Energy in Shm

8 MCQs2 revision cards9-step worked example

Source: NCERT Oscillations and WavesPYQ coverage: NEET 2020, 2021, 2022, 2023, 2024, 2025Official key: NTA-verifiedLast reviewed: May 2026

Lesson

The trap: aspirants know that total energy in SHM is E = ½kA². They then see a question where amplitude doubles and confidently answer "energy doubles." It quadruples. Energy scales as the square of amplitude — and this is a high-frequency NEET distractor.

The concept. In simple harmonic motion, mechanical energy continuously converts between kinetic and potential forms. At the mean position (x = 0), all energy is kinetic: KE = ½mω²A². At the extreme positions (x = ±A), all energy is potential: PE = ½kA². At any intermediate displacement x:

KE = ½k(A² − x²)
PE = ½kx²
Total E = KE + PE = ½kA² = ½mω²A²

The total is constant — independent of x, independent of time. This follows directly from the conservative nature of the restoring force (NCERT Class 11 Physics, Oscillations chapter, page 7).

Two critical facts from NCERT (Oscillations chapter, page 6): (1) KE and PE each oscillate at frequency 2ω (twice the oscillation frequency), and (2) their time-averaged values are equal, each ½E.

The amplitude-squared dependence is what catches students. Since E = ½kA², doubling A means E → 4E, not 2E. Tripling A means E → 9E. The period T = 2π√(m/k) does not depend on amplitude — but the energy emphatically does. This asymmetry is a common confusion (trap: period-amplitude independence does NOT mean energy-amplitude independence).

Watch-out for NEET: questions that change amplitude and ask for the new total energy, or ask at what displacement KE equals PE (answer: x = A/√2).

Practice MCQs

Select an option to see the explanation. Wrong answers show why your choice was tempting — and name the exact trap it exploits.

MCQ 1Easy RecallPractice

In simple harmonic motion, at which position is the kinetic energy maximum?

MCQ 2Easy RecallPractice

The total mechanical energy of a particle in ideal SHM (no damping) is:

MCQ 3Easy RecallPractice

In SHM, the kinetic energy and potential energy each oscillate with a frequency equal to:

MCQ 4Direct ApplicationPractice

A block on a frictionless surface oscillates on a spring with amplitude A and total energy E. If the amplitude is doubled to 2A (same spring, same mass), the new total energy is:

MCQ 5Direct ApplicationPractice

A particle in SHM has amplitude A. At what displacement from the mean position is the kinetic energy equal to the potential energy?

MCQ 6Direct ApplicationPractice

A spring-mass system has spring constant k = 200 N/m and oscillates with amplitude 0.10 m. The total energy of the system is:

MCQ 7CalculationPractice

A 0.50 kg block attached to a spring (k = 200 N/m) oscillates with amplitude 5.0 × 10⁻² m. What is the maximum speed of the block?

MCQ 8CalculationPractice

A particle executes SHM with amplitude A and angular frequency ω. At displacement x = A/2 from the mean position, the ratio of kinetic energy to potential energy is:

Quick recall before you leave

Worked Example

Pattern: Given a spring-mass system with changed amplitude, find the ratio of total energies. (Based on the amplitude-squared energy dependence — the core trap for this topic.)

1
Given
A spring-mass oscillator has mass m = 0.40 kg and spring constant k = 160 N/m. It oscillates first with amplitude A₁ = 0.050 m, then the amplitude is increased to A₂ = 0.15 m.
2
Required
Find the ratio E₂/E₁.
3
Concept
Total energy in SHM is E = ½kA². Energy depends on the square of amplitude. Spring constant and mass remain unchanged.
4
Formula
E = ½kA², so E₂/E₁ = A₂²/A₁².
5
Substitution
E₂/E₁ = (0.15)²/(0.050)² = 0.0225/0.0025
6
Calculation
E₂/E₁ = 9 Note: k and m are the same in both cases, so they cancel. The ratio depends purely on the amplitude ratio squared: (A₂/A₁)² = (0.15/0.050)² = 3² = 9.
7
Final answer
E₂/E₁ = 9. The new total energy is 9 times the original. Note on exact values: the ratio 3 (= 0.15/0.050) is exact arithmetic, and the squaring gives an exact integer 9. No sig-fig ambiguity arises.
8
Common trap
A student who thinks energy scales linearly with amplitude would answer E₂/E₁ = 3 (trap: period is amplitude-independent, but energy is NOT — it scales as A²). Another common error is confusing the energy ratio with the speed ratio (v_max ∝ Aω, so the speed ratio is 3, not 9).
9
Similar NEET-style question
A block oscillates on a spring with total energy 2.0 J and amplitude 4.0 × 10⁻² m. If the amplitude is halved to 2.0 × 10⁻² m (same spring), find the new total energy. [Answer: E_new = 2.0 × (1/2)² = 0.50 J] ---

Before solving, remember these

Formula

Energy in SHM

KE(t) = ½ m ω² A² sin²(ωt+φ); PE(t) = ½ k x² = ½ m ω² A² cos²(ωt+φ); Total E = ½ k A² = ½ m ω² A². Energy oscillates between KE and PE; total constant.

-- NCERT, p. 7

Key Fact

Phase relation in SHM

Velocity leads displacement by π/2: v = -A ω sin(ωt+φ). Acceleration leads displacement by π: a = -A ω² cos(ωt+φ) = -ω² x. KE max at x=0; PE max at x=±A.

-- NCERT, p. 6

Formulas

10 formulas — click to collapse

Beat frequency

f_{beat} = ∣ f_{1} - f_{2} ∣

When two waves of nearly equal frequencies superpose, amplitude oscillates at the difference frequency.

Symbol	Quantity	SI Unit
f_beat	beat frequency	Hz
f1, f2	superposed frequencies	Hz

Valid when

Linear superposition
f1, f2 close in value

Period of simple pendulum (small angle)

T = 2 π \frac{L}{g}

Period of simple pendulum of length L. Holds for small amplitudes (sin theta ~ theta).

Symbol	Quantity	SI Unit
T	period	s
L	pendulum length	m
g	gravity	m/s^2

Valid when

Small angular amplitude (typically <15°)
Massless string
Point bob

SHM displacement

x (t) = A cos (ω t + ϕ); ω = 2 π f

Displacement in simple harmonic motion. Velocity = -A*omega*sin(omega*t+phi); a = -omega^2 * x.

Symbol	Quantity	SI Unit
A	amplitude	m
omega	angular frequency	rad/s
phi	phase	rad
T	period	s
f	frequency	Hz

Valid when

Restoring force linear (F = -kx)
No damping

Total energy in SHM

E = \frac{1}{2} k A^{2} = \frac{1}{2} m ω^{2} A^{2}

Total mechanical energy is constant. Oscillates between KE (max at x=0) and PE (max at x=±A).

Symbol	Quantity	SI Unit
E	total energy	J
k	spring constant	N/m
A	amplitude	m
m	mass	kg
omega	angular frequency	rad/s

Valid when

Conservative SHM (no damping)
Elastic regime

Period of mass-spring oscillator

T = 2 π \frac{m}{k}

Period of horizontal spring with mass m, spring constant k. Independent of amplitude.

Symbol	Quantity	SI Unit
T	period	s
m	mass	kg
k	spring constant	N/m

Valid when

Hooke's law spring
No damping
Small enough amplitude to stay in elastic regime

Standing wave in closed-end pipe

f_{n} = \frac{( 2 n - 1 ) v}{4 L}

Pipe closed at one end has only odd harmonics: f, 3f, 5f, ...

Symbol	Quantity	SI Unit
f_n	n-th harmonic	Hz
v	sound speed	m/s
L	pipe length	m

Valid when

Closed at one end (open at other)
End correction neglected

Standing wave in open-open pipe

f_{n} = \frac{n v}{2 L}

Pipe open at both ends has all harmonics. Same formula as string.

Symbol	Quantity	SI Unit
f_n	n-th harmonic	Hz
v	sound speed	m/s
L	pipe length	m

Valid when

Open at both ends
End correction neglected

Standing wave frequencies on fixed-fixed string

f_{n} = \frac{n v}{2 L}

Allowed frequencies on string fixed at both ends. n=1 fundamental; harmonics 2f, 3f, ...

Symbol	Quantity	SI Unit
f_n	n-th harmonic	Hz
v	wave speed on string	m/s
L	string length	m
n	harmonic number	-

Valid when

String fixed at both ends
Wave speed v as defined above

Speed of sound in gas (Newton-Laplace)

v = \frac{γ P}{ρ}

Speed of sound in gas. Adiabatic index gamma, pressure P, density rho. Increases with sqrt(T).

Symbol	Quantity	SI Unit
v	speed of sound	m/s
gamma	adiabatic index	-
P	pressure	Pa
rho	density	kg/m^3

Valid when

Ideal gas
Adiabatic compression/expansion of sound waves

Wave speed on string

v = \frac{T}{μ}

Speed of transverse wave on string under tension T, linear mass density mu.

Symbol	Quantity	SI Unit
v	wave speed	m/s
T	tension	N
mu	linear mass density	kg/m

Valid when

Stretched uniform string
Small amplitude

Exam Traps & Common Mistakes

These are the exact patterns that cause wrong answers in NEET. Each trap includes when it triggers and how to avoid it.

6 items — click to collapse

Trap

Pendulum: mass-dependence claim

Category: Overthinking

Student writes T as depending on bob mass. Simple pendulum T = 2π√(L/g); independent of m.

When it triggers

Question changes pendulum bob mass and asks for new period.

How to avoid

Mass cancels in derivation (gravitational mass = inertial mass). Mass changes the bob's KE and PE proportionally; period unaffected.

Trap

SHM period: amplitude dependence claim

Category: Overthinking

Student claims SHM period depends on amplitude. For ideal SHM (Hooke's law spring or simple pendulum at small angle), period is INDEPENDENT of amplitude.

When it triggers

Question gives changes in amplitude and asks for new period.

How to avoid

T = 2π√(m/k) (spring) or 2π√(L/g) (pendulum, small angle) — neither depends on A. Only at large pendulum angles does T pick up a small amplitude correction.

Trap

Pipe harmonics: open vs closed end

Category: Similar Terms

Student includes even harmonics in a closed-end pipe. Closed pipe has only ODD harmonics (f, 3f, 5f, ...).

When it triggers

Question describes pipe closed at one end (e.g. resonance tube).

How to avoid

Open both ends: all harmonics, f_n = nv/(2L). Closed one end: odd only, f_n = (2n-1)v/(4L). Fundamental of closed pipe is HALF that of open pipe of same L.

Mistake

Includes bob mass in simple pendulum period formula.

Root cause: concept gap

Correction

Simple pendulum T = 2π√(L/g) — independent of mass. Equivalence of inertial and gravitational mass cancels m.

Mistake

Claims SHM period depends on amplitude.

Root cause: concept gap

Correction

Ideal SHM: T = 2π√(m/k) (spring) or 2π√(L/g) (pendulum, small angle) — no amplitude dependence. Doubling amplitude does not change period.

Mistake

Includes even harmonics in a closed-end pipe.

Root cause: concept gap

Correction

Closed-end pipe has only ODD harmonics (f, 3f, 5f, ...). Open-both-ends pipe has all (f, 2f, 3f, ...). Reason: closed end has displacement node and pressure antinode.

Past Year Questions

11 questions from NEET 2020, 2021, 2022, 2023, 2024, 2025. Answers verified against NTA official keys. — click to collapse

NEET 2025

A pipe open at both ends has a fundamental frequency f in air. The pipe is now dipped vertically in a water drum to half of its length. The fundamental frequency of the air column is now equal to: f

12f

22 3f

NTA Answer: Option 3(final)

NEET 2025

Two identical point masses P and Q, suspended from two separate massless springs of spring constants k 1 and k 2, respectively, oscillate vertically. If their maximum speeds are the same, the ratio (AQ/AP) of the amplitude AQ of mass Q to the amplitude AP of mass P is k k

22 k k 2 1

3k 1

4k 2 k k 2 1

NTA Answer: Option 1(final)

NEET 2024Revised key

 π If x =5sinπt + m represents the motion of a particle executing simple harmonic motion, the amplitude  3 and time period of motion, respectively, are

15 cm, 2 s

25 m, 2 s

35 cm, 1 s

45 m, 1 s

NTA Answer: Option 2(revised_final)

NEET 2024Revised key

If the mass of the bob in a simple pendulum is increased to thrice its original mass and its length is made half x its original length, then the new time period of oscillation is times its original time period. Then the value of 2 x is:

32 3

NTA Answer: Option 2(revised_final)

NEET 2023

The ratio of frequencies of fundamental harmonic produced by an open pipe to that of closed pipe having the same length is

12 : 1

21 : 3

33 : 1

41 : 2

NTA Answer: Option 1(final)

NEET 2023

The x-t graph of a particle performing simple harmonic motion is shown in the figure. The acceleration of the particle at t = 2 s is π2 π2

1– ms–2

2ms–2 8 16 π2 π2

3– ms–2

4ms–2 16 8

NTA Answer: Option 3(final)

NEET 2022

If the initial tension on a stretched string is doubled, then the ratio of the initial and final speeds of a transverse wave along the string is

11 : 2

21 : 1

32 : 1

41 : 2

NTA Answer: Option 4(final)

NEET 2022

Two pendulums of length 121 cm and 100 cm start vibrating in phase. At some instant, the two are at their mean position in the same phase. The minimum number of vibrations of the shorter pendulum after which the two are again in phase at the mean position is:

211

410

NTA Answer: Option 2(final)

NEET 2021

A spring is stretched by 5 cm by a force 10 N. The time period of the oscillations when a mass of 2 kg is suspended by it is

10.628 s

20.0628 s

36.28 s

43.14 s

NTA Answer: Option 1(final)

NEET 2020

U U U U A U B Sfl U U U 6 Hz fl 25. In a guitar, two strings A and B made of same flS U U – B fl U U material are slightly out of tune and produce beats , flS fl U of frequency 6 Hz. When tension in B is slightly decreased, the beat frequency increases to 7 Hz. 7 Hz – A fl 530 Hz , B If the frequency of A is 530 Hz, the original fl — frequency of B will be :

1537 Hz

2523 Hz

3524 Hz

4536 Hz

NTA Answer: Option 3(final)

NEET 2020

U fl U flS U fl U 29. The phase difference between displacement and U — acceleration of a particle in a simple harmonic motion is :

1zero

2rad 3

3rad 2

4rad 2

NTA Answer: Option 2(final)

How NEET usually asks this

6 recurring patterns from past papers — click to collapse

Two strings/forks slightly out of tune produce beats; find one frequency given the other and beat frequency.

Direct ApplicationEasy

Common distractors

treats beat as sum

Adds frequencies instead of subtracting

Given pendulum length and g, find T. Or given mass change, observe T unchanged.

Direct ApplicationEasy

Common distractors

expects mass dependence

Assumes T depends on bob mass

Open-open pipe (all harmonics) vs closed-end pipe (odd only). Compare frequency ratios.

InterpretationMedium

Common distractors

treats closed pipe like open

Includes even harmonics in closed pipe

Phase between displacement, velocity, acceleration in SHM. v leads x by π/2; a leads x by π.

InterpretationEasy

Common distractors

uses pi 2 instead of pi

Confuses v-x phase with a-x phase

Given spring extension/compression with given force, find k, then find period when given mass m. T = 2pisqrt(m/k).

Multi StepEasy

Common distractors

forgets 2pi factor

Drops 2*pi

uses amplitude in period

Believes T depends on amplitude

Given tension change, find new wave speed on string. v ∝ sqrt(T).

Direct ApplicationEasy

Common distractors

uses linear tension scaling

Treats v ∝ T not sqrt(T)

Test yourself on this topic with real past-paper questions:

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Energy in Shm

Lesson

Practice MCQs

Quick recall before you leave

Total energy in SHM

KE and PE at displacement x

Worked Example

Before solving, remember these

Formulas

Valid when

Valid when

Valid when

Valid when

Valid when

Valid when

Valid when

Valid when

Valid when

Valid when

Exam Traps & Common Mistakes

When it triggers

How to avoid

When it triggers

How to avoid

When it triggers

How to avoid

Correction

Correction

Correction

Past Year Questions

How NEET usually asks this

Common distractors

Common distractors

Common distractors

Common distractors

Common distractors

Common distractors

Free NEET study resources