Final answer:
The moment of inertia of a disk about an axis through its center can be calculated using the formula I = 0.5*m*R^2, where m is the mass of the disk and R is the radius of the disk. In this case, the mass is not given, but can be calculated using the surface mass density and the area of the disk. Plugging in the values, we can find the moment of inertia of the shaded area about the y axis.
Step-by-step explanation:
The moment of inertia of a disk about an axis through its center is given by the formula I = ½ mR^2, where m is the mass of the disk and R is the radius of the disk. In this case, the mass of the disk is not given, but the area and the surface mass density are given. We can use these values to calculate the mass of the disk.
The area of the disk is given by A = πR^2. The total mass of the disk is then given by m = πR^2ρ, where ρ is the surface mass density. Substituting this value for m into the moment of inertia formula, we get I = ½ (πR^2ρ)R^2 = ½ πR^4ρ.
Plugging in the given values, where R = 75 mm = 0.075 m and ρ = mR, we get I = ½ π(0.075^4)(mR) = ½ π(0.075^4)(m(0.075)). Evaluating this expression will give us the moment of inertia of the shaded area about the y axis.