Oscillations & Simple Harmonic Motion

Stage 2 – Page 2 | Long Answer Questions & Derivations

1. Derive the Expression for Displacement in SHM

Consider a particle executing SHM along a straight line. Let x be the displacement from mean position.

In SHM, acceleration is proportional to displacement and directed towards mean position.

a ∝ −x

Introducing proportionality constant ω²:

a = −ω²x

This is the defining equation of SHM. The solution of this equation is:

x = A sin(ωt + φ)

where A is amplitude and φ is phase constant.

2. Derive Velocity and Acceleration in SHM

Displacement:

x = A sin(ωt + φ)

Velocity is the time derivative of displacement:

v = dx/dt = Aω cos(ωt + φ)

Acceleration is the time derivative of velocity:

a = −Aω² sin(ωt + φ)

Since x = A sin(ωt + φ):

a = −ω²x

3. Expression for Time Period of a Spring–Mass System

Consider a mass m attached to a spring of force constant k.

Restoring force:

F = −kx

Using Newton’s second law:

m d²x/dt² = −kx

Comparing with SHM equation:

ω² = k/m

Time period:

T = 2π√(m/k)

4. Derive the Time Period of a Simple Pendulum

Consider a pendulum of length L making small angular oscillations.

Restoring force:

F = −mg sinθ ≈ −mgθ

Angular acceleration:

α = −(g/L)θ

Comparing with SHM equation:

ω² = g/L

Time period:

T = 2π√(L/g)

5. Energy in Simple Harmonic Motion

Kinetic Energy:

KE = ½mω²(A² − x²)

Potential Energy:

PE = ½mω²x²

Total Energy:

E = ½mω²A² (constant)

6. Draw and Explain Energy Graphs of SHM

KE maximum at mean position
PE maximum at extreme positions
Total energy remains constant

Board Tip:
Always draw neat graphs and label axes clearly.

How to Score Full Marks

Write stepwise derivations
Underline final formula
Mention assumptions (small angle)
Use neat diagrams

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