Explain the theoretical method for estimation of the centre of mass of a solid body.
Explain the theoretical method for estimation of the centre of mass of a solid body.
A solid body is made up of microscopic particle (molecules, ions, atoms) distributed continuously inside the body.
As shown in figure, suppose small mass $n$ made up of small mass element each having mass $\Delta m$ whose position vector is $\vec{r}=x \hat{i}+y \hat{j}+z \hat{k}$.
Whole solid body can be considered to be made up of such small mass elements. Let the solid body is made up of small mass elements $\Delta m_{1}, \Delta m_{2}, \ldots, \Delta m_{n}$ having position vectors $\overrightarrow{r_{1}}, \overrightarrow{r_{2}}, \ldots, \overrightarrow{r_{n}}$
respectively.
The position vector of centre of mass of the solid body is,
$\overrightarrow{\mathrm{R}}=\frac{\Delta m_{1} \vec{r}_{1}+\Delta m_{2} \overrightarrow{r_{2}}+\Delta m_{n} \overrightarrow{r_{n}}}{\Delta m_{1}+\Delta m_{2}+\ldots+\Delta m_{n}}$
$\therefore$ Coordinate of centre of mass,
$\mathrm{X}=\frac{\Sigma\left(\Delta m_{i}\right) x_{i}}{\Sigma \Delta m_{i}}$
$\mathrm{Y}=\frac{\Sigma \Delta m_{i} y_{i}}{\Sigma \Delta m_{i}} \text { and }$
$\mathrm{Z}=\frac{\Sigma \Delta m_{i} z_{i}}{\Sigma \Delta m_{i}} \quad[\text { where } i=1,2,3, \ldots, n]$
As no. of particles $n$ is more, $\Delta m_{i}$ is smaller the summation can be represented as an integration.
$\therefore \Sigma \Delta m_{i} \rightarrow \int d m=\mathrm{M}$
$ \Sigma \Delta m_{i} x_{i} \rightarrow \int x d m$
$ \Sigma \Delta m_{i} y_{i} \rightarrow \int y d m$
$ \Sigma \Delta m_{i} z_{i} \rightarrow \int z d m$
where $M$ is the total mass of the body.
The coordinate of centre of mass of solid body,
$\mathrm{X}=\frac{1}{\mathrm{M}} \int x d m$
$\mathrm{Y}=\frac{1}{\mathrm{M}} \int y d m$
$\mathrm{Z}=\frac{1}{\mathrm{M}} \int z d m$
Similar Questions
Mass is distributed uniformly over a thin square plate. If two end points of diagonal are $(-2, 0)$ and $(2, 2)$, what are the coordinates of centre of mass of plate ?
Mass is distributed uniformly over a thin square plate. If two end points of diagonal are $(-2, 0)$ and $(2, 2)$, what are the coordinates of centre of mass of plate ?
For the given uniform square lamina $ABCD$ whose centre is $O$ , pick incorrect statement
For the given uniform square lamina $ABCD$ whose centre is $O$ , pick incorrect statement
$(a)$ Centre of gravity $(C.G.)$ of a body is the point at which the weight of the body acts.
$(b)$ Centre of mass coincides with the centre of gravity if the earth is assumed to have infinitely large radius
$(c)$ To evaluate the gravitational field intensity due to any body at an external point, the entire mass of the body can be considered to be concentrated at its $C.G.$
$(d)$ The radius of gyration of any body rotating about an axis is the length of the perpendicular dropped from the $C.G.$ of the body to the axis of rotation.
Which one of the following pairs of statements is correct ?
$(a)$ Centre of gravity $(C.G.)$ of a body is the point at which the weight of the body acts.
$(b)$ Centre of mass coincides with the centre of gravity if the earth is assumed to have infinitely large radius
$(c)$ To evaluate the gravitational field intensity due to any body at an external point, the entire mass of the body can be considered to be concentrated at its $C.G.$
$(d)$ The radius of gyration of any body rotating about an axis is the length of the perpendicular dropped from the $C.G.$ of the body to the axis of rotation.
Which one of the following pairs of statements is correct ?
- [AIPMT 2010]
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How would you set a half wheel into uniform motion about an axis passing through the centre of mass of the wheel and perpendicular to its plane? Will you require external forces to sustain the motion ?
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