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Centre Of Mass Of A Solid Cone - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Position of centre of mass for solid cone is considered one of the most asked concept.

  • 3 Questions around this concept.

Solve by difficulty

QuestionEASY

Distance of the centre of mass of a solid uniform cone from its vertex is z0.  If the radius of its base is R and its height is h then z0 is equal to :

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Concepts Covered - 1

Position of centre of mass for solid cone

Have a look at the figure of a solid cone

            

Since it is symmetrical about y-axis  

So we can say that  its x_{cm}=0 and z_{cm}=0

Now we will calculate its y_{cm} which is given by

y_{cm} = \frac{\int y.dm}{\int dm}

So Take a small elemental disc of mass dm of radius r  at a vertical distance y from the bottom as shown in the figure.

        

So  dm=\rho dv=\rho (\pi r^2)dy

Here \rho =\frac{M}{V}=\frac{M}{\frac{1}{3}\pi R^2H}  

And from similar triangle

  \frac{r}{R}=\frac{H-y}{H}

r=(\frac{H-y}{H})R

 

y_{cm} = \frac{\int y.dm}{\int dm}

y_{cm} = \frac{1}{M}\int_{0}^{H} y.dm = \frac{1}{M}\int_{0}^{H}y \frac{3M}{\pi R^2H}(\pi r^2)dy = \frac{H}{4}

So,   \mathbf{y_{cm} = \frac{H}{4}} from bottom O

Or, Centre of Mass of a solid cone will lie at distance \frac{3h}{4} from the tip of the cone.

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Position of centre of mass for solid cone

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