Center Of Mass MCQ - Practice Questions with Answers

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 centre of mass

• For a system of N discrete particles

$$\begin{array}{rl}{x}_{cm}& =\frac{m_1{x}_1+m_2{x}_2\dots \dots \dots }{m_1+m_2\dots \dots }\\ y_{cm}& =\frac{m_1{y}_1+m_2{y}_2+m_3{y}_3\dots \dots \dots }{m_1+m_2+m_3\dots \dots }\\ z_{cm}& =\frac{m_1{z}_1+m_2{z}_2+m_3{z}_3\dots \dots \cdots }{m_1+m_2+m_3\dots \dots }\end{array}$$

Where $m_1{m}_2$. $$ are the mass of each particle $x_1,x_2$ $$ $y_1,y_2$ $$ $z_1,z_2$ are respectively $x_1$ $y,\mathrm{\&}z$ 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

$$x_{cm}=\frac{\int xdm}{\int dm},y_{cm}=\frac{\int ydm}{\int dm},z_{cm}=\frac{\int zdm}{\int dm}$$

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

3. Important points about the position of centre of mass 

• Its position is independent of the coordinate system chosen.

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

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

•  For symmetrical bodies having a 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 the 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 the 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 the body, and if not then its centre of mass and centre of gravity will be at two different locations.