M022 · Rigid-body statics

Center of Mass & Tipping

When an object could tip over any of several edges, which one goes first? The edge that needs the smallest lift of the mass center. Tipping about an edge raises the mass center until it sits directly above that edge, and only then does the body fall over. The edge with the smallest rise is the lowest barrier, so the body tips over it first.

Rearrange a rigid assembly of boxes and find its center of mass by hanging it from different points. Then compare how far the center sits inside the base with the energy barrier that gravity puts in the way of 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 $\text{—}$

Place the loads and test the balance

When an object could tip over any of several edges, which one goes first?

Gravity and support forcesOne fixed force scale per trialLeft box mass

m_L

8.00\,kg

Right box mass

m_R

8.00\,kg

Payload mass

m_c

12.00\,kg

Payload lateral position

x_c

0.20\,m

Payload depth position

z_c

0.00\,m

Mast length

H

1.00\,m

Base width

w

1.20\,m

Base depth

d

0.90\,m

Outward release tilt

\phi_0

6.00\,{}^\circ

Suspension station: four eyes

j

0.00\,

Angular noise bound

\epsilon

0.00\,{}^\circ

The rigid boxes, the eight-kilogram base and the two-kilogram mast are laid out in one geometry measured in metres. The connecting brackets have negligible mass. The floor is level, and static friction is large enough that nothing slides. The structure tips about the nearest support edge. The release tilt is capped just before a box corner would touch the floor. Each trial starts at rest from the chosen outward tilt and ends when the base lands back on the floor or a box first touches the floor somewhere new. Bouncing, rolling and structural failure are not modeled.

Hanging the object is a separate virtual trial. It uses detachable, massless eyes and a bearing that lets the object swing in one vertical plane only. Two angle readings in the side-to-side plane and two in the depth plane locate all three coordinates of the center of mass. Orange lines are noisy sensor readings. Teal markers show the reference computed independently from the mass model, and blue markers show the fit. The raw CSV holds only the angles and the known eye geometry. Changing the geometry, masses or noise clears the record; switching the eye or changing the release tilt keeps it.

The boxes are uniform, the floor is horizontal, nothing slides, and the rigid brackets have negligible mass. After release the body turns about one edge until it falls back onto its base or a box first touches the floor somewhere new; motion after that contact is not modeled. Hanging trials use a separate fixture whose bearing turns in one plane only.

Start with the essentials

Focus question: When an object could tip over any of several edges, which one goes first?
One-sentence intuition: While the center of mass lies over the base, tilting the body outward first lifts the center of mass.

Core mathematical model

The center of mass is a mass-weighted average

\mathbf r_C=\frac{\sum_i m_i\mathbf r_i}{\sum_i m_i}

The sum uses the center of every uniform box, the fixed base and mast included. The brackets have negligible mass, so they are left out.

Staying over the base

|x_C|\leq\frac w2,\qquad |z_C|\leq\frac d2

All four edges count. The Lab tips the body about the edge with the smallest signed margin.

Potential energy while tilting about an edge

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

Height is measured from the floor. The margin is positive when the center lies inside the chosen edge, and a positive angle tips the body outward.

The energy barrier to tipping

\phi_* =\arctan\frac\ell h,\qquad\Delta U=Mg\left(\sqrt{h^2+\ell^2}-h\right)

With a positive margin, the potential energy peaks when the center passes directly above the edge. With zero or negative margin there is no barrier to tipping outward.

Hanging angles locate the center

(q_C-a_q)\cos\alpha+(h_C-a_h)\sin\alpha=0

Each hanging trial, with a known eye and a measured angle, gives one line that the center must lie on. Trials in two vertical planes together fix all three coordinates by least squares.

Common difficulties

Height changes how easily it tips

Typical misconceptionTwo bodies with the same base and the same center position seen from above need the same energy to tip.

Better mental modelWith the same mass and margin, a higher center of mass lowers the energy barrier.

Hanging in one plane cannot find depth

Typical misconceptionTwo plumb lines in the side-to-side plane are enough to find all three coordinates.

Better mental modelTo find depth, you need trials in a second vertical plane.

Floor contact changes the motion

Typical misconceptionThe body keeps turning about the edge, right through the floor.

Better mental modelThe model stops when a box first touches the floor somewhere new and counts the energy absorbed there.

Run the experiment

01
Move mass in two directions
Change Payload lateral position and Payload depth position and watch the base footprint.
What to observe: The nearest edge can change, even when the body looks stable from the side.
02
Measure the center of mass
Hang the body from each eye, or press Measure all four eyes, and compare the fitted center with the model center.
What to observe: The measured lines cross near the model center; noise leaves a residual you can measure.
03
Tip past the peak
Choose Tall and vulnerable and release it from tilts below and above the barrier angle.
What to observe: Released at rest below the peak, the body falls back onto its base; above the peak it tips outward.
04
Start already unstable
Choose Lateral overhang or Depth overhang and release with zero tilt.
What to observe: With a negative margin, gravity turns the body outward with no push needed to start it.