NEET Physics - Chapter 16
Various Formulae of S.H.M. – Complete Notes, Revision, Important Questions & Downloads
Various Formulae of S.H.M. is a formula-application block where SHM in Physical Systems is solved by identifying inertia factor and spring/restoring factor before writing T = 2pi*sqrt(inertia/restoring). The textbook sequence here includes liquid in U-tube, floating cylinder, ball in hemispherical bowl, piston in cylinder, body in Earth tunnel, and torsional pendulum. NEET tests this topic through direct one-step numericals and option-elimination problems where the candidate must pick the correct model-specific period relation from closely spaced alternatives. A typical trap is using the spring-mass formula T = 2pi*sqrt(m/k) in a system that actually needs geometric parameters such as L, h, R, r, or rotational quantity I/C.
Formula LinkedApplication HeavyNCERT-Aligned
Expected QuestionsQ
1
Usually one formula-selection or direct substitution question from this formula cluster in mixed SHM papers.
Time Required⏱
2.5 h
1.5 h to memorize model-specific period formulas and 1 h for targeted numerical drills with unit checks.
Difficulty⚡
Medium
Each formula is short, but confusion between system geometry, mass term, and effective restoring constant causes frequent option mistakes.
NRI USA Curriculum GapUS
Moderate Bridge Needed
Many US high-school tracks treat oscillations by spring-pendulum examples, while NEET expects rapid switching across fluid, rotational, and Earth-tunnel SHM models.
11Subtopics
24Practice Questions
4Free Downloads
2.5 hPrep Time
Various Formulae of S.H.M. Weightage and Trend
Simple Harmonic Motion - Topic 20| NEET Year | Questions from this Topic | Bar | Marks |
|---|---|---|---|
| 2020 | 0 | 0 | |
| 2021 | 1 | 4 | |
| 2022 | 0 | 0 | |
| 2023 | 0 | 0 | |
| 2024 | 1 | 4 | |
| 2025 | 0 | 0 | |
| Topic-linked asks in recent NEET papers | 2 | 8 |
This block is tested as direct model-recognition: the stem gives a physical setup and asks the period expression.
Most errors come from inserting wrong geometric variable, such as L versus h in U-tube or R versus (R-r) in bowl motion.
Earth tunnel and torsional pendulum formulas are high-yield quick checks because they look different from spring-mass forms.
Most errors come from inserting wrong geometric variable, such as L versus h in U-tube or R versus (R-r) in bowl motion.
Earth tunnel and torsional pendulum formulas are high-yield quick checks because they look different from spring-mass forms.
0.3
Avg Questions / Year
8
Total Marks (6 yrs)
Direct
Pattern
Medium
Difficulty
5-Step Formula Application Routine
1
Classify the physical system first Before writing any equation, label the setup as fluid-column, buoyancy oscillation, rolling geometry, gas-piston, gravity tunnel, or torsional system so you select the correct period relation immediately.
2
Write restoring and inertia factors explicitly Use T = 2pi*sqrt(inertia/restoring) as the parent structure and map each symbol: for torsional pendulum inertia is I and restoring constant is C; for piston model restoring is pressure-area term.
3
Guard geometric substitutions For U-tube keep L as total liquid length and use L = 2h when needed; for hemispherical bowl keep (R-r) in the numerator instead of only R.
4
Run dimensional sanity check Confirm quantity inside square root has dimension of time squared; this catches common mistakes like forgetting division by g or mixing radius with diameter.
5
Finish with ratio-based speed practice Practice one-line ratio questions such as T1/T2 for changed liquid length or changed torsional constant so exam-time calculations stay fast and less error-prone.
Various Formulae of S.H.M. Download Kit
PDF · Cheat Sheet · MCQ Set · PYQFull Notes
Complete notes for SHM in Physical Systems with each model formula, variable map, and one worked NEET-style numerical per setup.
Download PDFFormula Sheet
One-sheet compact formula table covering U-tube, floating cylinder, bowl, piston, Earth tunnel, and torsional pendulum with symbol meanings.
Download PDFMCQ Practice
Practice set focused on formula selection, dimensional checks, and nearest-option elimination across mixed SHM system stems.
Download PDFPYQ Workbook
Year-tagged oscillations workbook with solved examples and a model-tag for each question to train fast formula recognition.
Download PDFSubtopics in Various Formulae of S.H.M.
2-Column Table| Column A | Column B |
|---|---|
Rapid Revision Cards
Concept → Trap → Example1) SHM in Physical Systems
Model mapUse model-specific period laws: U-tube T = 2pi*sqrt(L/2g), floating cylinder T = 2pi*sqrt(l/g), bowl motion T = 2pi*sqrt((R-r)/g), piston T = 2pi*sqrt(Mh/PA), Earth tunnel T = 2pi*sqrt(R/g), torsional pendulum T = 2pi*sqrt(I/C).
- First identify which restoring mechanism is active: buoyancy, gravity projection, gas compression, or torsion.
- Match each symbol to geometry before substitution, especially L versus h and R versus (R-r).
- Trap: treating all systems as m-k oscillator and forcing T = 2pi*sqrt(m/k) even when rotational or fluid parameters define restoring force.
Example (NEET-style)For a U-tube with total liquid length L = 0.80 m, T = 2pi*sqrt(L/2g) = 2pi*sqrt(0.80/19.6) about 1.27 s; using h = 0.80 m directly would overestimate period by sqrt(2).
Curriculum Gap: India vs USA
Two concrete preparation gaps to bridge for NEET readinessAP Physics 1/2 coverage vs NEET model switching
AP Physics usually emphasizes spring-block and pendulum prototypes, but this NEET topic expects quick switching among six different SHM physical systems in one chapter block.
- Build a one-page model-recognition chart listing each physical setup and its exact T-expression.
- Practice mixed MCQs where two options differ only by geometric term such as L/2g versus l/g.
US high-school algebra comfort vs NEET formula discipline
Many US classrooms discuss oscillations conceptually, while NEET asks strict formula substitution with unit consistency and symbol interpretation under time pressure.
- Train dimensional checks for every period formula to reject wrong options rapidly.
- Practice speed numericals involving Earth tunnel and torsional pendulum, which are less emphasized in standard US tracks.
NEET-style practice questions
6 MCQs1A liquid in a U-tube has total liquid length L = 0.98 m. Taking g = 9.8 m s^-2, the time period is closest to:U-tube model
1.0 s
1.4 s
2.0 s
2.8 s
For U-tube oscillation the relation is T = 2pi*sqrt(L/2g). Substituting L = 0.98 m and g = 9.8 m s^-2 gives T = 2pi*sqrt(0.98/19.6) = 2pi*sqrt(0.05) about 2pi*0.2236 about 1.40 s. So option 1.4 s is correct. Option 2.0 s usually comes from forgetting division by 2g and using L/g. Option 1.0 s appears when pi is dropped in fast arithmetic. Option 2.8 s appears if L is mistakenly doubled before substitution.
2In a hemispherical bowl of radius R, a small ball of radius r rolls without slipping near equilibrium. Which period formula is correct?Bowl geometry
T = 2pi*sqrt(R/g)
T = 2pi*sqrt((R-r)/g)
T = 2pi*sqrt((R+r)/g)
T = 2pi*sqrt(r/g)
The center of mass oscillates on an effective circular path whose radius is reduced from R to (R-r), so the given text formula is T = 2pi*sqrt((R-r)/g). Option A is a common near-miss where finite ball radius is ignored. Option C has no physical basis for restoring geometry. Option D incorrectly assumes only ball radius controls restoring acceleration. In NEET, this exact trap is used to test whether the candidate reads geometric definitions instead of applying a remembered shortcut.
3For a piston of mass M and area A in a cylinder of gas at pressure P and height h, SHM period is:Gas piston
T = 2pi*sqrt(Mh/PA)
T = 2pi*sqrt(PA/Mh)
T = 2pi*sqrt(M/PAh)
T = 2pi*sqrt(h/MPA)
The model-specific expression listed in the chapter is T = 2pi*sqrt(Mh/PA). The quantity PA behaves like restoring-force scale and appears in denominator. Option B is inverse and gives wrong dimension inside square root. Option C mixes h into denominator without model basis. Option D breaks dimensional consistency altogether. One fast check is to verify Mh/PA has time^2 dimension; only option A satisfies this structure from the given formula.
4A body moves in a frictionless tunnel through Earth along any chord. Its time period is independent of chord length and equals:Earth tunnel
2pi*sqrt(R/g), approximately 84.6 min
2pi*sqrt(g/R), approximately 84.6 min
2pi*sqrt(Rg), approximately 42.3 min
2pi*sqrt(2R/g), approximately 120 min
From the chapter formula, tunnel SHM period is T = 2pi*sqrt(R/g), numerically about 84.6 minutes for Earth. Option B inverts the ratio and gives incorrect dimensions. Option C multiplies R and g, which is dimensionally invalid for a period formula. Option D introduces an extra factor of sqrt(2) without physical basis. In exam conditions, the key cue is the stated independence from chord length, which uniquely points to the Earth-tunnel relation.
5A torsional pendulum has moment of inertia I and torsional constant C. If C becomes 4C while I is unchanged, the new period is:Torsional pendulum
T/4
T/2
2T
4T
For torsional pendulum, T = 2pi*sqrt(I/C). With C -> 4C, T' = 2pi*sqrt(I/4C) = (1/2)*2pi*sqrt(I/C) = T/2. Option B is correct. Option A assumes linear inverse dependence on C instead of inverse square-root dependence. Options C and D reverse the trend and would mean stronger restoring torque increases period, which is physically incorrect. This is a standard ratio question where exponent handling decides the answer quickly.
6A floating cylinder dips by length l in equilibrium. Which statement is correct for small vertical oscillations?Floating cylinder
T is proportional to sqrt(l) and equals 2pi*sqrt(l/g)
T is proportional to l and equals 2pi*l/g
T is proportional to 1/sqrt(l)
T is independent of l
For a floating cylinder in this chapter's SHM formula set, T = 2pi*sqrt(l/g). Therefore period grows with sqrt(l), not with l itself. Option A is correct. Option B corresponds to linear dependence and has wrong dimensions. Option C incorrectly predicts decreasing period with deeper dip length. Option D ignores the explicit l dependence in the formula. In objective papers, this question often appears as a proportionality check rather than full calculation.
Practice Questions
Click "Reveal Answer" after attempting1If in a U-tube the total liquid length changes from 0.5 m to 2.0 m, the ratio of new period to old period is:
1
2
4
1/2
👁 Reveal Answer
Correct option: 2. Since T = 2pi*sqrt(L/2g), period is proportional to sqrt(L). Therefore T2/T1 = sqrt(2.0/0.5) = sqrt(4) = 2. Option 1 would imply no dependence on liquid length, option 4 assumes linear dependence, and option 1/2 inverts the trend.
2For a torsional pendulum, I is increased by 44% while C remains unchanged. By what percentage does the period increase?
20%
44%
10%
50%
👁 Reveal Answer
Correct option: 20%. Using T = 2pi*sqrt(I/C), T is proportional to sqrt(I). If I becomes 1.44I, then new period T' = T*sqrt(1.44) = 1.2T. Hence increase is 20%. Option 44% is the common mistake of treating square-root relation as linear; options 10% and 50% do not follow from the ratio.
3A problem gives R = 1.0 m, r = 0.2 m for a ball in a hemispherical bowl. Taking g = 10 m s^-2, approximate period is:
1.78 s
2.50 s
0.89 s
3.14 s
👁 Reveal Answer
Correct option: 1.78 s. Formula is T = 2pi*sqrt((R-r)/g) = 2pi*sqrt(0.8/10) = 2pi*sqrt(0.08) about 2pi*0.283 = 1.78 s. Option 2.50 s typically comes from using R instead of (R-r); option 0.89 s comes from missing factor 2 in 2pi; option 3.14 s reflects rough pi misuse.
4For Earth tunnel SHM, if g is approximated as 9.8 m s^-2 and R = 6.37 x 10^6 m, which is nearest period?
42 min
60 min
84.6 min
120 min
👁 Reveal Answer
Correct option: 84.6 min. Use T = 2pi*sqrt(R/g). Substituting gives T about 2pi*sqrt(6.37x10^6/9.8) s about 2pi*806 s about 5065 s, which is about 84.4 min. So nearest listed value is 84.6 min. Other options are common memory errors from halving or doubling the known benchmark value.
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Frequently Asked Questions
Notes · Downloads · Revision · Important QuestionsWhy can I not use one universal spring formula for every SHM system in this topic?
Because each setup has a different restoring mechanism and effective inertia term. Spring-block problems use m and k directly, but U-tube uses liquid-column geometry, Earth-tunnel uses gravitational field variation with radius, and torsional pendulum uses rotational inertia I with torsional constant C. The parent idea is still T = 2pi*sqrt(inertia/restoring), but the symbolic form changes by system.
In U-tube questions, how do I decide between L and h in the period formula?
The textbook relation is written as T = 2pi*sqrt(L/2g) and also as T = 2pi*sqrt(h/g) because L = 2h for equal limbs. So both are equivalent only when L is total liquid length and h is undisturbed height in one limb. Most mistakes happen when students use h as total length or substitute L as single-limb height without converting.
What is the fastest way to avoid wrong options in hemispherical bowl numericals?
Immediately write the geometric term as (R-r), not just R. Then check whether the value inside square root has units of length divided by acceleration, which gives time squared. If an option ignores r, adds R+r, or produces wrong dimension, eliminate it before arithmetic. This two-step filter removes most distractors in less than 10 seconds.
How is the Earth tunnel period independent of which chord is chosen?
Inside Earth, gravitational force component along any tunnel chord is proportional to displacement from midpoint for small oscillations, so motion remains SHM with the same angular frequency based on R and g. That leads to T = 2pi*sqrt(R/g), independent of chord orientation. In exams, this independence is usually a clue that the Earth-tunnel formula should be recalled directly.
Why does increasing torsional constant C reduce the period in torsional pendulum?
A larger C means stronger restoring torque per unit angular displacement, so the system returns faster to equilibrium. Since T = 2pi*sqrt(I/C), period varies inversely with sqrt(C). This is exactly parallel to spring SHM where larger k reduces period, but here rotational inertia I replaces mass and torsional constant C replaces linear spring constant.
Is floating cylinder SHM mainly a buoyancy problem or a spring problem?
Physically it is a buoyancy-restored oscillation, but mathematically near equilibrium it behaves like SHM with an effective restoring coefficient, giving T = 2pi*sqrt(l/g) in the listed model. So treat it as a buoyancy-origin SHM formula, not as a literal metal spring system. This distinction helps when interpreting symbols and deriving proportionality questions.
How should I prepare this topic in the last week before NEET?
Do one formula-recall pass daily for six models, then solve short mixed sets where only one geometric or physical parameter changes. Focus on ratio questions and unit consistency instead of long derivations, because this topic is tested as fast objective application. Keep a compact chart with formula, variable meanings, and one frequent trap for each model.
What are the most common conceptual errors in this formula set?
The top errors are model confusion, wrong geometric substitution, and exponent mistakes in ratio updates. Students also mix translational and rotational inertia terms, especially in torsional cases. Another recurring mistake is accepting an option without dimensional verification. A quick dimension check and model label before substitution prevents most of these losses.
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