SHM Displacement Velocity Acceleration - UNSOLVED PRACTICE SET

Q7. Derive the expression for velocity of a particle executing SHM as a function of displacement. Show that v = ω√(A² – x²).

Q8. Show that in SHM, the acceleration is given by a = –ω²x. Explain the significance of the negative sign.

Q9. A particle in SHM has amplitude A and angular frequency ω. Write expressions for:

(i) Maximum velocity

(ii) Maximum acceleration

(iii) Velocity at displacement x

(iv) Acceleration at displacement x

Q10. Draw graphs of displacement, velocity, and acceleration versus time for SHM. Show how they are related in phase.

Q11. In SHM, at what displacement is the speed equal to half the maximum speed? Show your calculation.

Q12. Explain why the velocity-displacement graph in SHM is an ellipse. What do the semi-major and semi-minor axes represent?

Q13. Starting from x = A cos(ωt + φ), derive expressions for:

(i) Velocity as a function of time

(ii) Acceleration as a function of time

(iii) Velocity as a function of displacement

(iv) Acceleration as a function of displacement

For each case, discuss the phase relationship with displacement and identify the positions where each quantity is maximum, minimum, or zero. Draw the corresponding graphs.

Q14. A particle executes SHM with amplitude A = 10 cm and time period T = 4 s. At t = 0, the particle is at x = 5 cm and moving towards the mean position.

(i) Determine the phase constant and write the equation of motion.

(ii) Calculate the velocity and acceleration at t = 1 s.

(iii) Find the time when the particle first passes through the mean position.

(iv) Calculate the total distance traveled by the particle in the first 2 seconds.

Q15. The displacement-time graph for a particle in SHM is shown (sinusoidal curve with amplitude A and period T).

(i) Identify the points on the graph where velocity is maximum, zero, and minimum.

(ii) Identify the points where acceleration is maximum, zero, and minimum.

(iii) Sketch the corresponding velocity-time graph and acceleration-time graph.

(iv) Explain how these three graphs are related through differentiation.

Q16. A particle executes SHM along a straight line with amplitude 8 cm and frequency 2 Hz. At t = 0, the particle is at the mean position moving in the positive direction.

(i) Write the equation for displacement, velocity, and acceleration.

(ii) Calculate the displacement, velocity, and acceleration at t = 0.125 s.

(iii) Find the time when the particle first reaches x = 4 cm.

(iv) Calculate the speed of the particle when it is at x = 4 cm.

(v) Calculate the acceleration of the particle when it is at x = 4 cm.

Q17. A body executes SHM with time period 2 s and amplitude 5 cm.

(i) Calculate the maximum velocity and maximum acceleration.

(ii) Calculate the velocity when the displacement is 3 cm.

(iii) Calculate the acceleration when the displacement is 3 cm.

(iv) At what displacement is the speed equal to 80% of the maximum speed?

(v) At what displacement is the magnitude of acceleration equal to half the maximum acceleration?

Q18. In a school physics demonstration, a small ball is attached to a spring and oscillates vertically with amplitude 10 cm and frequency 1 Hz. A motion sensor records the position of the ball.

(i) Write the equation of motion, taking upward as positive and the mean position as origin. Assume the ball starts from the lowest point.

(ii) Calculate the maximum speed of the ball.

(iii) Calculate the speed of the ball when it is 5 cm above the mean position.

(iv) A student claims that the ball moves fastest at the top of its motion. Is this correct? Explain using your equations.

(v) Calculate the time taken by the ball to move from the mean position to a point 5 cm above it.