Sticking of Block With Accelerated Cart – Complete Notes, Revision, Important Questions & Downloads
Sticking of Block With Accelerated Cart covers the TOC subtopic Minimum Acceleration for Block to Stick. A small block of mass m is placed against the vertical face of a cart of mass M. The cart accelerates horizontally to the right. In the non-inertial frame, a pseudo-force ma acts on the block to the left, pressing it against the vertical cart wall. Friction (μ × normal force = μma) acts upward, supporting the block's weight mg. For the block not to slide down: μma ≥ mg → a ≥ g/μ. Minimum acceleration: a_min = g/μ. The minimum external force on the system: F_min = (M+m) × g/μ. NEET tests this as a direct substitution problem.
NEET Weightage — Sticking of Block With Accelerated Cart
Friction (Chapter 5)| NEET Year | Questions from this Topic | Bar | Marks |
|---|---|---|---|
| 2024 | 1 | 4 | |
| 2023 | 1 | 4 | |
| 2022 | 0 | 0 | |
| 2021 | 0 | 0 | |
| 2020 | 1 | 4 | |
| 2019 | 0 | 0 | |
| 6-Year Total (2019–2024) | 0–2 | 0–8 |
MINIMUM ACCELERATION: a_min = g/μ. At this minimum: friction exactly equals weight: f = μma_min = μm(g/μ) = mg ✓. If a < g/μ: f = μma < mg — block slides down. If a > g/μ: f = μma > mg — block is held up with excess friction (it is in static friction regime, not the limiting case). MINIMUM FORCE: For the entire (M+m) system to have acceleration a_min = g/μ: F_min = (M+m)×a_min = (M+m)g/μ.
NON-INERTIAL (CART) FRAME: In cart's frame, block is stationary (it's in the cart's frame). Pseudo-force on block = ma acting to the LEFT (opposite to cart's acceleration direction, which is to the right). This pseudo-force presses the block into the RIGHT wall of the cart. Normal force N = ma (from the wall on the block). Friction = μN = μma (upward). Weight = mg (downward). For equilibrium: μma = mg → a = g/μ. Both frames give the same result (as expected).
How to Solve the Block-on-Vertical-Cart-Face Problem
Step 1 — Identify the normal force The normal force on the block from the cart's vertical face = ma (the block must accelerate horizontally with the cart; only the wall can provide this force). Write N = ma.
Step 2 — Write the friction condition Friction supports the block: f = μN = μma (upward). Weight pulls block down: mg. For block not to slide: μma ≥ mg → a ≥ g/μ.
Step 3 — Find minimum acceleration and force a_min = g/μ. Minimum force on system: F_min = (M+m) × a_min = (M+m)g/μ.
Study Materials — Sticking of Block With Accelerated Cart
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Rapid Revision — Sticking of Block With Accelerated Cart
Concept → Trap → Example1) Minimum Acceleration for Block to Stick — Complete Analysis
Minimum Acceleration for Block to StickSETUP: A cart of mass M is on a smooth horizontal surface. A small block of mass m is placed against the vertical (left or right) face of the cart — the block touches the cart's wall, not the floor. The cart is pushed horizontally (to the right) by an external force F. The block and cart accelerate together at a = F/(M+m) (assuming they move as a unit). THE QUESTION: What is the minimum acceleration a such that the block does NOT slide down the cart's face? GROUND FRAME ANALYSIS: For the block (mass m) to accelerate horizontally at the same rate a as the cart, the cart's wall must push the block horizontally with a net horizontal force = ma. This is the NORMAL FORCE from the wall on the block: N = ma. Friction: between block and cart wall — since the block tends to slide downward (weight mg acts down), the kinetic/static friction acts UPWARD. Friction force f = μN = μma (upward, from cart wall on block). CONDITION FOR BLOCK TO STAY: Upward friction ≥ downward weight: f ≥ mg → μma ≥ mg → a ≥ g/μ. MINIMUM ACCELERATION: a_min = g/μ. AT THE MINIMUM: f = μm × (g/μ) = mg (friction exactly balances weight). Excess acceleration: if a > g/μ, friction > mg, block is held more firmly — static friction, not kinetic. MINIMUM FORCE: F_min produces a_min for the whole system: F_min = (M+m) × a_min = (M+m)g/μ.
- IMPORTANT: a_min = g/μ is independent of the block's mass m. Whether the block is a 1 kg block or a 100 kg block, the same minimum acceleration is required to hold it against the cart's face. This is analogous to stopping distance being independent of mass — the friction force (μma) and the weight (mg) both scale with m, so m cancels in the condition μma ≥ mg → a ≥ g/μ.
- WHAT IF BLOCK HAS MASS M (same as cart)? The block mass doesn't affect a_min = g/μ. However, the minimum FORCE does depend on total mass: F_min = (M+m)g/μ. If block mass m increases, F_min increases (need more force to accelerate a heavier system to the same a_min). Common confusion: 'bigger block needs more force' is TRUE for the force on the system, but NOT for the minimum acceleration.
- NON-INERTIAL FRAME SHORTCUT: In the cart's (non-inertial) frame, the block appears stationary. Three forces: weight mg (down), pseudo-force ma (horizontal, away from the direction of cart's acceleration — into the wall), and friction f (up). Equilibrium: horizontal: N = ma (pseudo-force balanced by normal); vertical: f = mg. At limit: f = μN = μma = mg → a = g/μ. This non-inertial frame analysis gives the same result and is sometimes faster to set up.
US Curriculum Gaps — Sticking of Block With Accelerated Cart
Topics in this section are in NEET but may be framed differently in US physics courses.Pseudo-Force Analysis in Non-Inertial Frames in AP Physics 1
AP Physics 1 covers Newton's laws in inertial frames and introduces the concept of fictitious forces qualitatively. The specific problem of a block on a vertical cart face — where the cart's acceleration provides the horizontal normal force — is not a standard AP Physics 1 problem type. NRI students need to explicitly practise identifying that the normal force on the block's vertical surface equals ma (not mg), as this is counterintuitive compared to floor-block problems where N = mg.
- AP Physics 1: pseudo-forces introduced qualitatively; block-on-vertical-cart-face not a standard problem
- NEET: a_min=g/μ is a direct-formula question — know the derivation via N=ma, f=μma, μma=mg
- Key insight: normal force on block = ma (horizontal), not related to gravity — vertical friction supports weight
Force Analysis on Objects in Accelerating Reference Frames in University Physics
Halliday & Resnick covers pseudo-forces and accelerating reference frames in the dynamics chapters. The block-on-vertical-cart configuration is sometimes presented as a worked example. However, NEET versions often include the minimum total force F = (M+m)g/μ calculation and comparison of two blocks of different masses (showing mass-independence of a_min), which are extensions beyond typical university textbook treatments.
- US university: pseudo-force in accelerating frames covered; vertical-cart-face occasionally as example
- NEET: compute a_min=g/μ and F_min=(M+m)g/μ in 30 seconds — practise until automatic
- Mass-independence of a_min is a conceptual test — two blocks of different masses need the same a_min
NEET-Style Practice — Sticking of Block With Accelerated Cart
4 QuestionsPractice Problems — Sticking of Block With Accelerated Cart
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Physics — Friction Revision Checklist
Use this section for quick chapter tracking before mocks, part tests, and final NEET revision.
FAQ — Sticking of Block With Accelerated Cart
Notes · Downloads · Revision · Important QuestionsWhat is the minimum acceleration for a block to stick to the vertical face of a cart?
Why does the normal force on the block equal ma?
How does the minimum force F_min relate to a_min?
Is a_min affected by the block's mass?
What happens if the cart decelerates (slows down) instead of accelerating?
What if the cart moves on a rough surface (not frictionless)?
If a = 2 × a_min, what is the friction force on the block?
Can I solve this using energy methods?
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