To Locate Center of Gravity

On discussing Newton’s law of universal gravitation, We have  seen a body is attracted to the earth by the force of gravity. This statement, however, says nothing about the point of application of the force, so we shall now discuss this ‘as I promised in our previous post’.

Definition of Center of gravity ‘mass’:

Any particular body, a stone, for example, may be regarded as being made up of a very large number of tiny equal particles, each of with is pulled towards the earth. The earth’s pull on the stone thus consists of a very large number of equal parallel forces. These will have a resultant which is equal to the total force of gravity on the stone, and it will act through a point G called the center of gravity.

Center of gravity of a body is defined as the point of application of the resultant force due to the earth’s attraction on it.

You can do it at home, with a ruler or pencil

A ruler can be made to balance on a finger-tip if the finger is placed immediately below the center of gravity. Under these conditions the ruler is in equilibrium under the action of two forces; the force of gravity acting vertically downwards and the equal and opposite reaction of the finger to the weight of the ruler acting upwards.

Should the center of gravity not be exactly above the finger, the force of gravity will have a turning moment about the finger and cause the ruler to topple over.

Note carefully the distinction we have made above between the force of gravity which acts on the ruler and the weight of the ruler which acts on the finger.

The essential point to grasp is that three forces all equal in magnitude are involved in the above discussion, namely:

1- the force of gravity acting on the ruler;

2- the weight of the ruler acting on the finger;

3- the reaction of the finger acting on the ruler.

While in this particular case, as indeed, in the case of all bodies at rest, the force of gravity on a body is equal to its weight on its support, this is not so for a body resting on an accelerating support.

The principle of the plumb line

When a body is freely suspended by a string and is at rest the force of gravity on it, acting vertically downwards, is balanced by an equal and opposite force or tension in the string.

Being flexible, the string therefore sets in a vertical direction. This is the principle of the plumb line, which consists of small leaden bob supported by a thin cord. From the earliest times, plumb lines and plumb rules have been used by builders for the purpose of testing the uprightness of walls, pillars, and so on during the course of construction.

To locate center of gravity by a balancing method

The center of gravity of a long thin object such as a ruler or billiard cue may be found approximately simply by balancing it on a straight-edge. The same method could also be used for a thin sheet or lamina of cardboard or metal, except that in this case it is necessary to balance in two positions.

A palette is balanced on the edge of a straight ruler in two directions AB and CD, and the lines of balance are marked with pencil lines.

Since the center of gravity, G, is situated on both lines, it must actually lie at their point of intersection. This may be checked by balancing once more in a third position EF.

The edge of the ruler should now pass through the intersection of the two previous lines.

To locate center of gravity by means of a plumb line

One of the best ways of finding the center of gravity of a body is by use of a plumb line. Suppose, for examples, we wish to find the center of gravity of an irregularly shaped piece of cardboard.

First of all, three small holes are made at well-spaced intervals round the edge of the card. A stout pin is then put through one of the holes and held firmly by clamp and stand so that the card can swing freely on it.

The card will come to rest with its center of gravity, G, vertically below the point of support.

The vertical line through the support can now be located by means of a plumb line.

A suitable plumb line is made from a length of cotton with a loop at one end and a weight tied at the other.

This is hung from the pin and the position of the cotton marked on the card by two small pencil crosses. These crosses are joined by a pencil line.

The experiment is repeated with the card suspended by one of the other holes. Since the center of gravity lies on both of the lines drawn on the card, it must be situated at their point of intersection.

As a check, the card is suspended by the third hole. It should be found that, within the limits of experimental error, the plumb line should pass through the intersection of the two lines.

The above method can be used to show that the center of gravity of a triangular lamina lies at the point of intersection of its medians. ( A median of a triangle is a line through one of the corners which bisects the opposite side).

Center of gravity of a stool or tripod

Sometimes the center of gravity of a body is not situated in the actual material of the body, but may be at a point in the air near by.

Examples are an iron tripod or a laboratory stool. The plumb line method can still be employed, but requires some more skill and ingenuity to carry out than in the case of a lamina.

It is necessary to suspend the stool or tripod in such a way that the plumb line positions can be located by lengths of cotton fixed with soft wax.

To weigh by means of a meter rule

A meter rule with a small hole drilled near its center can be used in conjunction with a single known weight to measure the weight of an object.

Having balanced the rule on a knitting needle, the known weight and the object to be weighed are suspended from the rule by cotton loops, one on either side of the pivot.

Their distances from the pivot are then adjusted until the rule once more balances horizontally.

Maximum accuracy will be obtained if these distances are made as large as possible. Since it is inevitable that a small error will be made when the distances of the weights from the pivot are measured, the percentage error in the final result will be less if large distances are used rather than small ones.

Simple example

Suppose that, in particular experiment, a 100 Newton weight placed 50 cm from the pivot balances an object of unknown weight W placed 20 cm from the pivot on the opposite side of the rule.

Applying the principle of moments

W × 20 = 100 × 50

therefore

W = 100 × 50 / 20 = 250 Newton.

To measure the weight of a meter rule by using a single  known weight

A meter rule is supported in a loop of thread hung from a clamp and stand and adjusted until it balances horizontally.

The loop will then be at the center of gravity, G, of the rule. If the rule is not of  uniform thickness ‘few rules are’ the center of gravity will not be at the 50 cm mark.

However, this will not affect the accuracy of the experiment so long as the correct position is noted and later used in the calculation.

Let’s suppose that a 100 N weight is now hung from the rule by thread at a point near one end. The position of the rule in the supporting loop is then adjusted until it again balances horizontally.

The rule is now in equilibrium under the action of two moments about the supporting loop, namely, that of its own force of gravity W acting through the center of gravity and that of the 100 N weight on the opposite side.

If d1 and d2 respectively are the distances of G and the 100 N weight from the loop, then equating moments we have

W × d1 = 100 × d2

whence the weight of the rule (equal to the force of gravity on it) is given by

W = (100 × d2 / d1) N.

Steelyard principle

The steelyard is a weighing device which has been used since Roman times.

It is particularly suitable for weighing objects which can be suspended from a hook, e.g., sacks of farm produce and carcasses of meat.

The steelyard consists of a graduated steel rod pivoted near one end and balanced by a heavy piece of metal. The object to be weighed is suspended from a hook.

The moment of the object about the pivot is balanced by that of a small rider weight which slides along the arm of the steelyard. This arm is calibrated to read the weight being measured directly in appropriate units.

Some types of platform weighing machines employ the steelyard principle. In these, the object is placed on a metal platform and its weight is transmitted to a balancing arm by a system of levers.

Coming up: Stable, unstable and neutral equilibrium “It will be a short article”. Thanks for visiting.