moment of inertia of a body about an axis 3m from its centre of mass is 8200kgm^2 find the moment of inertia of this body about the axis 5m away from its centre of mass and parallel to the above mentioned axis, mass of the body is 200kg

To find the moment of inertia of a body about an axis that is parallel to another axis and at a different distance from the center of mass, we can use the parallel axis theorem. This theorem is a fundamental principle in rotational dynamics that allows us to calculate the moment of inertia about any axis parallel to one through the center of mass.

Understanding the Parallel Axis Theorem

The parallel axis theorem states that the moment of inertia (I) about any axis parallel to an axis through the center of mass (I_cm) can be calculated using the formula:

I = I_cm + Md²

In this formula:

• I is the moment of inertia about the new axis.
• I_cm is the moment of inertia about the center of mass axis.
• M is the mass of the body.
• d is the distance between the two axes.

Given Values

From your question, we have the following values:

• Moment of inertia about the center of mass axis, I_cm = 8200 kg·m²
• Mass of the body, M = 200 kg
• Distance from the center of mass to the first axis = 3 m
• Distance from the center of mass to the second axis = 5 m

Calculating the Moment of Inertia

First, we need to find the distance (d) between the two axes. Since the second axis is 5 m away from the center of mass and the first axis is 3 m away, we can calculate:

d = 5 m - 3 m = 2 m

Now we can apply the parallel axis theorem:

I = I_cm + Md²

Substituting the known values into the equation:

I = 8200 kg·m² + (200 kg)(2 m)²

Calculating the second term:

(200 kg)(2 m)² = (200 kg)(4 m²) = 800 kg·m²

Now, we can add this to the moment of inertia about the center of mass:

I = 8200 kg·m² + 800 kg·m² = 9000 kg·m²

Final Result

The moment of inertia of the body about the axis that is 5 m away from its center of mass and parallel to the first axis is 9000 kg·m².