Question

Problem 24.3 The assembly is made from a steel hemisphere, rho st = 7. 80 Mg/m3 , and an aluminum cylinder, rho al = 2. 70 Mg/m3 . If the height of the cylinder is h = 180 mm, determine the location z of the mass center of the assembly.

Answer 1

Answer:

0.12 m    

Explanation:

The mass of the steel hemisphere is:

$m_h=\rho_s\times \frac{2}{3}\pi r^2\\m_h = 7.8\times 10^3 \times \frac{2}{3}\pi (0.16)^2\\m_h=66.9 kg$

Mass of the aluminium cylinder is:

$m_c=\rho_a\times \pi r^2 h\\m_c=2.7\times 10^3 \times \pi (0.08)^2(0.18) \\m_c =9.7 kg$

mass center of steel hemisphere from the bottom:

$z_1=r-3r/8\\z_1=0.16-(3\times 0.16/8) =0.1 m$

mass center of aluminium cylinder from the bottom:

$z_2=r+h/2 \\z_2=0.16+0.18/2=0.25 m$

center of mass is

$Z=\frac{m_hz_1+z_2m_c}{m_s+m_c}\\Z=\frac{66.9\times 0.1+9.76\times 0.25}{66.9+9.76} = 0.12 m$