A particular trophy has a solid 0.2m radius base with a mass of 1.2kg. Rising up from the center is a 0.01m radius solid rod with a mass of 0.4kg. Centered on top of this is a hollow sphere with a radius of 0.15m and a mass of 0.8kg. Calculate the total moment of inertia for this trophy around the vertical axis.
m1=1.2 kg, r1=0.2 m, - cylinder: J1=m1•r1²/2
m2 = 0.4 kg, r2=0.01 m – cylinder: J2=m2•r2²/2
m3=0.8 kg r3=0.15 m – hollow sphere: J3= 2m3•r3²/3
J=J1+J2+J3 = ...
To calculate the total moment of inertia for this trophy around the vertical axis, we need to consider the moment of inertia of each component separately and then add them up.
The moment of inertia of a solid object can be calculated using the formula:
I = (1/2) * m * r^2
where I is the moment of inertia, m is the mass, and r is the radius of the object.
For the base, moment of inertia is calculated as:
I_base = (1/2) * m_base * r_base^2
I_base = (1/2) * 1.2kg * (0.2m)^2
I_base = 0.12 kg·m^2
For the rod, moment of inertia is calculated as:
I_rod = (1/2) * m_rod * r_rod^2
I_rod = (1/2) * 0.4kg * (0.01m)^2
I_rod = 0.00002 kg·m^2
For the hollow sphere, the moment of inertia formula varies based on the axis of rotation. Since we are considering the vertical axis, we can use the formula:
I_sphere = (2/3) * m_sphere * r_sphere^2
I_sphere = (2/3) * 0.8kg * (0.15m)^2
I_sphere = 0.06 kg·m^2
Finally, we can add up all the individual moment of inertias to get the total moment of inertia:
Total moment of inertia = I_base + I_rod + I_sphere
Total moment of inertia = 0.12 kg·m^2 + 0.00002 kg·m^2 + 0.06 kg·m^2
Total moment of inertia ≈ 0.18002 kg·m^2
So, the total moment of inertia for this trophy around the vertical axis is approximately 0.18002 kg·m^2.