Centre of mass.md

Centre of mass

Centre of gravity, centre of mass and centroid

Centre of mass A point representing the mean position of matter in a system

Centre of gravity A point from which the entire weight of a system can be considered to act

In a uniform gravitational field, the centre of gravity is the same as the centre of mass.
Further, for small systems the difference between the centre of mass and centre of gravity of the system will be negligible.

Centroid The centre of mass of a geometric object with uniform density

Centre of mass of a system of point masses

Suppose that there is a a set of particles, $$m_1, m_2,..., m_n$$ placed at points $$(x_1, y_1), (x_2, y_2),..., (x_n, y_n)$$ in a plane.

If a uniform gravitational field acts perpendicular to the plane, the resultant weight is the sum of the weights of the individual particles, and acts through the centre of mass $$G(\overline{X}, \overline{Y})$$

Taking moments about the y-axis
$$(m_1g + m_2g _ ... m_ng)\overline{X} = m_1gx_1 + m_2gx_2 + ... + m_ngx_n $$

Hence
$$\overline{X} = \frac{m_1gx_1 + m_2gx_2 + ... + m_ngx_n}{m_1g + m_2g + ... + m_ng}$$

Cancelling by $$g$$

$$\overline{X} = \frac{m_1x_1 + m_2x_2 + ... + m_nx_n}{m_1 + m_2 + ... + m_n}$$

This can be expressed more cleanly with sigma notation

$$\overline{X} = \frac{\sum\limits_{i=1}^n m_i x_i}{\sum\limits _{i=1}^n m_i}$$

Resolving in each direction, and extending to three dimensions

$$ \begin{pmatrix} \overline{X} \ \overline{Y} \ \overline{Z} \end{pmatrix} = \frac{\sum\limits_{i=1}^n \begin{pmatrix} x_i \ y_i \ z_i \end{pmatrix}}{\sum\limits_{i=1}^{n} m_i} $$

Centre of mass of simple shapes

Line

A line of uniform density has a centre of mass situated at the mid point of the line

Rectangle

For a uniform rectangular lamina , the centre of mass is at the intersection of its diagonals.

Circle

The centre of mass of a uniform circular lamina is at the centre of the circle

Triangle

The centre of mass of a uniform triangular lamina is the point at which the medians of the triangle. The point divides each median in the ratio 2:1.

Semicircle

The centre of mass of a uniform circular lamina of radius $$r$$ lies on the line of symmetry at a distance $$h$$ from the diameter.

$$h = \frac{4r}{3\pi}$$

Centre of mass of composite shapes

Each of the components of a composite shape can be regarded as a point mass located at the centre of mass of the respective component, The centre of mass of the overall shape can then be found from this system of point masses.

Centres of mass of non uniform shapes

Shapes may be given with other shapes cut out of them.

The centre of mass of the shape can be found by taking moments around a point and equating the moment of the entire shape (Without the cutout) to the sum of the moments of the shape and the cutout around the same point.

Suspended shapes

When a lamina is suspended, the vertical from the point at which the lamina is suspended passes through the centre of mass of the the lamina. This can be used to find the angle between a part of the shape and the vertical.