Question

the centre of the masss of solid cone along the centre of base to vertex is at ????????????????

Answer 1

the centre of the mass of solid cone along the centre of base to vertex is at?

Solution:

h - height of the cone;

m - mass of the cone;

R - radius of the base of the cone;

First, we can find the density of the cone:

$$\rho = \frac {m}{V} = \frac {m}{\frac {\pi R ^ {2} h}{3}} = \frac {3 m}{\pi R ^ {2} h}$$

Radius dependence of the height $x$ :

$\mathrm{r(x) = R\cdot x / h}$ , where $\mathbf{x}$ - distance to the vertex

Now, let us split the cone on discs height dx. Volume of the disc on the height $dx$ will be:

$$\mathrm {d V} = \pi \mathrm {r} ^ {2} \mathrm {d x} = \frac {\pi \mathrm {R} ^ {2} \mathrm {x} ^ {2} \mathrm {d x}}{\mathrm {h} ^ {2}}$$

Mass of this disc will be:

$$\mathrm {d m} = \rho \cdot \mathrm {d V} = \frac {3 \mathrm {m x} ^ {2} \mathrm {d x}}{\mathrm {h} ^ {3}}$$

Position of the center of mass is determined by the sum of the $\mathrm{dm} \cdot \mathrm{x}$ divided by the total mass $\mathrm{m}$ :

$$x _ {\text {c e n t r}} = \frac {3}{h ^ {3}} \int_ {0} ^ {h} x ^ {3} d x = 3 \cdot \frac {h}{4}.$$

Hence, the center of mass is located at a distance $\frac{3}{4}$ the height of the cone of the vertex or $\frac{h}{4}$ of the center of base.

Answer: center of mass of the solid cone is located at a distance $\frac{3}{4}$ the height of the cone of the vertex or $\frac{1}{4} h$ of the center of base.