Intermediate physics / by W. Watson.

  • Watson, W. (William), 1868-1919.
Date:
1912
Catalogue details

Licence: In copyright

Credit: Intermediate physics / by W. Watson. Source: Wellcome Collection.

Provider: This material has been provided by UCL Library Services. The original may be consulted at UCL (University College London)

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    the centre of gravity of the whole triangle must lie somewhere along ad. In the same way if e is the mid point of the side ab, the centre of gravity must lie somewhere on the line eg. Hence g must be the centre of gravity. That is, the centre of gravity of a triangle is at the intersection of the lines joining the angular points to the mid points of the opposite sides. If in place of the triangle we had three equal masses m placed at the points A, B, and c, then the centre of gravity of B and c would be at d, and to find the centre of gravity of the three masses we may replace the masses at b and o by a mass of 2m at d, and then find the centre of gravity of this mass and the mass m at A. Thus the centre of gravity of the three masses must lie on the line ad. In the same way it is evident that the centre of gravity must lie on the line ce. Hence the centre of gravity of the three equal masses must lie at g, and must coincide with the centre of gravity of the triangle abc. Since G is the centre of gravity of a mass of %n at d and a mass of m at A, we have 2m x dg = m x ag, or AG = 2dg. That is the point G is at a third of the way up the line joining the mid point of a side to the opposite angle. In an exactly similar way it can be proved that the centre of gravity of a pyramid on a triangular base is on the line joining the apex to the centre of gravity of the base, and at a quarter of the way up this line. It is then an easy extension to show that the same rule gives the centre of gravity of a pyramid on any other base, and hence of a cone. If we know the centre of gravities and masses of the parts into which a body may be divided, it is easy to calculate the centre of gravity of the whole body, or if we know the centre of gravity of a given body and the centre of gravity of a portion, we can at once calculate the position of the centre of gravity of the remainder. The principle on which such calculation is founded is that if we take any axis, the product of the mass of the whole body into the distance of the centre of gravity from this axis is equal to the sum of the products obtained by multiplying the mass of each part into the distance of its centre of gravity from the axis. This result at once follows if we consider that the moment of the weight of the whole body about the axis must be equal to the sum of the moments of the weights of the parts about the same axis. As an example of the application of this method, let us calculate the position of the centre of gravity of a circular lamina from which a hole has been punched as shown in Fig. 25.