# Conveyor Belt & Block-on-Board Models - phase analysis, v–t graphs, heat

> Conveyor-belt and block-on-board friction problems: phase-by-phase forces, common speed, v–t graphs, relative sliding and heat, with practice.

- URL: https://qilingtools.com/en/physics/conveyor-belt
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-29
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

The two most common friction dynamics models in physics exams, as adjustable animations: a horizontal belt (block placed at rest, slower or faster than the belt, or sliding against it), an inclined belt (moving up or down, where μ versus tanθ decides what happens after the speeds match) and a block on a board (smooth or rough floor, force on the block or the board, force removed part-way). The solver works phase by phase (forces, accelerations, common speed or leaving, next phase), writes out the friction direction, accelerations and displacements of each phase, plots both v–t graphs with the relative displacement shaded, and reports the friction heat, the extra motor work and an energy-balance check.

## How to use

1. Choose horizontal belt, inclined belt or block on board and pick a classic case
2. Adjust belt speed, length, μ, initial velocities, angle, masses or force
3. Press Play, scrub the time slider, and read the v–t graph and phase analysis
4. Open Practice for random belt and board problems marked instantly

## Features

- Horizontal belt, inclined belt and block-on-board scenes with 14 classic presets
- Exact phase solver: common speed, reversal, sliding off, board stopping and force removal
- Forces, friction direction, accelerations, displacements and end condition for every phase
- v–t graphs of both bodies with the relative displacement shaded and a time scrubber
- Heat Q = f·s_rel, extra motor work, work by the force and an energy-balance check
- Critical pulling force for the board and random practice (time, heat, common speed, sliding distance, maximum force)

## FAQ

### Which way does friction act on a block on a belt?

Friction opposes motion relative to the belt. If the block is slower than the belt (including placed at rest) it slips backwards relative to the belt and friction drags it forward; if it is faster, friction slows it. Once the speeds match on a horizontal belt there is no friction. What matters is the velocity difference, not the block’s direction relative to the ground.

### What happens on an inclined belt after the speeds match?

Compare μ with tanθ. If μ ≥ tanθ the maximum static friction μmg cosθ can balance mg sinθ, so the block rides with the belt. If μ < tanθ it keeps slipping down relative to the belt and friction reverses, so on a downward belt the acceleration drops from g(sinθ + μcosθ) to g(sinθ − μcosθ). The classic case θ = 37°, μ = 0.5, v = 10 m/s, L = 16 m takes 1 s plus 1 s.

### How are the friction heat and the motor work found?

Heat Q = f·s_rel, where s_rel is the relative sliding distance (the area between the two v–t lines). The motor keeps the belt at constant speed against the block’s friction, doing W = f·s_belt. Energy conservation gives W_motor = ΔEk + ΔEp + Q. For a 1 kg block placed on a 2 m/s belt with μ = 0.1, Q = 2 J and the motor does 4 J, half of which becomes kinetic energy.

### How do I tell whether the block slips on the board?

Assume they move together, find the common acceleration a from the whole system, then find the static friction the block needs (for a force on the board, f = ma). If that f is no more than μ₁mg they move together; otherwise they slip and each body uses kinetic friction. The critical force on the board is F = (M + m)(μ₁ + μ₂)g.

### Why does the board sometimes stay still on a rough floor?

The sliding block pushes the board with at most μ₁mg, while the floor can hold the board with static friction up to μ₂(M + m)g. If μ₁mg ≤ μ₂(M + m)g the board does not move and the block decelerates alone until it stops or slides off. The phase analysis notes when the floor holds the board.

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