i) Calculate the force constant k.

(ii) Calculate the time period of oscillation.

(iii) Calculate the maximum speed of the body.

(iv) Calculate the total energy of the oscillation.

(v) Write the equation of motion if the body starts from the positive extreme position.

Q18. In a school physics lab, a student sets up an experiment to study SHM using a glider on an air track attached to two identical springs. The glider has mass 0.2 kg, and each spring has force constant 20 N/m. The glider is displaced 5 cm from equilibrium and released.

(i) Show that the motion is simple harmonic and calculate the effective force constant.

(ii) Calculate the time period of oscillation.

(iii) Calculate the maximum speed of the glider.

(iv) If the amplitude is doubled, how do the time period and maximum speed change? Explain.

(v) The student notices that the amplitude gradually decreases. What causes this, and what is this type of oscillation called?