Moment of Inertia of Curve without Prefix

[rock.freak667]

Homework Statement

Consider the curve y=(2/9)x^2 revolved around the y-axis. Find the moment of inertia about the y-axis

Homework Equations

$$I=\int r^2 dm$$

The Attempt at a Solution

So I considered a cylindrical element of width dy and radius x, so that it's volume is

$$dV=(\pi x^2)dy$$

Now the moment of inertia of this element about the y-axis is

$$dI= \frac{1}{2} x^2 \rho (\pi x^2)dy$$

so to get the moment of inertia of this entire curve, I just need to integrate like so

$$I= \int_{0}^{\frac{2x^2}{9}} \frac{1}{2} \rho \pi x^4 dy$$

Where $x=\sqrt{4.5y}$

is this correct??

 

[tiny-tim]

Hi rock.freak667!

Consider the curve y=(2/9)x^2 revolved around the y-axis. Find the moment of inertia about the y-axis

uhh? what are the limits?

and do you mean curve, so that you get a surface, or the area within the curve, so that you get a volume?

Now the moment of inertia of this element about the y-axis is

$$dI= \frac{1}{2} x^2 \rho (\pi x^2)dy$$

That's correct (assuming the question means a volume)

so to get the moment of inertia of this entire curve, I just need to integrate like so

$$I= \int_{0}^{\frac{2x^2}{9}} \frac{1}{2} \rho \pi x^4 dy$$

Where $x=\sqrt{4.5y}$

Noooo … your x⁴ has to be converted into y if you're integrating over y

(and where did you get those limits from??)

An alternative method (which doesn't assume knowledge of MI of a solid cylinder) is to use cylindrical shells of thickness dr ​

 

[rock.freak667]

Well I just made up the question but I will put the limits from x=0 to x=4 to get a volume.

But I am not sure how to do these questions as one example in my book has the method of considering an elemental disk like I did and another example has to use a triple integral. So I am not sure which one to use and when to use. Not even sure how to find the limits for the triple integral ones as well.

 

[tiny-tim]

Well I just made up the question but I will put the limits from x=0 to x=4 to get a volume.

ok, then your πρx⁴/2 has to be converted to 81/4 πρy², and the integral has to be from y = 0 to 32/9.

But I am not sure how to do these questions as one example in my book has the method of considering an elemental disk like I did and another example has to use a triple integral. So I am not sure which one to use and when to use. Not even sure how to find the limits for the triple integral ones as well.

The general rule is that if you have a symmetry, then use it.

For example, if there is spherical symmetry, use spherical shells.

If there's only rotational symmetry (ie, in one dimension), you can use discs, as you did (but it needs you to remember the MI for a disc), or cylindrical shells.

If there's no symmetry, you'll probably have to divide into into slices of thickness dz (say), and then divide each slice into strips of thickness dy (say) …

to find the limits, just draw a diagram, with x y z dy and dz marked, and it should be obvious what the limits are on each strip.