NEET Physics · Chapter 16
Differential Equation of S.H.M. – Complete Notes, Revision, Important Questions & Downloads
This topic has one core TOC subtopic, Mathematical Form of SHM, and it decides whether you can convert the restoring relation into a solvable second-order equation quickly. The core forms are d2y/dt2 = -omega^2 y and m(d2y/dt2) + ky = 0, with omega = sqrt(k/m), and NEET tests this through direct equation-identification, graph-slope interpretation, and restoring-condition checks. Questions often mix displacement, acceleration, and phase language in one line, so sign handling and dimensional checks must be automatic. For angular oscillations, the same structure appears as tau = -c theta, which maps to d2theta/dt2 + (c/I)theta = 0 under small-angle linearization.
Second-Order ODESign-Safe SolvingNCERT-Linked
Expected QuestionsQ
1
Usually one direct formula or model-identification ask, or one embedded step inside an SHM numerical each year cycle.
Time Required⏱
1.5 h
About 45 minutes for derivation chain and 45 minutes for equation-form recognition and elimination drills.
Difficulty⚡
Medium
Algebra is short, but errors come from missing the restoring sign, mixing linear and angular forms, or using wrong omega relation.
NRI USA Curriculum GapUS
Bridge Needed
Many US high-school tracks treat SHM more graphically; NEET expects immediate conversion from force or torque law to differential equation form.
4Subtopics
24Practice Questions
4Free Downloads
1.5 hPrep Time
Differential Equation of S.H.M. Weightage and Question Trend
Simple Harmonic Motion · Topic 9| NEET Year | Questions from this Topic | Bar | Marks |
|---|---|---|---|
| 2020 | 1 | 4 | |
| 2021 | 1 | 4 | |
| 2022 | 0 | 0 | |
| 2023 | 1 | 4 | |
| 2024 | 1 | 4 | |
| 2025 | 1 | 4 | |
| Topic-linked asks in the last 6 NEET sets | 5 | 20 |
Most questions test whether you identify SHM from a = -omega^2 y and reject non-linear restoring forms like F proportional to x^3.
A frequent one-correct pattern gives m and k and asks the equation or time period, so omega^2 = k/m must be substituted without sign loss.
Angular SHM appears as torque proportional to negative angular displacement, then mapped to d2theta/dt2 + (c/I)theta = 0.
A frequent one-correct pattern gives m and k and asks the equation or time period, so omega^2 = k/m must be substituted without sign loss.
Angular SHM appears as torque proportional to negative angular displacement, then mapped to d2theta/dt2 + (c/I)theta = 0.
0.8
Avg Questions / Year
20
Total Marks (6 yrs)
Mixed
Pattern
Medium
Difficulty
5-Step Differential-Equation Solving Routine
1
Lock the Restoring Law First Write force or torque as negative proportional to displacement variable before any algebra; this guarantees the correct sign of the second derivative equation.
2
Normalize to Standard ODE Form Convert to d2q/dt2 + omega^2 q = 0 form, where q is y or theta; this immediately exposes omega and lets you identify period and phase structure.
3
Check Dimensions Before Finalizing Confirm omega has unit s^-1 and omega^2 multiplies displacement variable; reject any option where coefficient dimensions are inconsistent.
4
Link with Kinematics Relations After obtaining standard form, connect to a = -omega^2 y and v-y ellipse relation to cross-check whether the model is physically restoring.
5
Handle Angular SHM Separately For tau = -c theta, divide by moment of inertia I to get d2theta/dt2 + (c/I)theta = 0; do not substitute spring-mass omega directly.
Differential Equation of S.H.M. Download Kit
PDF · Cheat Sheet · MCQ Set · PYQFull Notes
Concept notes on deriving SHM differential equations from restoring force and restoring torque, including linear and angular forms.
Download PDFFormula Sheet
One-page map of d2y/dt2 = -omega^2 y, m(d2y/dt2) + ky = 0, omega = sqrt(k/m), and angular analog forms.
Download PDFMCQ Practice
Equation-identification and coefficient-comparison MCQs that train fast recognition of SHM-compatible differential forms.
Download PDFPYQ Workbook
Year-tagged SHM equation and restoring-force questions with concise solution logic and error-analysis notes.
Download PDFSubtopics in Differential Equation of S.H.M.
2-Column TableRapid Revision Cards
Concept → Trap → Example1) Mathematical Form of SHM
Core relationDifferential equation of SHM is d2y/dt2 = -omega^2 y, equivalently d2y/dt2 + omega^2 y = 0, and for spring-mass system m(d2y/dt2) + ky = 0 with omega = sqrt(k/m).
- Use this form when restoring interaction is linear in displacement; if restoring term is not proportional to displacement, motion is not ideal SHM.
- Read coefficient of displacement in standard form to extract omega^2 directly, then infer period T = 2pi/omega without re-deriving from scratch.
- Trap: missing the negative restoring sign and writing d2y/dt2 = +omega^2 y, which predicts non-oscillatory exponential behavior instead of bounded oscillation.
Example (NEET-style)For m = 0.20 kg and k = 50 N/m, the equation is 0.20(d2y/dt2) + 50y = 0, so d2y/dt2 + 250y = 0 and omega = sqrt(250) rad/s. The displacement, velocity, and acceleration then remain phase-linked exactly as in SHM tables.
US Curriculum Gaps - Differential Equation of S.H.M.
The main bridge is translating verbal restoring-force statements into exact differential equation form under exam time pressure.AP Physics 1 to NEET Equation-Form Gap
AP Physics 1 often emphasizes qualitative SHM behavior, while NEET expects immediate conversion from F = -kx to m(d2y/dt2) + ky = 0 and coefficient interpretation.
- Practice 20 stems where force law is given in words and you must write the differential equation in under 30 seconds.
- Use a coefficient-to-omega drill: from d2y/dt2 + by = 0, identify omega = sqrt(b) with units check.
- Solve mixed questions where equation form is used to infer period and maximum acceleration in one chain.
Honors Physics to NEET Angular-SHM Gap
Many Honors Physics classes mention torsional oscillation conceptually, but NEET-style tasks require the full angular equation from torque and moment of inertia.
- Train the mapping tau = -c theta -> I(d2theta/dt2) + c theta = 0 -> omega = sqrt(c/I).
- Keep linear and angular variables separate to avoid substituting mass-spring formulas blindly.
- Do timed elimination sets with dimension-based rejection of incorrect equation options.
Concept IQ Check
4 NEET-style MCQs with Answers1A particle executes SHM under restoring force F = -kx. Which differential equation is correct for displacement y?Model identification
m(d2y/dt2) - ky = 0
m(d2y/dt2) + ky = 0
m(dy/dt) + ky = 0
m(d2y/dt2) + k = 0
From Newton's law, m(d2y/dt2) = F = -ky. Rearranging gives m(d2y/dt2) + ky = 0, which is the standard SHM differential equation and therefore option 2 is correct. Option 1 has wrong sign and would imply acceleration in the same direction as displacement, so it is not restoring. Option 3 uses first derivative and represents damping-like behavior, not simple harmonic motion. Option 4 is dimensionally inconsistent because the second term has no displacement variable. In NEET elimination, sign and dimensional consistency remove three options quickly.
2Given d2y/dt2 + 49y = 0, the angular frequency and time period are:Coefficient decoding
omega = 7 rad/s, T = 2pi/7 s
omega = 49 rad/s, T = 2pi/49 s
omega = 7 rad/s, T = 7/2pi s
omega = sqrt(49/2) rad/s, T = 2pi/sqrt(49/2) s
Compare with standard form d2y/dt2 + omega^2 y = 0. Here omega^2 = 49, so omega = 7 rad/s and T = 2pi/omega = 2pi/7 s. Option 2 treats coefficient itself as omega rather than omega^2. Option 3 inverts period relation incorrectly. Option 4 introduces an extra factor with no basis in the equation. This is a common NEET stem where one must read coefficient structure correctly before substituting formulas.
3For angular SHM with restoring torque tau = -c theta about fixed axis and moment of inertia I, the correct equation is:Angular analog
I(d2theta/dt2) - c theta = 0
I(d2theta/dt2) + c theta = 0
I(dtheta/dt) + c theta = 0
I(d2theta/dt2) + c = 0
Use rotational dynamics: I(d2theta/dt2) = tau = -c theta. Rearrangement gives I(d2theta/dt2) + c theta = 0, so option 2 is correct. Option 1 flips the restoring sign and breaks stability. Option 3 has first derivative and corresponds to damping-type behavior, not ideal SHM. Option 4 lacks the variable theta in restoring term, causing dimension mismatch. In exam conditions, writing tau proportional to negative angular displacement first prevents sign mistakes immediately.
4Which restoring-force law does NOT produce simple harmonic motion about x = 0?Condition check
F = -4x
F = -9x
F = -2x^3
F = -0.5x
SHM requires restoring force proportional to displacement and opposite in direction, i.e., F = -kx where k is constant. Options 1, 2, and 4 satisfy this linear condition and produce d2x/dt2 + (k/m)x = 0. Option 3 uses cubic dependence, so acceleration is not linear in displacement and the exact motion is oscillatory but not simple harmonic for finite amplitudes. This distinction is heavily tested in objective exams because students often equate any periodic motion with SHM without checking linearity of restoring law.
Practice Questions - Differential Equation of S.H.M.
Click "Reveal Answer" after attempting1A spring-mass system has m = 0.5 kg and k = 18 N/m. Select the correct differential equation for displacement y.
0.5(d2y/dt2) + 18y = 0
0.5(d2y/dt2) - 18y = 0
(d2y/dt2) + 9y = 0
(dy/dt) + 36y = 0
👁 Reveal Answer
Correct option: 1. For SHM, restoring force is F = -ky and Newton's law gives m(d2y/dt2) = -ky. Therefore 0.5(d2y/dt2) + 18y = 0. Option 2 has wrong restoring sign. Option 3 is equivalent only if we divide option 1 by 0.5 to get d2y/dt2 + 36y = 0, so option 3 is numerically wrong. Option 4 incorrectly uses first derivative and represents damping-like behavior instead of simple harmonic motion.
2Given the equation d2x/dt2 + 25x = 0, find time period.
2pi/5 s
5/2pi s
2pi/25 s
25/2pi s
👁 Reveal Answer
Correct option: 1. Compare with d2x/dt2 + omega^2 x = 0 to identify omega^2 = 25, so omega = 5 rad/s. Time period is T = 2pi/omega = 2pi/5 s. Option 2 inverts the formula. Option 3 treats coefficient as omega instead of omega squared. Option 4 is the reciprocal of the required expression and is dimensionally inconsistent for period.
3For angular oscillation, tau = -8 theta and I = 2 kg m^2. What is the standard equation and omega?
2(d2theta/dt2) + 8theta = 0, omega = 2 rad/s
2(d2theta/dt2) - 8theta = 0, omega = 2 rad/s
2(dtheta/dt) + 8theta = 0, omega = 4 rad/s
(d2theta/dt2) + 8theta = 0, omega = sqrt(8) rad/s
👁 Reveal Answer
Correct option: 1. Rotational equation is I(d2theta/dt2) = tau = -8theta, hence 2(d2theta/dt2) + 8theta = 0 and d2theta/dt2 + 4theta = 0. Therefore omega^2 = 4 and omega = 2 rad/s. Option 2 has wrong sign and non-restoring acceleration. Option 3 uses first derivative, which is not the SHM equation. Option 4 drops I without dividing consistently and gives incorrect omega from the resulting coefficient.
4A candidate writes d2y/dt2 = +omega^2 y for SHM. What is the most accurate correction?
Sign should be negative because acceleration must oppose displacement
Equation is correct for all oscillations
Replace second derivative with first derivative
Keep sign positive but square root omega
👁 Reveal Answer
Correct option: 1. In SHM, restoring acceleration is directed toward mean position, so acceleration and displacement have opposite signs. The correct equation is d2y/dt2 = -omega^2 y or d2y/dt2 + omega^2 y = 0. A positive sign would yield exponential-type solutions instead of bounded sinusoidal oscillation. Option 2 is false because not all oscillatory forms follow this relation. Options 3 and 4 do not fix the physical restoring condition and therefore are invalid.
NEET Physics · Differential Equation of S.H.M. Revision Checklist
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Differential Equation of S.H.M. FAQ
Notes · Downloads · Revision · Important QuestionsWhy is the sign negative in d2y/dt2 = -omega^2 y?
The negative sign encodes restoring behavior: when displacement is positive, acceleration is negative, and when displacement is negative, acceleration is positive. This opposite-sign relation forces the particle back toward equilibrium and produces bounded sinusoidal motion. Without the negative sign, solutions become exponential and the motion does not remain simple harmonic.
How do I identify omega quickly from a given differential equation?
Rewrite any valid SHM equation to standard form d2q/dt2 + omega^2 q = 0. The coefficient of q is omega^2, so omega is its positive square root in rad/s. Then period follows from T = 2pi/omega. This method is faster and safer than trying to re-derive from force law for each question.
What is the difference between m(d2y/dt2) + ky = 0 and d2y/dt2 + omega^2 y = 0?
They represent the same physics in different levels of normalization. Dividing m(d2y/dt2) + ky = 0 by m gives d2y/dt2 + (k/m)y = 0, so omega^2 = k/m. The first form is system-parameter form, while the second is compact dynamic form used for quick frequency and period extraction.
Can periodic motion exist even if the equation is not in SHM form?
Yes. Periodic motion is broader than simple harmonic motion. SHM is a special case that requires restoring acceleration linear in displacement. If force is non-linear, such as proportional to x^3, the motion can still oscillate but is not strictly SHM for finite amplitude, and its period can depend on amplitude.
How is angular SHM connected to linear SHM equations?
Angular SHM follows the same mathematical structure with variable replacement. For torque law tau = -c theta and rotational inertia I, equation becomes I(d2theta/dt2) + c theta = 0. After dividing by I, d2theta/dt2 + (c/I)theta = 0, so omega = sqrt(c/I). The analogy is exact under small-angle linear restoring conditions.
Which exam trap is most common in this topic?
The most common trap is selecting an equation with wrong restoring sign because options look algebraically similar. The second trap is confusing omega with omega squared when reading coefficients. In timed NEET settings, writing one-line standard form before evaluating options prevents both errors and improves elimination speed.
Do I need full ODE solution methods for NEET in this topic?
You usually do not need advanced differential-equation solving steps. NEET typically tests recognition of correct SHM equation, extraction of frequency or period, and physical interpretation of sign and proportionality. However, you should know that sinusoidal solutions satisfy d2q/dt2 + omega^2 q = 0 and why the equation represents oscillation.
How do I cross-check if my final equation is physically reasonable?
Apply three checks: restoring sign must be opposite to displacement, coefficient of variable must have unit s^-2 after normalization, and predicted behavior must be bounded oscillation around equilibrium. If any check fails, the equation is not valid SHM form. This three-check routine catches most objective-question mistakes immediately.
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