A rigid body resting on a surface can remain supported only while the contact forces can balance both its weight and its gravitational torque. This leads to a simple geometric criterion for tipping stability.

The support base is the region on the supporting surface over which the body can transmit compressive contact force. For a rectangular block resting on one face, the support base is that face. For a body supported at several separated contact points, the relevant base is the region spanned by those contacts.

In a uniform gravitational field, draw the vertical line of action of the body's weight through its center of gravity.

• If that line intersects the interior of the support base, the support can provide a resultant contact force that balances the weight without requiring the body to rotate about an edge.
• If the line passes exactly through an edge of the support base, the body is at the impending-tipping condition.
• If the line falls outside the support base, the weight produces an overturning torque about the nearest support edge and the original resting configuration cannot remain in static equilibrium without some additional external constraint.

Why the edge becomes the pivot

As a body approaches tipping, contact pressure becomes concentrated toward one edge. At the threshold, the resultant normal force acts through that edge, so it produces no torque about the edge itself.

The tipping criterion can therefore be found by examining the gravitational torque about the prospective pivot edge. When the weight line passes through the edge, its moment arm is zero. Beyond that point, gravity produces torque in the overturning direction.

Rectangular block on an incline

Consider a uniform rectangular block whose base length along the slope is $b$ and whose height normal to its base is $h$. Its center of gravity is at the geometric center.

Let the supporting plane be inclined by angle $\theta$ above the horizontal. Measured from the center of the base, the vertical weight line intersects the base a distance

$$x=\frac{h}{2}\tan\theta$$

downhill from the center.

The downhill edge lies at distance $b/2$. Impending tip occurs when

$$\frac{h}{2}\tan\theta_c=\frac{b}{2}.$$

Therefore

$$\boxed{\tan\theta_c=\frac{b}{h}}.$$

A wide, low block has a larger critical angle than a narrow, tall block because its center of gravity can shift farther horizontally before the weight line leaves the support base.

Example

A uniform block has

$$b=0.60,\mathrm m,\qquad h=1.20,\mathrm m.$$

Then

$$\tan\theta_c=\frac{0.60}{1.20}=0.50,$$

so

$$\theta_c=\tan^{-1}(0.50)\approx26.6^\circ.$$

If the block maintains contact without sliding, it reaches the tipping threshold at about

$$\boxed{26.6^\circ}.$$

Tipping and sliding are different limits

A body on a rough incline may slide before it tips, or tip before it slides. Sliding is controlled by the available frictional contact force, whereas tipping is controlled by the line of action of the resultant forces relative to the support base.

For example, with a Coulomb-friction model, the sliding threshold can be compared with the tipping threshold. The two criteria should be calculated separately; neither automatically implies the other.

More general support regions

For a three-dimensional object with several contacts on a horizontal surface, a common stability test uses the vertical projection of the center of gravity onto the support region. A larger support region and a lower center of gravity generally provide greater geometric margin against tipping.

This criterion is widely useful for furniture, vehicles, robots, cranes, structures, and human balance: stability depends not only on total weight, but on where its line of action falls relative to the available support.