Types of Equilibrium — IISER Physics MCQs with Solutions
Free IISER Physics Types of Equilibrium MCQs with step-by-step solutions (8 questions). Part of Work, Power and Energy. Practise online on Prepizo — no login needed.
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Questions with solutions
Q1 — Types of Equilibrium · easy · theory
In STABLE equilibrium, the potential energy of the body is at a:
A. Point of inflection
B. Constant value everywhere
C. Maximum
D. Minimum ✓ Correct
Solution: Stable: U minimum (displacement raises U, restoring force pulls back). Unstable: U maximum. Neutral: U constant.
Q2 — Types of Equilibrium · medium · theory
The mathematical conditions for STABLE equilibrium at x₀ are:
A. dU/dx = 0 and d²U/dx² < 0
B. dU/dx > 0
C. dU/dx = 0 and d²U/dx² > 0 ✓ Correct
D. d²U/dx² = 0 only
Solution: Equilibrium needs zero force (dU/dx = 0); a positive second derivative makes it a minimum ⇒ stable.
Q3 — Types of Equilibrium · easy · numerical
For U(x) = x² (J), the equilibrium at x = 0 is:
A. Neutral
B. Unstable
C. Stable ✓ Correct
D. Not an equilibrium
Solution: dU/dx = 2x = 0 at x = 0; d²U/dx² = 2 > 0 ⇒ minimum ⇒ stable (this is the spring potential).
Q4 — Types of Equilibrium · medium · numerical
For U(x) = x³ − 3x (J), the STABLE equilibrium position is:
A. x = +1 m ✓ Correct
B. x = 0
C. x = −1 m
D. x = ±√3
Solution: dU/dx = 3x² − 3 = 0 ⇒ x = ±1. d²U/dx² = 6x: positive at x = +1 (stable), negative at x = −1 (unstable).
Q5 — Types of Equilibrium · hard · numerical
For U(x) = x⁴/4 − x²/2 (J), the stable equilibrium positions are:
A. None exist
B. x = ±2
C. x = ±1 (x = 0 is unstable) ✓ Correct
D. x = 0 only
Solution: dU/dx = x³ − x = 0 ⇒ x = 0, ±1. d²U/dx² = 3x² − 1: at 0 it is −1 (max, unstable); at ±1 it is +2 (minima, stable) — a double-well potential.
Q6 — Types of Equilibrium · medium · numerical
For U(x) = 2x³ − 6x (J), the equilibrium at x = +1 m is:
A. Unstable
B. Not an equilibrium point
C. Stable ✓ Correct
D. Neutral
Solution: dU/dx = 6x² − 6 = 0 at x = ±1; d²U/dx² = 12x = +12 > 0 at x = 1 ⇒ U minimum ⇒ stable.
Q7 — Types of Equilibrium · hard · theory
A particle sits in a potential well of depth U₀ with total energy E. It can escape to infinity only if:
A. E exceeds the potential barrier (E > U at the well edge) ✓ Correct
B. The well is symmetric
C. E < 0
D. It is exactly at the minimum
Solution: Classically the particle is confined between the turning points where E = U(x); it escapes only where its energy tops the barrier — the basis of bound vs free states.
Q8 — Types of Equilibrium · hard · numerical
For U(x) = (x² − 1)² (J), the positions of stable and unstable equilibrium are:
A. All three stable
B. Only x = +1 is an equilibrium
C. Stable at x = 0, unstable at x = ±1
D. Stable at x = ±1, unstable at x = 0 ✓ Correct
Solution: dU/dx = 4x(x² − 1) = 0 ⇒ x = 0, ±1. U(±1) = 0 (minima → stable); U(0) = 1 (local max → unstable) — a symmetric double well.