MHT-CET Physics · Mechanical Properties of Solids

Springs and Elastic Energy

A spring stretched by x pulls back with kx and stores ½kx²; its constant grows as it is cut shorter, springs in series add their extensions, and energy conservation links a falling body to the compression it produces.

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Why this matters

6 PYQs, one HARD. Four are springs — the energy for a larger stretch, the constant of a cut piece, two springs in a chain, and a ball dropped onto a platform on a spring. Two follow an impact: a ball's speed after it rebounds, and the rise in temperature of a bullet that stops in a wall. Two cards.

Concept 1 of 2: Spring Constant, Series Springs and Elastic Energy

A spring obeys F = kx and stores U = ½kx², so three times the stretch stores nine times the energy. A spring's constant is inversely proportional to its length: cut a spring of constant K into pieces L₁ = NL₂, and the longer piece has constant K(N + 1)/N. Two springs hung in a chain carry the same force and add their extensions, f/k₁ + f/k₂. A ball of mass m dropped from height s onto a spring platform that sinks h loses mg(s + h) of potential energy into ½kh².

Definition

$F = kx$ , $U = \tfrac{1}{2}kx^2$ (stretch × 3 ⇒ U × 9).
$k \propto \dfrac{1}{L}$ : $L_1 = NL_2$ ⇒ $k_1 = \dfrac{K}{N}(N + 1)$ .
Chain (series): total extension $f\left(\dfrac{1}{k_1} + \dfrac{1}{k_2}\right)$ .
Drop onto a spring: $mg(s + h) = \tfrac{1}{2}kh^2$ ⇒ $k = \dfrac{2mg(s + h)}{h^2}$ .

Springs

F = kx, \qquad U = \tfrac{1}{2}kx^2, \qquad k \propto \frac{1}{L}

Worked example

A spring of constant 600 N/m is cut into two pieces with lengths in the ratio 2 : 1. Constant of each piece?

Practice this conceptself-check · 1 quick reps

The same idea in a real exam question:

MHT-CET · 2023 · 15th May Shift 2 · Q50Hard

Example 1 · Mechanical Properties of Solids · Elasticity, Springs, and Energy in Strained Solids

A spring has length

L

and force constant

K

. It is cut into two springs of length

L_1

and

L_2

such that

L_1 = NL_2

(N is an integer). The force constant of spring of length

L_1

Thinking a shorter spring is softer

Each part of a spring stretches by its share of the total. A shorter piece stretches less under the same force, so it is STIFFER: k ∝ 1/L.

Leaving out the compression in the drop height

The ball falls s before touching the platform and h more while compressing it: mg(s + h), not mgs.

Drill 3 more on spring constant, series springs and elastic energy

Concept 2 of 2: Rebounds and Heat From an Impact

A ball falling h hits at √(2gh) and leaves at e times that. A bullet that stops turns its kinetic energy ½MV² into heat; if a fraction f of it warms the lead, fMV²/2 = MJsΔT, so ΔT = fV²/(2Js) and does not depend on the mass.

Definition

Rebound speed $= e\sqrt{2gh}$ (20 m, e = 0.4 ⇒ 8 m/s).
Bullet heated by a fraction f of its KE: $\Delta T = \dfrac{fV^2}{2Js}$ (75% ⇒ $\dfrac{3V^2}{8Js}$ ).

Impact

v_{\text{rebound}} = e\sqrt{2gh}, \qquad \Delta T = \frac{fV^2}{2Js}

Worked example

A ball dropped from 5 m rebounds with e = 0.6. Rebound speed and height? (g = 10 m/s²)

Practice this conceptself-check · 1 quick reps

The same idea in a real exam question:

MHT-CET · 2024 · 11th May Shift 2 · Q1Easy

Example 2 · Mechanical Properties of Solids · Elasticity, Springs, and Energy in Strained Solids

A ball kept at 20 m height falls freely in vertically downward direction and hits the ground. The coefficient of restitution is 0.4. Velocity of the ball after first rebound is

g = 10 \text{ ms}^{-2}

Applying e to the height

e scales the SPEED. The rebound height is e² times the drop height.

Drill 1 more on rebounds and heat from an impact

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Spring Constant, Series Springs and Elastic Energy
Springs
$F = kx, \qquad U = \tfrac{1}{2}kx^2, \qquad k \propto \frac{1}{L}$
Rebounds and Heat From an Impact
Impact
$v_{\text{rebound}} = e\sqrt{2gh}, \qquad \Delta T = \frac{fV^2}{2Js}$

Watch out for (3)

Thinking a shorter spring is softer
→ Spring Constant, Series Springs and Elastic Energy
Leaving out the compression in the drop height
→ Spring Constant, Series Springs and Elastic Energy
Applying e to the height
→ Rebounds and Heat From an Impact

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