Physics · Mechanics

Static Equilibrium / Centre of Mass

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Wrap-up

What you learned

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Read next: Rotational Inertia / Moment of InertiaMass-distribution response

Key takeaway

The combined centre of mass is the single line where the total weight can be treated as acting.
Static equilibrium needs both vertical force balance and torque balance; equal total up and down forces are not enough by themselves.
A support region can hold the plank only while the combined centre-of-mass line falls between its edges.
Narrowing or shifting the support changes the stability margin, not the centre of mass by itself.

Common misconception

Force balance alone does not guarantee stability; a leftover torque or an impossible negative reaction means the plank tips.

Force balance alone is not enough. Equal upward and downward forces can still leave a net torque that starts rotation.

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Reference and support

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Equation snapshotBalance and support window · Vertical force balanceShow 3 equations

Keep the weighted-average centre of mass, force balance, and support-region rule visible while you move the load.

Balance and support window
$x_{CM} = \frac{\sum m _{i} x _{i}}{\sum m _{i}}$
Treat the total weight as acting through this weighted-average position.
Vertical force balance
$R_{L} + R_{R} = W$
The support reactions must add up to the total weight.
Support-region stability rule
$x_{L} \leq x_{CM} \leq x_{R}$
The vertical line through the combined centre of mass must stay between the support edges if both reactions are to remain upward.

Worked examples

Solve the live balance

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Frozen walkthrough

Step through the frozen example

Frozen walkthrough

Read the current plank state rather than an abstract diagram. First find where the combined weight acts and how large it is. Then check whether the current support region can provide physically possible reactions and still leave a positive margin before tipping.

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Example 1 of 2

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?

m_{c}

Cargo mass

3 kg

x_{c}

Cargo position

0.8 m

m_{plank}

Plank mass

4 kg

1. Write the weighted-average 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. Insert 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 line and total weight

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.

Test the support-region rule

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

Make a prediction before you reveal the next step.

Narrow the support while keeping the same cargo and support centre. Predict what changes: the total weight, the centre of mass, or only the margin to the nearest edge.

Check your reasoning against the live bench.

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

Static stability depends on geometry as well as force balance. The total weight line can stay in exactly the same place while the allowed support region around it becomes smaller.

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Keep this idea moving

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Mass-distribution responseMechanics

Rotational Inertia / Moment of Inertia

Keep the same total mass and torque, then slide equal masses inward or outward to see why moment of inertia makes some rotors much harder to spin up than others.

Support-to-contact bridgeMechanics

Rolling Motion

Roll a sphere, cylinder, hoop, or custom mass distribution down one incline and see how rolling without slipping ties translation, rotation, and rotational inertia to the same honest run.

Carry rotational ideas forwardMechanics

Angular Momentum

Treat angular momentum as rotational momentum on one compact rotor where mass radius and spin rate stay tied to the same readouts, response maps, and same-L conservation story.

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Rotational Mechanics

Static Equilibrium / Centre of Mass appears later in this track, so it is cleaner to start from the beginning first.

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