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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.

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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

Why it behaves this way

Explanation

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.

Frozen walkthrough

Step through the frozen example

Frozen walkthrough
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.

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View plans
Frozen valuesUsing frozen parameters

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

Cargo mass

3 kg

Cargo position

0.8 m

Plank mass

4 kg

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

Use with the plank midpoint at .

2. Substitute the live masses and cargo position

With , , and , the total mass is .

3. Read the combined weight location

That gives . The total supported weight is then .

Combined centre of mass and total weight

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?

Make a prediction before you reveal the next step.

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 against the live bench.

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?

Use the live bench to test the result before moving on.

Accessibility

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.