Question

calculate the moment of a uniform solid cone about an axis through its centre. the cone has mass M and altitude h, the radius of circular base is R ?

Answer 1

Task:

Calculate the moment of a uniform solid cone about an axis through its centre. The cone has mass $m$ and altitude $H$ , the radius of circular base is $R$ ?

Solution:

Split the cone into disks with thickness $dh$ . Radius of such disk:

$$r = \frac {R h}{H},$$

$R$ - the radius of circular base,

$H$ -altitude,

$h$ -distance between the top of the cone and the disk.

$$d m = \rho V = \rho \cdot \pi r ^ {2} d h;$$ $$d J = \frac {1}{2} r ^ {2} d m = \frac {1}{2} \pi \rho r ^ {4} d h = \frac {1}{2} \pi \rho \left(\frac {R h}{H}\right) ^ {4} d h;$$ $$\begin{array}{l} J = \int_ {0} ^ {H} d J = \frac {1}{2} \pi \rho \left(\frac {R}{H}\right) ^ {4} \int_ {0} ^ {H} h ^ {4} d h = \frac {1}{2} \pi \rho \left(\frac {R}{H}\right) ^ {4} \left. \frac {h ^ {5}}{5} \right| _ {0} ^ {H} = \\ = \frac {1}{1 0} \pi \rho R ^ {4} H = \left(\rho \cdot \frac {1}{3} \pi R ^ {2} H\right) \frac {3}{1 0} R ^ {2} = \frac {3}{1 0} m R ^ {2}. \\ \end{array}$$

Answer:

$$J = \frac {3}{1 0} m R ^ {2}$$