** Next:** Relation Between Torque and
**Up:** Rotational Equilibrium and Dynamics
** Previous:** Another Condition for Equilibrium

**Definition:** Center of Gravity

The **center of gravity** of a collection of masses is the point
where all the weight of the object can be considered to be
concentrated. If
(*x*_{cg},*y*_{cg}) are the coordinates of the
centre of gravity
of a collection of point masses *m*_{1} , *m*_{2} ,
etc, located at coordinates ( *x*_{1},*y*_{1} ), ( *x*_{2},*y*_{2} ),
respectively,
then:

(m_{1} + m_{2} + ..)gx_{cg}
| = |
m_{1}gx_{1} + m_{2}gx_{2} +...
| (5) |

(m_{1} + m_{2} + ..)gy_{cg}
| = |
m_{1}gy_{1} + m_{2}gy_{2} +...
| (6) |

Solving for the *x* -coordinate of the center of
gravity:

x_{cg} = .
| (7) |

y_{cg} = .
| (8) |

**Note: **

- Eqs(8.6,8.6) also imply that the torque due to
gravity about the
center of gravity is zero. The force of gravity,
*F*=*mg*, acts through the center of gravity so there is no moment arm and therefore no torque due to gravity about the center of gravity. - An extended body can be treated in exactly the same way simply by considering it to be made up of a collection of smaller masses ``stuck together".
- The centre of gravity is not necessarily inside the object.
- In problems involving extended bodies and gravity, one can impose the equilibrium condition by assuming that the entire weight of the bodies acts through the centre of gravity.
- In order to balance an object against gravity with a single force, that force must lie in a vertical line that runs through the centre of gravity.
- For objects with symmetry, the center of gravity is always located along the axis of symmetry.
- The center of gravity of an extended body or system of masses is also distinguished by the fact that it will remain at rest or moving at constant velocity unless the body is acted on by a net external force.

10/9/1997