Center Of Mass - Practice Questions & MCQ

1. Definition-

• Centre of mass of a body is defined as a single point at which the whole mass of the body or system is imagined to be concentrated and all external forces are applied there.

• It is the point where if a force is applied it moves in the direction of the force without rotating.

2. x, y, & z coordinates of the centre of mass

• For a system of N discrete particles

$
\begin{aligned}
x_{c m} & =\frac{m_1 x_1+m_2 x_2 \ldots \ldots}{m_1+m_2 \ldots \ldots} \\
y_{c m} & =\frac{m_{1 y_1}+m_2 y_2+m_3 y_3 \ldots \ldots \ldots}{m_1+m_2+m_3 \ldots \ldots} \\
z_{c m} & =\frac{m_1 z_1+m_2 z_2+m_3 z_3 \ldots \ldots \ldots}{m_1+m_2+m_3 \ldots \ldots}
\end{aligned}
$

Where $m_{1,} m_2$ $\qquad$ are mass of each particle and $x_1, x_2$ $\qquad$ $y_1, y_2$ $\qquad$ $z_1, z_2$ are respectively x , $y, \& z$ coordinates of particles.
- It is the unique point where the weighted relative position of the distributed mass sums to zero.
- Centre of Mass of a continuous Distribution

$
x_{c m}=\frac{\int x d m}{\int d m}, \quad y_{c m}=\frac{\int y d m}{\int d m}, z_{c m}=\frac{\int z d m}{\int d m}
$

Where $d m$ is mass of small element. $x, y, z$ are the coordinates of dm part.

3. Important points about position of centre of mass 

• Its position is independent of the coordinate system chosen.

• Its position depends upon the shape of the body and distribution of mass.

And depending on this it may lies inside of the body as well as outside the body.

•  For  symmetrical bodies having the homogenous distribution of mass ,the centre of mass coincides with the geometrical centre of the body.

• It changes its position only under the translatory motion whereas there is no effect on its position because of rotatory motion of the body.

4. Centre of gravity-

• Centre of gravity of a body is a point, through which the resultant of all the forces experienced by various particles of the body due to the attraction of earth, passes irrespective of the orientation of the body.

• If the body is located in a uniform gravitational field,then the centre of mass coincides with the centre of gravity of body, and if not then  its centre of mass and centre of gravity will be at two different locations.