Distance of the centre of mass of a solid uniform cone from its vertex is `z_0` . If the radius of its base is R and its height is h then `z_0` is equal to:

Options

A. `(3h)/(4)`
B. `(5h)/(8)`
C. `(3h^(2))/(8R)`
D. `(h^(2))/(4R)`

Answer

A

Explanation

All distances have to be taken from `Z_(0)` (the reference line) From similar triangle

`(z)/(r)=(h)/(R)`
mass of small element of radius r
`dm=(M)/((1)/(3)piM^(2)h)xxpir^(2)dz`
`=(3M)/(piR^(2)h)xxpi((Rz)/(h))^(2)dz`
`dm=(3M)/(h^(3))xxz^(2)dz`
Centre of mass lies on the Z-axis Take limit from 2 = 0 to Z =h
`Z_(cm)=(1)/(M)int_(z=0)^(z=h)z.dm=(3)/(h^(3))int_(0)^(h)z^(3).dz`
`=(3h)/(4)`