Question

$1000{\text{m}}^3$ of sand is being poured on the ground. It accumulates on the ground in the shape of a conical pyramid. If the radius of the base of pyramid is $5\text{m}$, where is the centre of mass of the pile of sand located?

• A. $\frac{90}{\pi }\text{m}$ from the centre of the base.
• B. $\frac{30}{\pi }\text{m}$ from the centre of the base.
• C. $\frac{12}{\pi }\text{m}$ from the top of the pyramid.
• D. $\frac{3}{\pi }\text{m}$ from the top of the pyramid.

Answer 1

The correct option is B $\frac{30}{\pi }\text{m}$ from the centre of the base.

Sand will pile up in the form of a solid cone
Given, radius of cone's base $=R=5\text{m}$
Volume of solid cone $V=\frac{1}{3}\pi R^{2}H\Rightarrow 1000{\text{m}}^3=\frac{1}{3}\times \pi \times 5^2\times H\Rightarrow H=\frac{3000}{25\pi }=\frac{120}{\pi }\text{m}$


$\because$ The solid cone is symmetrical about an axis passing through its vertex and centre of the base, i.e $y$ axis in the shown figure, so its $\text{COM}$ will lie along $y$ axis.
COM of solid cone (measured from centre of base) is given by $y=\frac{H}{4}$
i.e $y_{\text{CM}}=\frac{H}{4}=\frac{30}{\pi }\text{m}$

$\Rightarrow$$\text{COM}$ of the pile of sand will lie at a distance of $\frac{30}{\pi }\text{m}$ from the centre of base