Rotational inertia of square about axis perpendicular to its plane

[member 731016]

Homework Statement

Please see below

Relevant Equations

Please see below

For this problem,

How do we calculate the moment of inertia of (2) and (3)?

For (3) I have tried,

$I_z = \int r^2 \, dm$
$ds = r$ $d\theta$
$\lambda = \frac {dm}{ds}$
$\lambda$ $ds = dm$
$\lambda r$ $d\theta = dm$
$I_z = \lambda \int r^3 d\theta$
$I_z = \lambda \int_0^{2\pi} r^3 d\theta$

EDIT:
I know how to find the moment of inertia of (3) (shown in post ##2), however, how do I find the moment of inertia of the square about z-axis?

Many thanks!

 

[Frabjous]

Look at the I_z equation and think about what it means for each of the cases. You do not need to calculate anything.

 

[member 731016]

Look at the I_z equation and think about what it means for each of the cases. You do not need to calculate anything.

Thank you for your reply @Frabjous!

I see what you mean for (3)! So
$I_z = \int r^2 dm$
$I_z = r^2 \int dm$ (since a circle has a constant radius)
$I_z = mr^2$

However, how would I find the moment of inertia of the square?

Many thanks!

 

[Frabjous]

Thank you for your reply @Frabjous!

I see what you mean for (3)! So
$I_z = \int r^2 dm$
$I_z = r^2 \int dm$ (since a circle has a constant radius)
$I_z = mr^2$

However, how would I find the moment of inertia of the square?

Many thanks!

You don’t need to. Notice that the square fits inside the circle. What does this imply?

 

[member 731016]

You don’t need to. Notice that the square fits inside the circle. What does this imply?

Thank you for your reply @Frabjous!

It implies that $I_{square} < I_{circle}$ (since mass elements are either the same radius or closer than the circle) so $I_{square} < mr^2$. However, how do we calculate the square's specific moment of inertia?

Many thanks!

 

[Frabjous]

Notice that each side of the square will have the same inertia. Take two points, like (r,0) and (0,r), and write down the equation of the line that connects them.

 

[haruspex]

$I_z = \lambda \int_0^{2\pi} r^3 d\theta$

Fwiw, you can get I_z that way. Just do the integral (trivial) then substitute for λ using m and r.

 

[member 731016]

Notice that each side of the square will have the same inertia. Take two points, like (r,0) and (0,r), and write down the equation of the line that connects them.

Thank you for your reply @Frabjous !

The equation I got was $y = -x + r$

Thank you!

 

[erobz]

Thank you for your reply @Frabjous!

It implies that $I_{square} < I_{circle}$ (since mass elements are either the same radius or closer than the circle) so $I_{square} < mr^2$. However, how do we calculate the square's specific moment of inertia?

Many thanks!

Rotate it by 90 degrees. Then just focus on the top horizontal segment. Multiply the result by 4.

 

[member 731016]

Fwiw, you can get I_z that way. Just do the integral (trivial) then substitute for λ using m and r.

Thank you for your reply @haruspex!

Sorry, how do you get $r$ in terms of $\theta$?

Thank you!

 

[haruspex]

Thank you for your reply @haruspex!

Sorry, how do you get $r$ in terms of $\theta$?

Thank you!

In (3), r is constant.

 

[member 731016]

In (3), r is constant.

Whoops thank you @haruspex! I see now :)

 

[member 731016]

Rotate it by 90 degrees. Then just focus on the top horizontal segment. Multiply the result by 4.

Thank you for your reply @erobz!

Oh ok, so maybe I use Pythagorean theorem?

Thank you!

 

[erobz]

Thank you for your reply @erobz!

Oh ok, so maybe I use Pythagorean theorem?

Thank you!

That’s the idea. You know the diagonal of the square $2r$, the rest you can figure out with that info.

 

[nasu]

You can do this without extra integrals. You can calculate the moment of inertia of one of the sides around the center by using the parallel axis theorem. You know the moment of inrtia about an axis going through the middle of the side. $I=\frac{1}{12}ma^2$ where m is the mass of one of the sides (1/4 of the total mass) and a is the side of the square. Then just add the $m(a/2)^2$ to get the moment about the center of the square (a/2 is the distance between the two centers). Once you know the moment of one side, you multiply by 4.

 

[member 731016]

Thank you both @erobz and @nasu for your replies!

I will try again with those hints, and let you know how I get on.

Thank you!

 

[member 731016]

So far @erobz and @nasu I have got

$I = \int r^2 dm$
$I = \int R^2 + x^2 dm$ Where $2R$ is the length of the diagonal which means that each side as a length $2(2)^{1/2}$
$$ I = \lambda \int_{-(2)^{1/2}R}^{(2)^{1/2}R} R^2 + x^2 dx $$
$I = \lambda R^2 + \lambda \int_{-(2)^{1/2}R}^{(2)^{1/2}R} x^2 dx$

Is everything so far correct?

Many thanks!

 

[erobz]

So far @erobz and @nasu I have got

$I = \int r^2 dm$
$I = \int R^2 + x^2 dm$ Where $2R$ is the length of the diagonal which means that each side as a length $2(2)^{1/2}$
$I = \lambda \int_{-(2)^{1/2}R}^{(2)^{1/2}R} R^2 + x^2 dx$
$I = \lambda R^2 + \lambda \int_{-(2)^{1/2}R}^{(2)^{1/2}R} x^2 dx$

Is everything so far correct?

Many thanks!

Check your algebra for $r^2$. I do not get $r^2 = R^2 + x^2$. Remember $2R$ is the squares diagonal. (that's not the only problem, but it's the first problem I see).

P.S. use the $ sign delimiters for the latex. It formats more readable with integrals and limits.

 

[member 731016]

Check your algebra for $r^2$. I do not get $r^2 = R^2 + x^2$. Remember $2R$ is the squares diagonal. (that's not the only problem, but it's the first problem I see).

P.S. use the $ sign delimiters for the latex. It formats more readable with integrals and limits.

Thank you for your reply @erobz!

Sorry, what do you mean about r?

Where R is a constant and x is a variable in the y-direction.

Many thanks?

 

[erobz]

Thank you for your reply @erobz!

Sorry, what do you mean about r?
View attachment 322246
Where R is a constant and x is a variable in the y-direction.

Many thanks?

 

[member 731016]

View attachment 322247

Thank you for your reply @erobz!

 

[nasu]

$\lambda R^2$ does not have the right dimension to be a term in the moment of inertia.

 

[member 731016]

$\lambda R^2$ does not have the right dimension to be a term in the moment of inertia.

True, thanks for pointing that out @nasu!

 

[member 731016]

$\lambda R^2$ does not have the right dimension to be a term in the moment of inertia.

Are you saying that I can't take $\lambda R^2$ out of the integral since it is a constant in the space integral?

Many thanks!

 

[member 731016]

View attachment 322247

So the new integral is so far,

$$ I = \lambda \int_{-(2)^{1/2}R}^{(2)^{1/2}R} 2R^2 + x^2 dx $$
$$ I = \frac {M}{4(2)^{1/2}R} \int_{-(2)^{1/2}R}^{(2)^{1/2}R} 2R^2 + x^2 dx $$

Many thanks!

 

[haruspex]

Are you saying that I can't take $\lambda R^2$ out of the integral since it is a constant in the space integral?

Many thanks!

You can't just take it out of the integral unchanged. How do you integrate a constant wrt x?

 

[member 731016]

You can't just take it out of the integral unchanged. How do you integrate a constant wrt x?

Thank you for your reply @haruspex! I completely forgot! Integrating $R^2$ wrt to x is $R^2x$

Thank you!

 

[erobz]

So the new integral is so far,

$$ I = \lambda \int_{-(2)^{1/2}R}^{(2)^{1/2}R} 2R^2 + x^2 dx $$
$$ I = \frac {M}{4(2)^{1/2}R} \int_{-(2)^{1/2}R}^{(2)^{1/2}R} 2R^2 + x^2 dx $$

Many thanks!

The function for square root in latex is \sqrt{} Also, I wouldn't substitute for $\lambda$ just yet, its just more symbols to juggle IMO.

 

[nasu]

So the new integral is so far,

$$ I = \lambda \int_{-(2)^{1/2}R}^{(2)^{1/2}R} (2R^2 + x^2) dx $$
$$ I = \frac {M}{4(2)^{1/2}R} \int_{-(2)^{1/2}R}^{(2)^{1/2}R}( 2R^2 + x^2 )dx $$

Many thanks!

You need to put some parantheses inside the integral. Otherwise it does not make sense.

 

[member 731016]

The function for square root in latex is \sqrt{} Also, I wouldn't substitute for $\lambda$ just yet, its just more symbols to juggle IMO.

Ok thank you @erobz , that is good advice!