# Moment of Inertia of a body with linearly increasing density?

• easwar2641993
In summary, a thin wire of length L and uniformly density ρ is bent into a circular loop with center at O. The moment of inertia of the loop about a tangential axis lying in the plane of the loop is 3ρL^5/8π^2. However, the correct answer is 3ρL^3/8π^2. This may be due to a misunderstanding of the dimensions of the wire or the given line density ρ.
easwar2641993
A thin wire of length L and uniformly density ρ is bent into a circular loop with center at O.The moment of inertia of it about a tangential axis lying in the plane of loop is.
Ans : Mass M is not given,but ρ is given. So M=ρL3->(1) (L3 means L cube,no idea how to post it in that manner!). For a circular loop, we all know the formula for that condition is (3/2)MR square. Applying this in equation for equation (1) will give (3/2)ρL3 x (L square/4∏ square)
This will give 3ρL raised to 5 / 8∏ square.

But my teacher told the answer is 3ρL cube/8∏square.
This is not a home work.This is just a practice for me.
In this,where is my mistake?

The wire is thin, not a cube, its volume is not L3 (can be written as [noparse]L3[/noparse], but L^3 is fine, too. Or use LaTeX: $L^3$ becomes $L^3$).
I would expect that ρ is given as line density (kg/m), or the cross-section of the wire has to be given.

## 1. What is the moment of inertia of a body with linearly increasing density?

The moment of inertia of a body with linearly increasing density is a measure of the object's resistance to rotational motion. It takes into account both the mass and distribution of that mass within the object.

## 2. How is the moment of inertia calculated for a body with linearly increasing density?

The moment of inertia for a body with linearly increasing density can be calculated by integrating the density function over the volume of the object, multiplied by the square of the distance from the axis of rotation.

## 3. Does the moment of inertia change for a body with linearly increasing density as it rotates?

Yes, the moment of inertia changes as the body rotates, as the distribution of mass changes with respect to the axis of rotation. This is why objects with more mass concentrated farther from the axis of rotation have a larger moment of inertia.

## 4. What factors affect the moment of inertia of a body with linearly increasing density?

The main factors that affect the moment of inertia of a body with linearly increasing density are the mass of the object, the distribution of that mass, and the axis of rotation. The shape and size of the object also play a role in determining its moment of inertia.

## 5. How is the moment of inertia of a body with linearly increasing density useful in physics and engineering?

The moment of inertia is a crucial concept in understanding rotational motion and is used in many areas of physics and engineering. It is particularly important in calculating the torque and angular acceleration of an object, as well as predicting its behavior when subject to external forces and collisions.

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