A uniform sphere of radius R=30cm is made of a material of density 5000kg/m3.find the moment of inertia about an axis through the center of the sphere.
To find the moment of inertia (I) of a uniform sphere about an axis through its center, we can use the formula:
I = (2/5) * m * r²
where m is the mass of the sphere and r is its radius.
First, let's find the mass (m) of the sphere. We know that the density (ρ) of the material is 5000kg/m³. The mass is given by the formula:
m = ρ * V
where V is the volume of the sphere. The volume of a sphere is given by:
V = (4/3) * π * r³
Now, let's substitute the values:
V = (4/3) * π * (30cm)³ = (4/3) * π * (0.3m)³ ≈ 0.1131 m³
m = 5000kg/m³ * 0.1131m³ ≈ 565.5 kg
Next, let's calculate the moment of inertia (I) using the formula mentioned earlier:
I = (2/5) * m * r² = (2/5) * 565.5kg * (0.3m)² ≈ 30.09 kg.m²
Therefore, the moment of inertia of the uniform sphere about an axis through its center is approximately 30.09 kg.m².