Minimum Mass Hung From String – Complete Notes, Revision, Important Questions & Downloads
Minimum Mass Hung From String covers the TOC subtopic Horizontal and Inclined Cases. In the horizontal case, mass m₁ lies on a rough horizontal surface connected via a string over a frictionless pulley to a hanging mass m₂. The minimum m₂ to just start m₁ moving is m₂ = μm₁. In the inclined case where m₁ is on a rough incline, the minimum hanging mass becomes m₂ = m₁[sinθ + μcosθ]. NEET tests both configurations as direct numericals — given m₁, μ, and surface geometry, find m₂_min; or given m₂ and m₁, find μ.
NEET Weightage — Minimum Mass Hung From String
Friction (Chapter 5)| NEET Year | Questions from this Topic | Bar | Marks |
|---|---|---|---|
| 2024 | 0 | 0 | |
| 2023 | 1 | 4 | |
| 2022 | 0 | 0 | |
| 2021 | 1 | 4 | |
| 2020 | 0 | 0 | |
| 2019 | 0 | 0 | |
| 6-Year Total (2019–2024) | 1–2 | 4–8 |
INCLINED CASE: m₁ on rough inclined plane (angle θ), m₂ hanging. At limiting condition for m₁ to just start moving UP the incline. T must overcome gravity component (m₁gsinθ down the incline) AND limiting friction (μm₁gcosθ down the incline). From m₂: T = m₂g. From m₁ on incline: T = m₁gsinθ + μm₁gcosθ = m₁g[sinθ + μcosθ]. So minimum m₂ = m₁[sinθ + μcosθ].
REVERSE FORMULA (finding μ): From horizontal case: μ = m₂/m₁. From inclined case: m₂ = m₁[sinθ + μcosθ] → μ = [m₂/m₁ − sinθ]/cosθ = [m₂/(m₁cosθ)] − tanθ. Both are direct NEET calculation formulas. Check: when θ = 0, inclined formula reduces to m₂ = μm₁ (horizontal case). At θ = angle of repose (μ = tanθ), sinθ + μcosθ = 2sinθ → m₂ = 2m₁sinθ for just-motion on rough incline at repose angle.
How to Solve Minimum Mass Hung Problems for NEET
Step 1 — Identify surface geometry and direction of impending motion Is m₁ on horizontal or inclined surface? If inclined, is m₁ being pulled UP or DOWN the incline by m₂? For standard NEET problems, m₂ tries to pull m₁ UP the incline. This means both gravity component (m₁gsinθ, down slope) AND friction (μm₁gcosθ, opposing upward motion so also down slope) both oppose the tension.
Step 2 — Apply limiting equilibrium (T = F_l at threshold) At minimum m₂, the system is exactly at the threshold of motion (static limiting condition). Set T = m₂g (from m₂'s equation, since m₂ just barely not accelerating). Set T = F_l (net resistant force on m₁). For horizontal: F_l = μm₁g. For incline up: F_l = m₁gsinθ + μm₁gcosθ. Equate: m₂g = F_l and solve for m₂.
Step 3 — Check angle-of-repose condition as a sanity check For θ = 0 (horizontal), inclined formula gives m₂ = m₁[0 + μ×1] = μm₁ ✓. As θ → 90°, sinθ → 1 and cosθ → 0, giving m₂ → m₁ (gravity component alone must be overcome, friction negligible on vertical surface). At θ = angle of repose (tanθ = μ): m₂ = m₁[sinθ + tanθ×cosθ] = m₁×2sinθ.
Study Materials — Minimum Mass Hung From String
PDF · Cheat Sheet · MCQ Set · PYQSubtopics — Minimum Mass Hung From String
2-Column Table| Column A | Column B |
|---|---|
Rapid Revision — Minimum Mass Hung From String
Concept → Trap → Example1) Horizontal and Inclined Cases — Minimum Hanging Mass Formula
Horizontal and Inclined CasesSETUP: mass m₁ on a surface, connected via massless string over a frictionless pulley to mass m₂ hanging vertically. HORIZONTAL CASE: Normal force on m₁ = m₁g. Limiting friction F_l = μ_s × m₁g = μm₁g. String tension T = m₂g (from m₂'s equilibrium when just at threshold). At limiting condition: T = F_l → m₂g = μm₁g → m₂ = μm₁. This is the minimum mass to just start motion. Coefficient can be found: μ = m₂/m₁. INCLINED CASE (m₁ on incline angle θ, m₂ pulls m₁ UP the slope): Normal force N = m₁gcosθ. Friction force (opposing upward motion, directed DOWN slope): F_l = μN = μm₁gcosθ. Gravity component along incline (opposing upward motion, directed DOWN slope): m₁gsinθ. For m₁ on incline: T = m₁gsinθ + F_l = m₁gsinθ + μm₁gcosθ = m₁g[sinθ + μcosθ]. From m₂: T = m₂g. At threshold: m₂g = m₁g[sinθ + μcosθ] → m₂ = m₁[sinθ + μcosθ]. Reverse formula: μ = [m₂/m₁ − sinθ]/cosθ = m₂/(m₁cosθ) − tanθ. CHECK: θ = 0 → m₂ = m₁[0 + μ×1] = μm₁ ✓ (reduces to horizontal case).
- INCLINED CASE — m₂ pulling m₁ DOWN slope (m₁ on incline, m₂ hangs, m₁ about to slide DOWN): Here gravity component aids downward motion. Friction now acts UP the slope (opposing downward motion). For m₁ about to slide down: m₁gsinθ − T − μm₁gcosθ = 0 → T = m₁g[sinθ − μcosθ]. Since T = m₂g: m₂ = m₁[sinθ − μcosθ]. NOTE: For m₁ to actually slide down, we need sinθ > μcosθ (tanθ > μ), i.e., angle exceeds angle of repose. If tanθ < μ, the body won't slide down under its own weight even without m₂.
- NORMAL FORCE ON INCLINE: The most common error is using N = m₁g instead of N = m₁gcosθ. On incline, N = mgcosθ because the weight component perpendicular to incline surface is mgcosθ (the surface cancels only the perpendicular component of gravity). Using N = mg gives friction = μmg (wrong) and leads to m₂ = m₁[sinθ + μ], which is incorrect.
- STRING AND PULLEY ASSUMPTIONS: The derivation assumes (a) string is massless and inextensible, (b) pulley is frictionless. If pulley has friction, the tension on the two sides of the string differs. If rope has mass, the analysis must include rope's weight. NEET problems typically state 'frictionless pulley' and 'massless string' — confirm these before applying m₂ = m₁[sinθ + μcosθ] directly.
US Curriculum Gaps — Minimum Mass Hung From String
Topics in this section are in NEET but may be framed differently in US physics courses.Limiting Friction Threshold in AP Physics 1
AP Physics 1 covers the Atwood machine and inclined plane with friction as separate topics. The limiting equilibrium condition T = μN at the threshold of motion is covered, but the combined result m₂ = m₁[sinθ + μcosθ] is typically derived per-problem rather than memorised. For NEET speed, commit both the horizontal formula (m₂ = μm₁) and inclined formula (m₂ = m₁[sinθ + μcosθ]) to memory.
- AP Physics 1: derives limiting condition from FBD each time; no named 'minimum mass' result
- NEET: direct formula recall of m₂ = μm₁ (horizontal) and m₂ = m₁[sinθ + μcosθ] (inclined) expected
- Reverse formula μ = m₂/(m₁cosθ) − tanθ is equally testable in NEET — AP students may not have this memorised
Pulley-and-Incline Friction Systems in University Physics
Halliday & Resnick covers pulley-string-incline systems in Chapter 6 (Force and Motion II). However, the specific scenario of finding the minimum hanging mass to just initiate motion is not presented as a named formula. NRI students should focus on the limiting equilibrium setup: at the instant of 'just about to move,' static friction reaches its maximum value μsN, and the system is in equilibrium (net force = 0 on each mass).
- US textbooks: solve pulley-incline problems using component equations, not named minimum-mass formulas
- NEET: requires direct formula application; m₂ = m₁[sinθ + μcosθ] is a commonly tested result
- Coefficient derivation μ = m₂/(m₁cosθ) − tanθ is a common NEET MCQ sub-type not emphasized in US courses
NEET-Style Practice Questions — Minimum Mass Hung From String
4 QuestionsPractice Problems — Minimum Mass Hung From String
Click "Reveal Answer" after attempting👁 Reveal Answer
👁 Reveal Answer
👁 Reveal Answer
👁 Reveal Answer
Physics — Friction Revision Checklist
Use this section for quick chapter tracking before mocks, part tests, and final NEET revision.
FAQ — Minimum Mass Hung From String
Notes · Downloads · Revision · Important QuestionsWhat is the minimum hanging mass formula on a horizontal surface?
What is the minimum hanging mass formula on an inclined surface?
How does the formula change if m₁ is being pulled DOWN the incline?
Why is normal force mgcosθ on an incline, not mg?
What does 'frictionless pulley' mean for this problem?
How do you find θ given all other values?
What happens if m₂ exceeds the minimum value?
What is the condition for the incline formula to be valid?
NEET NRI Counseling & Admission eBook Download
A practical guide covering sponsor rules, document checklist, verification traps, NRI quota reality, and step-by-step counselling flow. Designed to prevent last-minute rejections and wrong choice filling.
Schedule Trial Session For NEET Prep
Get a short diagnostic + study roadmap: syllabus gaps (NCERT vs U.S. curriculum), weak chapters, and the exact weekly plan needed to improve accuracy under time.