# Moment of inertia of this shape about the x-axis

• jisbon
In summary, the conversation discusses a question involving an integral and the incorrectness of the answer. The equation in question is ##I=\int y^2 dm##, and the conversation goes on to discuss the correct equation and boundaries of integration. There is a discrepancy in the final answer, with one person suggesting a value of -88.2 while the other suggests 37.8g/cm^2. The conversation ends with a potential solution involving a different mass element and a revised integral.
jisbon
Homework Statement
Calculate moment of inertia of a 2d plane rotating about x axis. Mass per unit area of plate = ##1.4g/cm^2## , total mass = 25.2g
Relevant Equations
##I=\int y^2 dm##

I've attempted this question, but the answer seems to be incorrect. Here's my workings:
##I=\int y^2 dm## - standard equation
##dM = \mu * dy * x## - take small slice and find mass of it
##x = 4y-16## - convert equation in terms of x to sub in later
##dM = \mu * dy * 4y-16##
##I=\int y^2 \mu * dy * 4y-16##
##I=\mu\int y^2(4y-16) dy##
##I=\mu\int 4y^3-16y^2 dy##
##I=\mu \int_{0}^{3} 4y^3-16y^2 dy## = 12.6, which is not the answer.
Any clues why?

Something wrong in your final calculation of the integral, I found it to be
##\int_0^3 (4y^3-16y^2 )dy=-63## and if ##\mu=1.4## then the final value should be -88.2. The minus sign can be explained (because you took ##x=4y-16## while i believe it is ##x=16-4y##).

Delta2 said:
Something wrong in your final calculation of the integral, I found it to be
##\int_0^3 (4y^3-16y^2 )dy=-63## and if ##\mu=1.4## then the final value should be -88.2. The minus sign can be explained (because you took ##x=4y-16## while i believe it is ##x=16-4y##).
The answer seems to be 37.8g/cm^2 though :/ Any ideas on this?

At the moment I can't spot any other mistake. Are we sure that the equation of y(x) as well as the boundaries of integration (from y=0 to y=3 or from x=4 to x=16) are correct?

Delta2 said:
At the moment I can't spot any other mistake. Are we sure that the equation of y(x) as well as the boundaries of integration (from y=0 to y=3 or from x=4 to x=16) are correct?
Equation of y is y= -1/4x+4 , so it should be correct :/

I think I spotted a mistake, the mass element should be ##dM=\mu(x-4)dy## so the final integral should be ##\mu\int_0^3 (12y^2-4y^3)dy##

Last edited:

## What is moment of inertia?

Moment of inertia is a physical property of a rigid body that determines how resistant it is to changes in rotational motion. It is also known as rotational inertia or angular mass.

## How is moment of inertia calculated?

Moment of inertia is calculated by multiplying the mass of an object by the square of its distance from the axis of rotation. This is also known as the radius of gyration.

## What is the significance of calculating moment of inertia about the x-axis?

Calculating moment of inertia about the x-axis is important because it helps determine the distribution of mass of an object along the x-axis. This information is useful in predicting the object's rotational motion and stability.

## How does the shape of an object affect its moment of inertia about the x-axis?

The shape of an object greatly affects its moment of inertia about the x-axis. Objects with a larger mass and greater distance from the axis of rotation have a higher moment of inertia, while objects with a smaller mass and shorter distance have a lower moment of inertia.

## Can the moment of inertia of an object change?

Yes, the moment of inertia of an object can change if its mass or distance from the axis of rotation is altered. This can also be affected by the shape of the object, as different shapes have different moments of inertia even with the same mass and distance from the axis.

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