How to determine rectangular and polar radii of gyration?

• sagarbhathwar
In summary: Does that make it clearer ?In summary, the problem is to determine the rectangular and polar radii of gyration of a shaded area about the given axes. The equations used are dIx = y2dA and dA = ydx. The solution involves finding the double integral of dIx = y2dydx and integrating from 1 to 2. The radius of gyration about the y-axis is correctly calculated as 1.705, but the calculation for the x-axis is incorrect due to a missing factor of 1/3. The correct answer for the radius of gyration about the x-axis is 0.75.
sagarbhathwar

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dIx = y2dA
Ix = ∫y2dA

dA = ydx
A = ∫(dA

The Attempt at a Solution

dIx = y2dA
=(x6/16) *(x3/4) dx
=(x9/64)dx
Ix =(integrating from 1 to 2) ∫dIx = 1023/640

dA = ydx
A = (integrating from1 to 2)∫(x3/4)dx = 15/16

kx = √(Ix/A) = 1.705

I am getting the radius of gyration about y-axis correctly but that baout x-axis is wrong.
The actualy answer given is 0.75. I am unable to figure out where I am going wrong.

Yes: work it out dA bit more and you'll see you miss a factor 1/3 in Ix.

By the way, you confuse me with ##k_x =\sqrt{I_x/A} = 1.705##; I get ##1.306\;##; a typo ?​

Last edited:
Yes. Sorry I forgot to take the square root so yea a typo.
But to be honest, I don't get what you are saying. Could you be more specific in where I am going wrong?

##dI_x = y^2 dA## needs to be worked out to ##dI_x = y^2 dy dx##: it's a double integral. You seem to treat it as a single one.

1) How do I calculate the rectangular radius of gyration?

The rectangular radius of gyration is calculated by taking the square root of the moment of inertia of the object divided by its mass. This can be represented by the formula: Rectangular Radius of Gyration = √(Moment of Inertia / Mass).

2) What is the formula for determining the polar radius of gyration?

The formula for calculating the polar radius of gyration is similar to the rectangular radius of gyration. It is also equal to the square root of the moment of inertia divided by the mass, but it involves the moment of inertia about the polar axis instead. The formula can be written as: Polar Radius of Gyration = √(Moment of Inertia about Polar Axis / Mass).

3) Can the rectangular and polar radii of gyration be equal?

Yes, it is possible for the rectangular and polar radii of gyration to be equal for certain objects. This occurs when an object has symmetrical mass distribution and its moment of inertia is equal about both the rectangular and polar axes.

4) How do I determine the moment of inertia of an object?

The moment of inertia of an object can be calculated by summing up the products of the mass of each particle and the square of its distance from the axis of rotation. This can also be represented by the formula: Moment of Inertia = ∑(Mass x Distance^2).

5) Are there any other methods for determining the rectangular and polar radii of gyration?

Yes, there are other methods for calculating the rectangular and polar radii of gyration, such as using integration or using computer software. However, the basic formula mentioned above is the most commonly used method for determining these values.

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