§ 13.3NCERT Class 11 · Physics · Chapter 13
The Spring-Block System & The Simple Pendulum
Core Principle: In any physical oscillating system, the time period of small oscillations depends solely on two fundamental physical quantities:
T = 2π √(Inertia Factor / Spring Factor) = 2π √(m / k)
The time period is completely independent of amplitude (isochronous property of SHM).
1. The Spring-Block Oscillator
Consider a block of mass m connected to a massless spring of stiffness constant k:
- Horizontal Spring-Block: The restoring force is F = −kx.
T = 2π √(m / k) • f = (1 / 2π) √(k / m)
- Vertical Spring-Block: Gravity elongates the spring by an equilibrium extension x0 = mg / k. When further displaced by x, the net restoring force is −kx.
T = 2π √(m / k) = 2π √(x0 / g)Crucial Concept: Gravity does NOT alter the time period of a vertical spring-block system! It merely shifts the mean position downwards by x0 = mg/k.
2. Cutting and Combinations of Springs
1. Cutting of Springs
For any uniform spring, the product of spring constant and length is invariant:
k · L = Constant
If a spring of constant k is cut into two parts in the ratio m : n, their new spring constants are:
k1 = k (m + n) / m • k2 = k (m + n) / n
Example: If cut into two equal halves (1:1), each half has stiffness 2k!
2. Combinations of Springs
- Series Combination: Springs connected end-to-end (same restoring force F, displacements add up: x = x1 + x2):
1 / keq = 1 / k1 + 1 / k2 ⇒ keq = (k1 k2) / (k1 + k2)
- Parallel Combination: Springs share the same displacement (restoring forces add up: F = F1 + F2):
keq = k1 + k2
3. The Simple Pendulum
A simple pendulum consists of a point mass (bob) of mass m suspended from a rigid, frictionless support by a light, inextensible string of length l:
Derivation for Small Angular Amplitude (θ ≤ 4°–5°):
The restoring torque about the suspension point O is:
The restoring torque about the suspension point O is:
τ = −mg · (l sinθ) ≈ −mglθ (since sinθ ≈ θ in radians)
Using τ = Iα = (ml2)α:
ml2 α = −mglθ ⇒ α = −(g / l) θ = −ω2 θ
Therefore, the motion is angular SHM with:
ω = √(g / l) ⇒ T = 2π √(l / g)
4. Second's Pendulum & Pendulum in Accelerating Frames
| Physical Situation | Effective Acceleration geff | Time Period T' | Effect on Time Period |
|---|---|---|---|
| Second's Pendulum (Stationary) | g = 9.8 m/s2 | T = 2.0 s | Length l ≈ 0.993 m ≈ 1 metre |
| Lift accelerating UPWARDS with a | geff = g + a | T' = 2π√[l / (g + a)] | T decreases (clock runs FAST) |
| Lift accelerating DOWNWARDS with a | geff = g − a | T' = 2π√[l / (g − a)] | T increases (clock runs SLOW) |
| Lift in FREE FALL (a = g) | geff = 0 | T' → ∞ | Pendulum does not oscillate (weightless) |
| Cart accelerating HORIZONTALLY with a | geff = √(g2 + a2) | T' = 2π√[l / √(g2 + a2)] | T decreases (oscillates about tilted equilibrium) |
Classic NEET Trap • Leaking Hollow Pendulum Bob:
A hollow spherical bob filled with water oscillates. As water leaks out slowly from a hole at the bottom:
- Initially, as water drains, the centre of mass (CM) of the system shifts downward → effective length l increases → T increases!
- As the last drops drain out, the CM shifts back up to the geometric centre of the empty sphere → effective length l decreases → T decreases back to its original value!
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