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# Center of Gravity

by Ron Kurtus (25 August 2010)

The center of gravity (CG) of an object is the balance point around which there are equal moment arms of length times weight. The object can act as if all its weight was concentrated at the CG.

You can find the center of gravity mathematically by taking the average or mean distribution of the weight of the object. You can also find the CG experimentally by either using a plumb line or finding the balance point.

Note: Some textbooks confuse center of gravity with center of mass (CM). Finding the center of gravity requires that the object is under the influence of gravity, while center of mass is the center of a mass distribution. Although CG is often at the same location as the CM, they are completely different concepts.(

See Gravitation and Center of Mass for more information.)

Applications include the fact that free rotation of an object is always around its center of gravity and that an object will tip over when the center of gravity lies outside the supporting base of the object. Also, the greatest force is applied through the center of gravity.

Questions you may have include:

- How can you calculate the center of gravity?
- What experiments can determine the center of gravity?
- What are some applications of the CG?

This lesson will answer those questions. Useful tool: Units Conversion

## Calculation of the center of gravity

Calculation of the center of gravity is based on the fact that a torque exerted by the weight of a system is the same as if it's total weight were located at the center of gravity. This point is the average or mean of the distribution of the moment arms of the object.

The calculation of the center of gravity of an object involves the summation of the weights times their separations from a starting point divided by the total weight of the object.

In the case of a highly irregular object, the weights can consist of individual particles or even atoms. Calculus is then used to integrate the product of these weights and the differential separations.

If the object is made up of regular parts, such as squares or circles, you can use the fact that each has a CG at its geometric center. This is seen in the illustration below:

Calculating CG of weights

The center of gravity in the illustration is at the following separation from the arbitrary zero-point:

CG = (aM + bN + cP)/(M + N + P)

For example, if:

**a =**1 ft**b =**4 ft**c =**8 ft**M =**1 lb**N =**2 lb**P =**4 lb

CG =(1*1 + 4*2 + 8*4)/(1 + 2 + 4)

CG =41/7

CG =5.9 ft from the zero point

The approximate CG is shown in the illustration.

## Finding CG through experimentation

The center of gravity for objects with regular shapes, such as squares, cubes, circles and spheres, is at their geometric centers. For other shapes or configurations, you need to either use experimentation or calculation. It is easier to determine the center of gravity of many objects by experimentation than by calculation.

### Plumb line technique

When an object is suspended so that it can move freely, its center of gravity is always directly below the point of suspension.

You can find the center of gravity in an object experimentally by hanging it from several points and using a plumb line to mark the vertical line. The intersection of two or more vertical lines from the plumb line is the center of gravity for the object.

Measuring CG with plumb line

This procedure is relatively easy for a flat object. However, it can be more difficult if the object has some shape in the three dimensions.

### Balance point

For some objects, you can find its balance point through experimentation. For the object in the illustration above, the balance point could easily be found. However, it would be difficult to find the balance point for an object with curved surfaces.

Weight balances on sharp edge

## Applications

A lower center of gravity helps prevent an object—such as an automobile—from tipping over. Objects will spin about their CG. Also, the center of gravity provides the greatest impact in a collision.

### Tipping point

When you tilt an object on an edge, it will tip over only when the center of gravity lies outside the supporting base of the object.

Object tips over when CG passes pivot point

Objects that are heavier toward the bottom have a lower center of gravity and are thus more difficult to tip over. Automobiles and trucks have a lower center of gravity to improve their stability.

### CG below balance point

An interesting application is when the center of gravity is below the balance or pivot point. The object will readily balance and may oscillate or swing back and forth in that position.

CG below balance point

There are a number of novelty items and toys that use this principle.

### Spins about center of gravity

If you throw an object in the air with a spin on it, the object will rotate around its center of gravity as it follows its path.

A ball has its center of gravity at its center, so it will simply spin as it is thrown in the air. The ball will follow a parabolic path until it hits the ground.

Likewise, if you threw a baseball bat in the air, it too would rotate about its center of axis, and that axis would follow a parabolic path similar to that of the ball.

Bat follows parabolic path as it spins

### Hitting a baseball

The center of gravity is a point where all of the weight of the object is concentrated.

CG is best location to hit the ball

When you swing a baseball bat, the best location to hit the ball is at the bat's center of gravity. At that point, the maximum force is applied to the ball for a given swing.

## Summary

The center of gravity (CG) is where all of the weight of an object appears to be concentrated. This point is the average distribution of the weight of the object. The center of gravity of an object can also be found experimentally.

Free rotation of an object is always around its center of gravity. An object will tip over when the CG lies outside the object's support. The greatest force is applied through he center of gravity.

Observe to learn more

## Resources and references

### Websites

**Center of gravity and Rotational variables** - Boston University Physics

**Center of gravity of an aircraft** - Wikipedia

**Finding the Center of Gravity** - NASA

**Center of Gravity experiment** - Exploratorium

### Books

**Top-rated books on Simple Gravity Science**

**Top-rated books on Advanced Gravity Physics**

## Questions and comments

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## Where are you now?

## Center of Gravity