# Non-uniform Inertia of a Cylinder: I = (3MR^3)/5

• LASmith
In summary, the moment of inertia of a cylinder with increasing linear density about a longitudinal axis through the centre is given by the formula I=(3MR^3)/5, where M is the total mass of the cylinder and ρ0 is the constant density at the cylinder axis. To find M in terms of ρ0, an integral can be set up to solve for the total mass.
LASmith

## Homework Statement

A cylinder with radius R and mass M has density that increases linearly with radial distance r from the cylinder axis, ie. $\rho$=$\rho$$_{0}$(r/R), where $\rho$$_{0}$ is a positive constant. Show that the moment of inertia of this cylinder about a longitudinal axis through the centre is given by I=(3MR$^{3}$)/5

## Homework Equations

I=$\int$r$^{2}$.dm
volume = 2$\pi$rL.dr

## The Attempt at a Solution

I=$\int$r$^{2}$$\rho$.dv
=$\int$(r$^{3}$$\rho$$_{0}$/R.)dv
=$\int$(r$^{3}$$\rho$$_{0}$/R.)(2$\pi$rL).dr

integrate between 0 and R to obtain
2$\rho_{0}$$\pi$R$^{4}$L/5

However, I do not understand how to express this without using the term $\rho_{0}$

LASmith said:
However, I do not understand how to express this without using the term $\rho_{0}$
Find an expression for M in terms of ρ0.

Doc Al said:
Find an expression for M in terms of ρ0.

I realize this, however as the density is not constant, I am unsure of how to do this.

LASmith said:
I realize this, however as the density is not constant, I am unsure of how to do this.
Set up an integral to solve for the total mass, just like you set one up for the rotational inertia.

Once you get M in terms of ρ0, you can rewrite your answer in terms of M instead of ρ0.

.I would approach this problem by first understanding the concept of inertia. Inertia is the resistance of an object to change its state of motion. It is directly related to the mass and distribution of the object. In this case, we are dealing with the inertia of a cylinder, which is a measure of how difficult it is to change its rotational motion.

To calculate the moment of inertia of a cylinder, we use the formula I = (MR^2)/2, where M is the mass of the cylinder and R is the radius. However, in this scenario, the density of the cylinder is not constant, but increases as we move away from the axis of rotation. This means that the mass of the cylinder is not evenly distributed, and we need to take that into account when calculating the moment of inertia.

To do this, we can divide the cylinder into infinitesimally thin concentric rings, each with a thickness of dr. The mass of each ring can be calculated using the formula dm = \rho(r)2\pirL.dr, where \rho(r) is the density at a distance r from the axis of rotation. We can then integrate this expression from 0 to R to obtain the total mass of the cylinder.

I = \int_{0}^{R} \rho(r)2\pirL.dr

Now, to simplify the expression, we can substitute \rho(r) with \rho_{0}(r/R), as given in the problem statement. This allows us to express the moment of inertia as:

I = \int_{0}^{R} \rho_{0}(r/R)2\pirL.dr

= \rho_{0}2\piL \int_{0}^{R} rdr

= \rho_{0}2\piL \frac{r^2}{2} \Big|_{0}^{R}

= \rho_{0}\piLR^2

= \frac{2}{5}M \piR^2

= \frac{2}{5} \cdot \frac{M}{\piR^3} \piR^4

= \frac{2}{5} \cdot \frac{M}{V} \cdot V \piR^4

= \frac{2}{5} \cdot \frac{M}{V} \cdot \frac{V}{\piR^3} \pi

## 1. What is Non-uniform Inertia of a Cylinder?

The Non-uniform Inertia of a Cylinder refers to the distribution of mass within a cylinder that affects its resistance to changes in motion. It is a measure of the object's rotational inertia or its tendency to resist changes in rotation.

## 2. What is the formula for calculating Non-uniform Inertia of a Cylinder?

The formula for calculating Non-uniform Inertia of a Cylinder is I = (3MR^3)/5, where I is the inertia, M is the mass of the cylinder, and R is the radius of the cylinder.

## 3. How does Non-uniform Inertia of a Cylinder differ from uniform Inertia?

Uniform Inertia refers to an object having a constant distribution of mass, while Non-uniform Inertia refers to an object having an uneven distribution of mass. This means that different parts of the object will have different resistances to changes in motion.

## 4. What factors affect the Non-uniform Inertia of a Cylinder?

The Non-uniform Inertia of a Cylinder is affected by the mass and distribution of the mass within the cylinder. A cylinder with a larger mass or a larger radius will have a greater inertia.

## 5. What is the significance of the Non-uniform Inertia of a Cylinder?

The Non-uniform Inertia of a Cylinder is significant in understanding the rotational motion of objects. It helps calculate the amount of force needed to change the rotational motion of a cylinder and is important in designing objects that require stable and controlled rotation.

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