2.5.7.2. Center of Gravity

Table of Contents

Meaning and Physical Idea

The center of gravity is the point where the total gravitational effect on an object can be treated as if all the object's weight acted there. This idea is especially useful in static equilibrium, because gravity usually acts on every tiny part of an object, but for torque calculations we often replace all those small forces with one single force, the total weight.

If an object is in a uniform gravitational field, like near Earth's surface for ordinary everyday objects, the center of gravity is at the same location as the center of mass. In that common case, the weight $W = mg$ acts downward through that point.

For most problems near Earth's surface, the center of gravity and the center of mass are treated as the same point.

Why It Matters in Equilibrium

In static equilibrium, both the net force and the net torque must be zero. The center of gravity matters because the object's weight creates torque about supports, pivots, or contact points. Knowing where the center of gravity is tells us how strongly gravity tends to rotate the object.

If the line of action of the weight passes through a pivot, gravity produces no torque about that pivot. If it does not pass through the pivot, the torque is

$$
\tau = r_\perp W
$$

where $r_\perp$ is the perpendicular distance from the pivot to the vertical line through the center of gravity.

To find the torque due to weight, use the perpendicular distance from the pivot to the line of action of the weight, not just the straight line distance to the point.

Center of Gravity in a Uniform Gravitational Field

For an extended object made of many small masses $m_i$, each part has weight $m_i g$. If $g$ is the same everywhere in the object, the center of gravity has coordinates

$$
x_{\text{cg}} = \frac{\sum m_i x_i}{\sum m_i}, \qquad
y_{\text{cg}} = \frac{\sum m_i y_i}{\sum m_i}
$$

This is the same formula used for the center of mass.

For a continuous body,

$$
x_{\text{cg}} = \frac{1}{M}\int x \, dm, \qquad
y_{\text{cg}} = \frac{1}{M}\int y \, dm
$$

where $M$ is the total mass.

Common Locations for Simple Objects

For symmetric objects of uniform density, the center of gravity lies at the geometric center. For irregular objects, it may not be obvious and must be found from symmetry, calculation, or experiment.

Object	Center of gravity location
Uniform rod	Midpoint
Uniform rectangular plate	Geometric center
Uniform solid sphere	Geometric center
Uniform circular disk	Geometric center
Right triangle plate of uniform density	At the centroid, one third of the way from each side toward the opposite vertex

Suspended Objects and Plumb Lines

A practical way to find the center of gravity of a flat irregular object is to suspend it from one point and let it hang freely. The center of gravity must lie somewhere on the vertical line below the suspension point. If the object is then suspended from another point, a second vertical line is found. The intersection of the two lines gives the center of gravity.

This works because, when hanging at rest, the torque due to gravity must be zero. That happens only when the center of gravity lies directly below the support point.

Rendering scientific sketch…

Finding the center of gravity of an irregular plate

Center of Gravity and Support

An object standing on a surface remains in equilibrium only if the vertical line through its center of gravity falls inside its base of support. If that line falls outside the base, the object tips over.

This gives an important physical rule. A lower center of gravity usually makes an object harder to tip, because a larger tilt is needed before the line of action of the weight moves outside the support region.

Rendering scientific sketch…

Center of gravity and tipping

Difference Between Center of Gravity and Center of Mass

Although these two ideas are often identical in elementary mechanics, they are not always exactly the same. The center of mass depends only on how mass is distributed. The center of gravity depends on how gravitational force is distributed.

If the gravitational field changes noticeably from one part of the object to another, the center of gravity can differ from the center of mass. For example, for a very large object in a nonuniform gravitational field, different parts of the object experience slightly different gravitational forces.

Center of mass depends on mass distribution.
Center of gravity depends on weight distribution.
They are equal only when the gravitational field is uniform across the object.

Using Center of Gravity in Torque Problems

When solving equilibrium problems, the weight of the whole object is often applied at the center of gravity. This simplifies the calculation of torques from distributed mass.

For example, for a uniform horizontal beam of length $L$ and mass $M$, supported at one end, the beam's weight acts at its center of gravity, which is at $L/2$ from the end. The torque due to the beam's own weight about the support is

$$
\tau = Mg \frac{L}{2}
$$

if the beam is horizontal.

This replacement is valid because the distributed gravitational forces produce the same net torque as a single force $Mg$ acting at the center of gravity.

Summary

The center of gravity is the effective point where the weight of an object acts. In ordinary mechanics near Earth's surface, it is usually the same as the center of mass. It is essential in static equilibrium because it determines the torque due to gravity and helps predict whether an object will remain balanced or tip over.

Key facts:
$$
W = mg
$$
For uniform gravity,
$$
\text{center of gravity} = \text{center of mass}
$$
An object is stable on a surface only if the vertical line through its center of gravity stays within the base of support.

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