Oscillations & Simple Harmonic Motion (SHM)

IIT / JEE Tough Objective Paper – 2

Q1. Variable Force SHM

A particle of mass m moves under force F = −kx³. Small oscillations about equilibrium have angular frequency:

A) √(k/m)
B) √(3k/m)
C) √(6k/m)
D) Motion is not SHM

Correct Answer: D

IIT Insight: Restoring force must be ∝ x, not x³ → NOT SHM.

Q2. Two SHMs Superposition

Two SHMs of same frequency and amplitude A act along same line. Phase difference = π/2. Resultant amplitude is:

A) A
B) √2 A
C) 2A
D) A/√2

Correct Answer: B) √2 A

Trick: Resultant amplitude = √(A² + A²) when phase = 90°

Q3. Energy Ratio (JEE Advanced)

At what displacement is kinetic energy three times potential energy?

Answer:

KE = 3PE ⇒ Total energy = 4PE ⇒ PE = (1/4)E ⇒ x²/A² = 1/4 ⇒ x = A/2

Correct Answer: x = A/2

Q4. SHM with Moving Mean

If the mean position of SHM itself moves with constant velocity, the motion observed is:

A) Still SHM
B) Periodic but not SHM
C) Non-periodic
D) Uniform motion

Correct Answer: B

IIT Concept: Restoring force still acts, but about a shifting mean → not pure SHM.

Q5. Block on Rough Surface

A block attached to a spring oscillates on rough horizontal surface. Which remains constant?

A) Amplitude
B) Total energy
C) Time period
D) Maximum velocity

Correct Answer: C

Trick: Friction reduces energy & amplitude, not time period (small damping).

Q6. Time to Go Mean → Extreme

Time taken to move from mean to extreme position is:

Answer: T/4

IIT Shortcut: Mean → Extreme = quarter cycle always

Q7. Phase Confusion Trap

If x = A cos(ωt + π/3), initial position is:

x(0) = A cos(π/3) = A/2

Correct Answer: A/2

Q8. Max Acceleration Condition

Acceleration becomes maximum when:

A) v = max
B) x = 0
C) x = ±A
D) KE = PE

Correct Answer: C

Q9. Graph Based (Advanced)

Slope of displacement–time graph is zero at:

Answer: Extreme position

Why? Velocity = slope → zero at extreme

Q10. Dimensional Trap

Which quantity has dimension of angular frequency?

A) √(k/m)
B) √(m/k)
C) k/m
D) mk

Correct Answer: A

IIT MASTER TAKEAWAY:
• Check proportionality first
• Use energy shortcuts
• Phase ≠ position blindly
• Mean & extreme rules never fail

IIT Tough Objective Paper – 2 Completed ✅

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