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Sandbox Physics

M022 · Rigid-body statics

Center of Mass & Tipping

Rearrange a rigid assembly, locate its mass center with virtual suspension trials and compare support geometry with the gravitational tipping barrier.

Interactive modelCenter of Mass & Tipping
Total mass—\text{—}
Model mass center: lateral—\text{—}
Model mass center: height—\text{—}
Model mass center: depth—\text{—}
Nearest edge margin—\text{—}
Gravitational energy barrier—\text{—}
Barrier tilt—\text{—}
Initial outward edge torque—\text{—}
First new floor contact—\text{—}
Plumb fit: lateral—\text{—}
Plumb fit: height—\text{—}
Plumb fit: depth—\text{—}
Equilibrium status—\text{—}
Rotation inertia about selected axis—\text{—}
Recorded observations—\text{—}
Fit residual RMS—\text{—}
Current rotation—\text{—}
Angular speed—\text{—}
Mechanical energy balance error—\text{—}
Energy absorbed at terminal stop—\text{—}

Physics tutorial

A mass center defines both stability and a barrier

BackgroundA composite body can tip although every component is individually symmetric.

Why it mattersMIT extended-object mass centers and OpenStax stability connect hanging geometry with support contact.

Start with the essentials

Focus question
Which edge gives way first?
One-sentence intuition
Inside the support polygon, outward rotation first raises the mass center.

Core mathematical model

Mass-weighted center

rC=∑imiri∑imi\mathbf r_C=\frac{\sum_i m_i\mathbf r_i}{\sum_i m_i}

Uniform box centers include the fixed base and mast. Negligible-mass brackets do not enter the mass sum.

Support geometry

∣xC∣≤w2,∣zC∣≤d2|x_C|\leq\frac w2,\qquad |z_C|\leq\frac d2

All four edge margins matter. The smallest signed margin selects the trial rotation edge.

Potential about an edge

U(ϕ)=Mg[hcos⁡ϕ+ℓsin⁡ϕ]U(\phi)=Mg\left[h\cos\phi+\ell\sin\phi\right]

Height is measured from the floor, margin is positive inside the selected edge, and positive angle tips outward.

The gravitational barrier

ϕ∗=arctan⁡ℓh,ΔU=Mg(h2+ℓ2−h)\phi_* =\arctan\frac\ell h,\qquad\Delta U=Mg\left(\sqrt{h^2+\ell^2}-h\right)

For positive margin, the center passes vertically above the edge at maximum potential. Nonpositive margin has no outward barrier.

Angular plumb observations

(qC−aq)cos⁡α+(hC−ah)sin⁡α=0(q_C-a_q)\cos\alpha+(h_C-a_h)\sin\alpha=0

Each known eye and measured planar angle defines one line constraint. Combining two vertical planes locates three coordinates by least squares.

Common difficulties

Height changes susceptibility

Typical misconceptionEqual footprint and projected center imply equal energy barriers.

Better mental modelFor equal mass and margin, a taller center reduces the gravitational barrier.

One photograph misses depth

Typical misconceptionTwo lateral-plane plumb lines locate all three coordinates.

Better mental modelDepth requires a second vertical-plane trial.

Contact changes the dynamics

Typical misconceptionAn edge-rotation solution continues through the floor.

Better mental modelThis model stops at the first new box contact and accounts for absorbed energy.

Run the experiment

  1. 01

    Move mass in two directions

    Move the payload laterally and in depth while observing the footprint.

    What to observe: The nearest edge can change even when a side view looks stable.
  2. 02

    Measure the mass center

    Suspend from each eye or measure all four; compare fitted and model centers.

    What to observe: Raw lines intersect near the model center; noise adds a measurable residual.
  3. 03

    Cross the barrier

    Try a tall assembly with release tilts below and above the barrier angle.

    What to observe: Starting at rest below the peak returns to the base; above it tips outward.
  4. 04

    Start already unstable

    Choose a lateral or depth overhang and release with zero tilt.

    What to observe: A negative margin gives outward gravity torque without a starting kick.