IIT / JEE ADVANCED – ULTRA-EXTREME PROBLEMS
Oscillations & Simple Harmonic Motion (SET–5)
Problem 1: SHM with Time-Dependent Equilibrium (Topper Trap)
A mass m is attached to a spring of constant k and oscillates horizontally. The support of the spring moves as y = vt with constant velocity. Find the time period of oscillation observed from ground frame.
Solution:
Let x be displacement from instantaneous equilibrium. Restoring force depends only on extension relative to moving support.
Equation of motion:
m d²x/dt² = −kx
Hence time period remains:
T = 2π √(m/k)
Answer: T = 2π √(m/k)
Elite Insight:
Constant velocity changes position, not acceleration → no change in frequency.
Problem 2: SHM with Gradually Increasing Mass
A spring–mass system oscillates with mass increasing slowly due to rain falling uniformly. How does frequency vary with time?
Solution:
f = (1/2π) √(k/m)
As mass increases, frequency decreases gradually.
Answer: Frequency decreases with time
Exam Trap:
Students assume symmetry like sand leakage → opposite effect here.
Problem 3: Maximum Rate of Change of Energy
During SHM, at what displacement is the rate of change of kinetic energy maximum?
Solution:
Rate of change of KE = Power = Fv
F = kx, v = ω√(A² − x²)
P ∝ x√(A² − x²)
Maximum occurs at:
x = A/√2
Answer: x = A/√2
Problem 4: SHM Validity under Arbitrary Force
Force on a particle is given by:
F(x) = −k(x − x³)
Is the motion SHM for small oscillations?
Solution:
For small x, x³ ≪ x
So F ≈ −kx
Hence restoring force ∝ displacement.
Answer: Yes, SHM for small oscillations
Topper Rule:
Always expand force near equilibrium point.
Problem 5: Energy Ratio with Phase Velocity
At an instant, phase angle θ satisfies tanθ = 2. Find the ratio of kinetic to potential energy.
Solution:
In SHM:
KE/E = sin²θ
PE/E = cos²θ
tanθ = 2 ⇒ sin²θ / cos²θ = 4
Therefore:
KE : PE = 4 : 1
Answer: 4 : 1
Problem 6: Frequency Matching (Resonance Concept)
A system executes SHM with natural frequency ω₀. A periodic force of frequency ω is applied. For which condition does amplitude become maximum?
Answer: ω = ω₀
This is the condition of resonance.
SET–5 FINAL TAKEAWAYS
- Acceleration (not velocity) controls oscillation nature
- Mass variation directly alters frequency
- Maximum rates occur at balance points
- Small-x approximation is a powerful tool
- Phase relations unlock energy ratios
IIT / JEE Tough Problems – SET–5 Completed ✅
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🔹 Simple Harmonic Motion (SHM) — Core Series
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