# Polar moment of inertia/polar radius of gyration via integration

• drunknfox
In summary, to determine the polar moment of inertia and polar radius of gyration of a shaded area with respect to point P, you need to use the equations Jp = Ix + Iy, Ix = &int y^2dA, and Iy = &int X^2dA. To find the integral of y^2 dA, you need to express the boundaries of the region in terms of x and y, and then integrate using the limits of the region.
drunknfox

## Homework Statement

Determine the polar moment of inertia and the polar radius of gyration of the shaded area shown with respect to point P.

http://imgur.com/8Kc1S

Jp = Ix + Iy
Ix = &int y^2dA
Iy = &int X^2dA

## The Attempt at a Solution

A = 2(a/2)(a) + (2)(1/2)(a/2)(a) = 3a^2/2

Jp = Ix + Iy

Ix = &int y^2dA = ? I am having trouble with this next step. If someone could please help me with it and explain to me what's suppose to integrated I would be eternally grateful

I have 2 &int a-0

Firstly be warned are specific to planar objects of unit area or density 1/area.

Integrating y^2 dA involves expressing y and dA in terms of your independent variable (variable of integration) and its differential.

Looking at your region (a trapazoid?) it would appear that your best bet is to express the width at a point as a function of height i.e. express left and right line boundaries in terms of x as a function of y.

Left boundary x =f(y)= p y + q,
Right boundary x = g(y) = r y + s.

You are basically, in the Riemann sum, slicing the object up into horizontal strips with thickness dy and width x_right - x_left = g(y)-f(y), and so its area is:

dA = [g(y)-f(y)] dy

With this integrate y^2 dA between the appropriate limits.

## 1. What is the concept of polar moment of inertia?

The polar moment of inertia is a measure of an object's resistance to rotational movement about its axis. It takes into account the distribution of mass around the axis and is calculated by integrating the square of the distance from the axis for all the infinitesimal particles that make up the object.

## 2. How is polar moment of inertia different from moment of inertia?

Moment of inertia is a measure of an object's resistance to rotational movement about any axis, while polar moment of inertia specifically considers rotational movement about the object's axis. In other words, moment of inertia is a general term, while polar moment of inertia is a specific type of moment of inertia.

## 3. How is polar moment of inertia calculated?

To calculate the polar moment of inertia, one must integrate the square of the distance from the axis for all the infinitesimal particles that make up the object. This integration is usually done using calculus, and the resulting value represents the polar moment of inertia of the object.

## 4. What is the significance of polar moment of inertia in engineering?

Polar moment of inertia is an important concept in engineering, particularly in designing structures that are subjected to torsional loads. It helps engineers determine the amount of torque that can be applied to a structure before it begins to rotate. It is also used in the design of rotating machinery, such as engines and turbines.

## 5. How does polar radius of gyration relate to polar moment of inertia?

The polar radius of gyration is the distance from the axis at which the entire mass of the object could be concentrated to produce the same polar moment of inertia as the object itself. In other words, it is a measure of how the mass is distributed around the axis. It is directly proportional to the polar moment of inertia, meaning that as the polar radius of gyration increases, so does the polar moment of inertia.

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