consider a uniform solid hemisphere of mass m and radius r. mi of body about axis passing through com and inclined at 37 degree is

To find the moment of inertia (I) of a uniform solid hemisphere about an axis passing through its center of mass (COM) and inclined at 37 degrees, we can break down the problem into manageable steps. The moment of inertia is a measure of an object's resistance to rotational motion about an axis, and for a solid hemisphere, we can derive it using known formulas and principles of geometry.

Understanding the Geometry

A solid hemisphere can be visualized as half of a sphere. The moment of inertia about an axis through the center of mass of a solid hemisphere can be calculated using the formula:

• I_COM = (2/5) m r² for a solid sphere.

However, since we are dealing with a hemisphere, we need to adjust this formula. The moment of inertia about the axis through the flat circular face (the base of the hemisphere) can be derived from the moment of inertia of a full sphere.

Calculating the Moment of Inertia

The moment of inertia of a solid hemisphere about an axis through its flat face is given by:

• I_flat = (2/5) m r² + m (r²/4) = (7/10) m r².

Here, the first term is the moment of inertia about the COM, and the second term accounts for the parallel axis theorem, which states that I = I_COM + md², where d is the distance from the COM to the new axis. For a hemisphere, this distance is r/4.

Inclined Axis Calculation

Now, since the axis is inclined at 37 degrees, we need to apply the perpendicular axis theorem. The moment of inertia about an inclined axis can be calculated using the following relation:

• I_inclined = I_flat cos²(θ) + I_{perpendicular} sin²(θ).

In this case, we can consider the moment of inertia about the perpendicular axis (which is the same as the flat face) and the flat face moment of inertia. The moment of inertia about the axis perpendicular to the flat face is:

• I_{perpendicular} = (7/10) m r².

Now substituting θ = 37 degrees:

• cos(37°) = 0.8 and sin(37°) = 0.6.

Thus, we can calculate:

• I_inclined = (7/10) m r² (0.8)² + (7/10) m r² (0.6)².

Calculating this gives:

• I_inclined = (7/10) m r² (0.64 + 0.36) = (7/10) m r² (1) = (7/10) m r².

Final Result

Therefore, the moment of inertia of the uniform solid hemisphere about the inclined axis at 37 degrees is:

I_inclined = (7/10) m r².

This result shows how the geometry and orientation of the axis affect the moment of inertia, illustrating the principles of rotational dynamics in a clear and structured manner.