# Connected Bodies - Whole-System vs Isolation, Pulleys, Stacked Blocks, Spring Cut

> Connected bodies with Newton’s 2nd law: strings, pushes, pulleys, stacked-block slipping, spring vs string instant acceleration, elevator readings.

- URL: https://qilingtools.com/en/physics/connected-bodies
- 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

This tool animates the connected-body problems that come up again and again with Newton’s second law: two blocks tied by a string or pushed side by side on a level floor or slope, an Atwood machine and a table-plus-hanging-mass pulley, a block stacked on a board (finding the force at which they start to slip), what springs and strings do the instant a thread is cut or a force removed, and apparent weight in an elevator. Each model follows the routine of finding a for the whole system and then isolating one body for the internal force, with the numbers substituted.

## How to use

1. Pick a tab: pulled or pushed pair, pulleys, stacked blocks, instant acceleration or elevator
2. Choose a classic case or adjust masses, force, slope, friction and connection type
3. Play the animation and read the force arrows, the a–F graph or the scale reading graph alongside the analysis
4. Open Practice for random connected-body problems with instant marking

## Features

- String or push, level or slope: a from the whole system, T by isolation, showing T = m_B F/(m_A+m_B) for equal μ
- Different μ changes the internal force; a slack string or separating blocks are detected
- Atwood machine and table pulley: acceleration, tension and the T < m₂g comparison, with a static check
- Stacked blocks with the force on the top or bottom: start-moving force F₀, slipping threshold F_c and an a–F graph
- Instant acceleration: two hanging balls, a ball held by slanted and horizontal lines, a spring-linked pair; switch spring/string, slow-motion simulation
- Elevator scale reading over a three-phase trip, separating heavier, lighter and weightless, plus random practice

## FAQ

### When should I treat the bodies as one system and when should I isolate one?

When the bodies share the same acceleration you can treat them as one system; internal forces such as tension and contact forces cancel in pairs, so F_ext = (m₁ + m₂)a gives the acceleration quickly. To find an internal force you must isolate one body, usually the one with the fewest forces (often the one the external force does not act on), and apply F_net = ma to it. The tool’s analysis always follows this whole-system-then-isolation pattern.

### Why does the tension not depend on μ or the slope when both blocks have the same μ?

Whole system: F − μ(m_A + m_B)g cosθ − (m_A + m_B)g sinθ = (m_A + m_B)a. Isolating the rear block B: T − μm_Bg cosθ − m_Bg sinθ = m_Ba. Substituting a gives T = m_B F/(m_A + m_B): friction and the downhill pull are shared in proportion to mass and cancel out. With different μ values this no longer holds, which you can see by unlocking μ_A and μ_B in the tool.

### How do you find the force at which stacked blocks start to slip?

Find the largest acceleration the unpulled body can get. If the force pulls the bottom board, the top block is driven only by static friction, so a_max = μ₁g; then for the whole system F − μ₂(m + M)g = (m + M)a gives F_c = (μ₁ + μ₂)(m + M)g. If the force pulls the top block, the board’s largest acceleration is (μ₁mg − μ₂(m + M)g)/M, and if that is not positive the board never moves. The tool plots both accelerations against F and marks F₀ and F_c.

### Why does a spring force stay the same when a thread is cut, while a string force can jump?

A spring’s force depends on its stretch, and the stretch cannot change in an instant, so the force keeps its old value. A light string (or rod) hardly stretches, so its force can change immediately and must be re-analysed. For example, with ceiling–thread–A–spring–B at rest, cutting the thread leaves the spring still pulling A down with m_Bg, so A accelerates at (m_A + m_B)g/m_A and B at 0. Replace the spring by a thread and both fall at g.

### Do heavier and lighter readings depend on velocity or acceleration?

Only on acceleration. For a person on a scale in an elevator, N − mg = ma with up as positive. With acceleration upward N > mg (reading heavy), whether speeding up going up or slowing down going down. With acceleration downward N < mg (reading light), and with a = g downward N = 0 (weightless). Gravity on the person never changes; only the force on the support, the apparent weight, does.

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