Oscillations & Simple Harmonic Motion (SHM)

Stage 1 – Page 4 | Time Period of SHM (Core Systems)

1. Time Period – Conceptual Meaning

Time Period (T) is the time taken by a particle to complete one full oscillation.

In SHM, the time period depends on:

Restoring force mechanism
Inertia of the system
Geometry of motion

Very Important:
Time period does NOT depend on amplitude (ideal SHM).

2. SHM of a Mass–Spring System

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

Restoring force:

F = −kx

From Newton’s second law:

m d²x/dt² = −kx

Comparing with SHM equation:

d²x/dt² = −ω²x

We get:

ω = √(k/m)

Therefore, time period:

T = 2π√(m/k)

Depends on mass and spring constant
Independent of gravity

IIT Trap:
Changing orientation (vertical/horizontal) does NOT change time period.

3. Simple Pendulum – Physical Idea

A simple pendulum consists of:

Point mass (bob)
Light, inextensible string
Fixed support

For small angular displacement (θ small):

sinθ ≈ θ

This approximation is the key condition for SHM.

4. Time Period of Simple Pendulum

Restoring torque:

τ = −mgL sinθ

For small θ:

τ ≈ −mgLθ

Using rotational equation:

I d²θ/dt² = −mgLθ

For point mass:

I = mL²

Therefore:

ω = √(g/L)

Time period:

T = 2π√(L/g)

Depends on length and gravity
Independent of mass

Must Remember:
Pendulum shows SHM only for small angles.

5. Comparison: Spring vs Pendulum

Spring: restoring force ∝ displacement
Pendulum: restoring torque ∝ angular displacement
Spring T depends on m
Pendulum T depends on g

6. Effect of Changing Parameters

Increase m → T increases (spring)
Increase k → T decreases
Increase L → T increases (pendulum)
Increase g → T decreases

Board Favourite:
Explain effect of parameters on time period.

7. Real-Life Applications

Shock absorbers
Timekeeping devices
Seismographs
Vibration isolation

8. Why This Page is Rank-Defining

Direct formulas asked
Conceptual comparison tested
Multiple traps hidden

Topper Rule:
Always ask: what provides restoring force?

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