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MechanicsIntroStarter track

Concept module

Static Equilibrium / Centre of Mass

Shift one support region under one loaded plank and see how centre of mass, support reactions, and torque balance decide whether the object stays stable or tips.

Interactive lab

Keep the stage, graph, and immediate control feedback in one working view.

Drag the cargo block or the support center. The plank mass, combined centre of mass, support region, reaction forces, response graphs, compare mode, and challenge checks all read the same static-balance model.

Graphs

Switch graph views without breaking the live stage and time link.

Torque about the support center

For the current cargo mass, cargo position, and support width, this response curve shows where the support center makes the weight torque vanish.

support center x (m): -1.3 to 1.3torque about support center (N m): -128 to 128

tau_supportcurrent setup

Hover or scrub to link the graph back to the stage.support center x (m) / torque about support center (N m)

Controls

Adjust the physical parameters and watch the motion respond.

Cargo mass

m_{c}

3 kg

Changes how strongly the movable block shifts the combined centre of mass and the total supported weight.

Cargo position

x_{c}

0.8 m

Slide the cargo left or right along the plank to shift the combined centre of mass.

Support center

x_{s}

0.3 m

Move the support region left or right under the plank without changing the weight distribution.

Support width

w

1.4 m

Narrow or widen the support region to change the stability margin before tipping.

More tools

Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.

Show

More presets

Presets

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

Graph readingPrompt 1 of 2

Notice on the support-torque graph that the zero crossing happens when the support center sits under the combined centre of mass.

Try this

Hover across the response curve, then drag the support center until the marker lands on the zero line.

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

m_{c}

Cargo mass

3 kg

Heavier cargo pulls the combined centre of mass more strongly toward its own position and raises the total supported weight.

Graph: Torque about the support centerGraph: Stability margin vs cargo positionOverlay: Weight linesOverlay: Combined centre of mass

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Use the live prompt to keep the balance story disciplined. Each prompt points at the same static model from a different angle: centre of mass, torque balance, support reactions, or support-region geometry.

Graph readingPrompt 1 of 2

Graph: Torque about the support center

Notice on the support-torque graph that the zero crossing happens when the support center sits under the combined centre of mass.

Try this

Hover across the response curve, then drag the support center until the marker lands on the zero line.

Why it matters

It keeps static equilibrium tied directly to the torque language you already used on the pivoted bar.

Control: Support centerGraph: Torque about the support centerOverlay: Combined centre of massOverlay: Torque armEquation

x_{s}

Equation

x_{c}

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

3 visible

Overlay focus

Weight lines

Shows the plank weight and cargo weight separately before combining them into one effective weight line.

What to notice

The separate weight arrows explain why mass placement matters before you collapse them into one combined centre of mass.
Heavier cargo does two things at once: it raises the total weight and pulls the combined weight line toward itself.

Why it matters

Static-equilibrium reasoning is cleaner when you can see where the total weight came from instead of treating the centre of mass like magic.

Control: Cargo massControl: Cargo positionGraph: Torque about the support centerGraph: Stability margin vs cargo positionEquation

m_{c}

Equation

x_{c}

Challenge mode

Use the same static bench to prove that you can read centre of mass, torque balance, and support reactions together instead of as separate rules.

0/3 solved

ConditionStretch

3 of 8 checks

Same centre of mass, different cargo

Open compare mode. Keep Setup A on Support under load, then tune Setup B so a heavier cargo placed closer in lands on the same combined centre of mass.

Compare modeGraph-linkedGuided start2 hints

Suggested start

Clone Setup A into compare mode, then make Setup B heavier but closer in.

Matched

Open the Torque about the support center graph.

Torque about the support center

Matched

Keep the Combined centre of mass visible.

Matched

Keep the Support region visible.

Pending

Stay in compare mode while editing Setup B.

Explore

Pending

Keep Setup A on the reference centre-of-mass position.

Pending

Make Setup B clearly heavier than Setup A.

Pending

Move the heavier cargo closer to the middle.

Pending

Match Setup B to the same combined centre of mass.

The checklist updates from the live simulation state, active graph, overlays, inspect time, and compare setup.

A 4 kg plank carries 3 kg of cargo at x = 0.8 m. The total mass is 7 kg and the combined centre of mass is at x_CM = 0.34 m. The support region runs from -0.4 m to 1 m, so the current reactions are R_left = 32.2 N and R_right = 36.4 N. The combined centre of mass stays 0.66 m inside the support region, so both support reactions remain positive.

Equation detailsDeeper interpretation, notes, and worked variable context.

Combined centre of mass

The total weight can be treated as acting through the weighted-average position of the mass distribution.

x_{CM} = \frac{\sum m _{i} x _{i}}{\sum m _{i}}

m_{c}

Cargo mass 3 kg

x_{c}

Cargo position 0.8 m

Vertical force balance

The support reactions together must hold up the total weight.

R_{L} + R_{R} = W

m_{c}

Cargo mass 3 kg

x_{s}

Support center 0.3 m

w

Support width 1.4 m

Torque balance for static equilibrium

Even if the forces add to zero, the supported object is not in static equilibrium unless the torques also cancel.

\sum τ = 0

x_{c}

Cargo position 0.8 m

x_{s}

Support center 0.3 m

Support-region stability rule

The combined centre-of-mass line must fall inside the support region if both reactions are going to stay upward and physically possible.

x_{L} \leq x_{CM} \leq x_{R}

x_{c}

Cargo position 0.8 m

x_{s}

Support center 0.3 m

w

Support width 1.4 m

Progress

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Let the live model runChange one real controlOpen What to notice

Try this setup

Copy the live bench state and reopen this concept with the same controls, graph, overlays, and compare context.

Stable links

Starter track

Step 2 of 50 / 5 complete

Rotational Mechanics

Earlier steps still set up Static Equilibrium / Centre of Mass.

1. Torque2. Static Equilibrium / Centre of Mass3. Rotational Inertia / Moment of Inertia4. Rolling Motion+1 more steps

Previous step: Torque.

Start from track beginning

Short explanation

What the system is doing

Static equilibrium means two things are true at once: the upward and downward forces balance, and the torques about any point balance as well. For supported objects, those two conditions are easiest to read by tracking where the combined weight acts.

This bench stays bounded on purpose. One plank has its own weight, one movable cargo block shifts the mass distribution, and one adjustable support region can move or narrow. The same live state drives the stage, support reactions, response graphs, worked examples, challenge checks, and quick test, so centre-of-mass reasoning never drifts away from the torque language introduced in Torque.

Key ideas

01The combined centre of mass is the point where the total weight can be treated as acting: $x_{\mathrm{CM}} = \dfrac{\sum m_i x_i}{\sum m_i}$.

02Static equilibrium needs both $\sum F_y = 0$ and $\sum \tau = 0$. On this bench the support reactions and the weight must balance without any leftover turning tendency.

03Stability depends on support region, not on support point alone. If the vertical line through the combined centre of mass falls outside the support region, one required reaction would have to become negative and the plank tips.

04Changing support width does not move the centre of mass by itself. It changes the safety margin before tipping.

Live balance checks

Solve the exact state on screen.

Use the live plank state you are actually looking at. The first example builds the combined centre of mass from the current mass placement, and the second checks whether the current support region can actually hold that weight in static equilibrium.

Live valuesFollowing current parameters

For the current plank and cargo, where is the combined centre of mass and what total weight does the support need to hold?

m_{c}

Cargo mass

3 kg

x_{c}

Cargo position

0.8 m

m_{plank}

Plank mass

4 kg

1. Start from the weighted-average centre-of-mass rule

Use

x_{CM} = \frac{m _{plank} x _{plank} + m _{c} x _{c}}{m _{plank} + m _{c}}

with the plank midpoint at

x_{plank} = 0

2. Substitute the live masses and cargo position

With

m_{plank} = 4 kg

m_{c} = 3 kg

, and

x_{c} = 0.8 m

, the total mass is

7 kg

3. Read the combined weight location

That gives

x_{CM} = 0.34 m

. The total supported weight is then

W = M g = 68.6 N

Combined centre of mass and total weight

x_{CM} = 0.34 m, W = 68.6 N

The cargo shifts the combined centre of mass to the right, so the total weight now acts to the right of the plank midpoint.

Support-region checkpoint

Can you make the support region narrower without changing the total weight or the centre of mass, but still keep the plank stable?

Prediction prompt

Shrink the support width while keeping the same cargo and support center. Predict what changes first: the total weight, the centre of mass, or just the margin before tipping.

Check your reasoning

Yes, as long as the combined centre of mass still lands inside the support region. Narrowing the support region does not change total weight or the centre of mass by itself; it only reduces the margin before tipping.

Static stability is a geometry question built on top of force and torque balance. The total weight line can stay exactly where it was while the allowed support region around it shrinks.

Common misconception

If the total upward force equals the total downward force, the object must be stable.

Force balance alone is not enough. Equal and opposite forces can still leave a net torque that makes the object start rotating.

For a supported object, the combined centre of mass must also project inside the support region. Otherwise the reactions needed to keep the plank flat are not physically possible.

Quick test

Reasoning

Question 1 of 3

Answer from the same centre-of-mass and support-region logic the bench is using. These questions check whether you can separate mass distribution, support geometry, force balance, and torque balance cleanly.

The combined centre-of-mass line falls just to the right of the support region. Which statement is best defended?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

The simulation shows a horizontal plank with its own weight and one movable cargo block. A support region sits under the plank and can move left or right or become narrower or wider. Optional overlays can draw separate weight lines, one combined centre-of-mass line, the support region and stability margin, support reactions, and the torque arm from the support center to the combined centre of mass.

Changing cargo mass, cargo position, support center, or support width updates the plank drawing, the reaction readout, the response graphs, the prediction prompts, the worked examples, and the challenge checks from the same static-equilibrium model.

Graph summary

The support-torque graph is a response sweep against support center and shows where the torque about the support center crosses zero for the current cargo state. The support-reactions graph sweeps support center again and plots the left and right reactions required for static equilibrium, including when one would need to go negative.

The cargo-stability graph sweeps cargo position for the current support geometry and reports the margin to the nearest support edge. Positive margin means the combined centre-of-mass line is inside the support region, and negative margin means tipping.

Carry balance ideas forward

Keep moving through the concept map.

These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.

Mass-distribution responseMechanics

Static Equilibrium / Centre of Mass

See each variable before you move it.

Weight lines

Same centre of mass, different cargo

Rotational Mechanics

What the system is doing

Solve the exact state on screen.

For the current plank and cargo, where is the combined centre of mass and what total weight does the support need to hold?

The combined centre-of-mass line falls just to the right of the support region. Which statement is best defended?

Keep moving through the concept map.

Rotational Inertia / Moment of Inertia

Rolling Motion

Angular Momentum