For a body,the center of volume is defined as | vedclass

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(A) The center of mass $\vec{R}_{cm}$ is defined as $\frac{\int \vec{r} \, dm}{\int dm}$.For a body of uniform density $\rho$,the mass of a small volu

Vedclass · Accepted Answer

For a body of uniform density,the center of volume is the same as the center of mass.

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(A) The center of mass $\vec{R}_{cm}$ is defined as $\frac{\int \vec{r} \, dm}{\int dm}$.For a body of uniform density $\rho$,the mass of a small volume element $dV$ is $dm = \rho \, dV$.Substituting this into the center of mass formula:$\vec{R}_{cm} = \frac{\int \vec{r} \, \rho \, dV}{\int \rho \, dV}$.Since $\rho$ is constant for a uniform body,it can be taken out of the integral:$\vec{R}_{cm} = \frac{\rho \int \vec{r} \, dV}{\rho \int dV} = \frac{\int \vec{r} \, dV}{\int dV}$.This expression is exactly the definition of the center of volume provided in the question.Therefore,for a body of uniform density,the center of volume is the same as the center of mass.

For a body,the center of volume is defined as $\frac{\int \vec{r} \, dV}{\int dV}$ over the complete body,where $dV$ is a small volume element of the body and $\vec{r}$ is the position vector of that small volume from the origin. Which of the following statements is correct?

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