Any object can be assumed to be composed of a very large number of point masses. Gravitational forces on all the particles can be considered to be parallel, all of which combine to produce a resultant force, the weight of the body. Centre of gravity of an object is that point where the total weight can be imagined to act, *i.e.* a single force (weight) acting at the centre of gravity produce exactly the same result as having gravity act on all the point masses constituting the body.

Position of centre of gravity can be calculated from the expression

If gravitational field is uniform over the body, *g* cancels out in the above expressions, and the expression reduced to that for centre of mass of the body.

If *g* is constant, the centre of gravity coincides with the centre of mass.

- A rigid body has a fixed centre of mass independent of the gravitational field; the location of the C.G. however depends on the gravitational field.
- Only if there is variation in acceleration due to gravity CM and CG differ e.g. Mount Everest.
- Consider a system of two particles of masses
*m*1 and*m*2 separated by a distance*r*. Suppose they start to move towards each other due to their mutual attraction (attractive force may be electrical, gravitational, etc.). Since particles starts from rest, and**<**

It follows

… (i)… (ii)

… (iii)

… (iv)

We can also deal this equation in two separate direction.

*X *– Direction

If and

**Þ**

**Y – Direction **

**If
and
**

Þ

Let us find the centre of mass of a uniform semi-circular ring of radius R and mass M.

Consider the centre of the ring as origin. Consider a differential element of length *dl* of the ring whose radius vector makes an angle **q** with the x-axis. If the angle subtended by the length d*l* is** d****q**at the centre then,** d l = Rd**

Let **l** be the mass per unit length.

Then mass of this element is** dm = ****l****Rd****q**

Now let us try to find the centre of mass of solid hemisphere.

Consider an elementary disc at an angle**q** and of width **d****q**mass of the disk

or *y = R* cos **q**

**1. **

**2. **

**3. **

**4. **

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