Oscillations & Simple Harmonic Motion
Phase 1 – Page 2 | Multi-Concept IIT/JEE Solved Problems
Problem 9: Velocity–Displacement Relation
A particle executes SHM with amplitude A and angular frequency ω. Find the ratio of velocities at displacements x = A/2 and x = A/√2.
Solution:
Velocity in SHM:
v = ω√(A² − x²)
At x = A/2:
v₁ = ω√(A² − A²/4) = ωA√(3/4)
At x = A/√2:
v₂ = ω√(A² − A²/2) = ωA√(1/2)
Ratio:
v₁ : v₂ = √3 : √2
Problem 10: Energy at Given Displacement
A particle executes SHM with total energy E. Find its kinetic energy when displacement is A/2.
Solution:
Total energy:
E = ½mω²A²
Potential energy at x = A/2:
PE = ½mω²(A²/4) = E/4
Kinetic energy:
KE = E − E/4 = 3E/4
Problem 11: Phase Difference
What is the phase difference between velocity and acceleration in SHM?
Solution:
- Velocity leads displacement by π/2
- Acceleration is opposite to displacement
Phase difference between velocity and acceleration = π/2
Problem 12: Change in Time Period (Pendulum)
Length of a simple pendulum is increased by 21%. Find percentage change in time period.
Solution:
T ∝ √L
New length = 1.21L
T' = √1.21 T = 1.1T
Percentage increase = 10%
Problem 13: Mean Position Crossing
How many times does a particle cross mean position in one complete SHM?
Solution:
In one oscillation, the particle crosses mean position twice.
Answer: 2 times
Problem 14: Graph-Based Concept
Which graph is a straight line in SHM?
- A) x vs t
- B) v vs t
- C) a vs x
- D) KE vs t
Answer: C
Because:
a = −ω²x
Problem 15: Identifying Correct SHM Equation
Which of the following represents SHM?
- A) a = +5x
- B) a = −5x
- C) a = constant
- D) a ∝ v
Answer: B
IIT Trap:
Negative sign is mandatory for restoring force.
Problem 16: Maximum Acceleration
Find the maximum acceleration of a particle executing SHM.
Solution:
Acceleration:
a = −ω²x
Maximum at x = A:
amax = ω²A
Problem 17: Energy Graph Interpretation
At what position are kinetic and potential energies equal?
Solution:
KE = PE ⇒ x = A/√2
Answer: x = A / √2
IIT Thinking Summary (Page 2)
- Use proportionality instead of full formulas
- Energy method saves time
- Graph questions are concept-based
- Negative sign decides SHM
Phase 1 – Page 2 Completed Successfully ✅
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🔹 Simple Harmonic Motion (SHM) — Core Series
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- SHM – Final Exam Day Checklist
- SHM Part 2
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🔹 Simple Harmonic Motion (SHM) — Extended Series (30–56)
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