- #36

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No worries @erobz! What was the mistake?erobz said:I goofed. Should be ##\frac{\sqrt{2}}{2} R##

Many thanks!

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In summary: 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!I think you might want to check the limits of integration. Remember that x is the distance from the center of the square. Also, I think you will be able to simplify the integrals a bit more as you go along. Good

- #36

member 731016

No worries @erobz! What was the mistake?erobz said:I goofed. Should be ##\frac{\sqrt{2}}{2} R##

Many thanks!

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- #37

erobz

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##a^2 + a^2 = R^2##Callumnc1 said:

##2a^2 = R^2##

##\implies a = \frac{R}{\sqrt{2}} = \frac{\sqrt{2}}{2} R##

Just carless algebra...late at night for me.

- #38

member 731016

No worries! We all make algebra mistakes! Thank you for your reply @erobz!erobz said:##x^2 + x^2 = R^2##

##2x^2 = R^2##

##\implies x = \frac{R}{\sqrt{2}} = \frac{\sqrt{2}}{2} R##

Just carless algebra...late at night for me.

So, I'm just trying to understand how you got ## y\hat j = x\hat j##. I have drawn a grey square below.

I'm trying not to use circular reasoning, but did you get ## y\hat j = x\hat j## because ##\vec r## will always form a right isosceles triangle since it is a square so ##\theta = 45##?

Many thanks!

- #39

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You should be able to work that out yourself. R is the distance from the centre of the square to one corner. How far is it from the centre of the square to the centre of one side? If x is a displacement along a side from its midpoint, what is the range of x?Callumnc1 said:

- #40

member 731016

Thank you for your reply @haruspex!haruspex said:You should be able to work that out yourself. R is the distance from the centre of the square to one corner. How far is it from the centre of the square to the centre of one side? If x is a displacement along a side from its midpoint, what is the range of x?

I was trying to consider cases where ##\vec r ≠ R## for when the position vector is pointing somewhere along to the mass element on the sides not to the corners.

When ##\vec r = R##, I can see that since it square then each side of the right triangle will have equal sides of length ##x##.

I think the range of x for that case would be from ##0## at the midpoint to ##\frac {R}{\sqrt{2}}## at the centre of one side.

Dose this special case hold for ##\vec r ≠ R##?

Many thanks!

- #41

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Why do you want to solve this using vectors? It's straightforward using scalars and the parallel axis theorem.Callumnc1 said:Thank you for your reply @haruspex!

I was trying to consider cases where ##\vec r ≠ R## for when the position vector is pointing somewhere along to the mass element on the sides not to the corners.

When ##\vec r = R##, I can see that since it square then each side of the right triangle will have equal sides of length ##x##.

I think the range of x for that case would be from ##0## at the midpoint to ##\frac {R}{\sqrt{2}}## at the centre of one side.

Dose this special case hold for ##\vec r ≠ R##?

Many thanks!

Edit: rereading your post, I don’t understand it. How are you defining ##\vec r## and x?

- #42

erobz

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- #43

erobz

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The vector ## \vec{r}## varies, in both magnitude and direction, but for that horizontal segment It will always have the form:Callumnc1 said:No worries! We all make algebra mistakes! Thank you for your reply @erobz!

So, I'm just trying to understand how you got ## y\hat j = x\hat j##. I have drawn a grey square below.

View attachment 322331

I'm trying not to use circular reasoning, but did you get ## y\hat j = x\hat j## because ##\vec r## will always form a right isosceles triangle since it is a square so ##\theta = 45##?

Many thanks!

$$ \vec{r} = x ~{\hat i} + \frac{\sqrt{2}}{2} R ~{\hat j}$$

- #44

- #45

member 731016

Thank you for your reply @haruspex !haruspex said:Why do you want to solve this using vectors? It's straightforward using scalars and the parallel axis theorem.

Edit: rereading your post, I don’t understand it. How are you defining ##\vec r## and x?

Sorry, let me try to explain it better.

##\vec r_1 = \sqrt {x^2 + y_1^2} ## since it has a ##x\hat i## and ##y_1\hat j## component. However since ##y_1 = x## in this special case then ##\vec r_1 = \sqrt {2x^2} = R ##

For this other vector their y-component is not ##x##

##\vec r_2 = \sqrt {x^2 + y_2^2} ≠ R## (cannot equal R since ##\vec r_2 < \vec r_1 ##)

##\vec r_3 = \sqrt {x^2 + y_3^2} ≠ R##

So what I was saying is that we should consider the general case where,

##\vec r = \sqrt {x^2 + y^2} ##

Where x is a constant equal to ##\frac {\sqrt {2}}{2}R## and y is a variable position vector component in the y-direction.

However, if we choose a different coordinate system, such as what @erobz has done in post #43

Then now y a constant equal to ##\frac {\sqrt {2}}{2}R## and x is variable position vector component in the y-direction.

I think the confusing thing about this problem is choosing the right side to find the moment of inertia for and defining the right coordinate system. What do you think?

How would you solve this using the parallel axis theorem out of curiosity?

Many thanks!

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- #46

member 731016

- #47

member 731016

Thank you @erobz!erobz said:

I think I now understand where you get that constant from now. So basically, you assume that both sides of the triangle are equal which means that angle must be 45 degrees.

So ##\sin45 = x/R## which gives your result.

Is that how you did it?

Many thanks!

- #48

member 731016

Thank you for that @erobz!erobz said:The vector ## \vec{r}## varies, in both magnitude and direction, but for that horizontal segment It will always have the form:

$$ \vec{r} = x ~{\hat i} + \frac{\sqrt{2}}{2} R ~{\hat j}$$

- #49

erobz

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Yes, that will always be the case for a square.Callumnc1 said:Thank you @erobz!

I think I now understand where you get that constant from now. So basically, you assume that both sides of the triangle are equal which means that angle must be 45 degrees.

View attachment 322347

So ##\sin45 = x/R## which gives your result.

Is that how you did it?

Many thanks!

- #50

nasu

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For one side of the square, the moment of inertia relative to the center is just $$\int_{-L/2}^{L/2} [(L/2)^2+x^2]\lambda dx =\lambda L (L/2)^2+\lambda \frac{x^3}{3}\Big|_{-L/2}^{L/2} =\lambda L (L/2)^2+\lambda \frac{2L^3}{3\times 8}$$

I considered the x axis parallel to the side I am looking at.

Using ##M=\lambda L## you get

$$I_{side} =m\frac{L^2}{4}+m \frac{L^2}{12}$$

The last term is the moment of inertia for an axis going through the center of the bar (look it up in a table) and the first term is what you add by using the parallel axis theorem. You could write this right away, without any integral.

Then the monemt of the square is just ##I_{square}=4I_{side}##.

Only now is time to see what happens if ##L=R\sqrt{2}##.

- #51

member 731016

Thank you for your reply @erobz !erobz said:Yes, that will always be the case for a square.

- #52

member 731016

Thank you very much for that @nasu!nasu said:

For one side of the square, the moment of inertia relative to the center is just $$\int_{-L/2}^{L/2} [(L/2)^2+x^2]\lambda dx =\lambda L (L/2)^2+\lambda \frac{x^3}{3}\Big|_{-L/2}^{L/2} =\lambda L (L/2)^2+\lambda \frac{2L^3}{3\times 8}$$

I considered the x axis parallel to the side I am looking at.

Using ##M=\lambda L## you get

$$I_{side} =m\frac{L^2}{4}+m \frac{L^2}{12}$$

The last term is the moment of inertia for an axis going through the center of the bar (look it up in a table) and the first term is what you add by using the parallel axis theorem. You could write this right away, without any integral.

Then the monemt of the square is just ##I_{square}=4I_{side}##.

Only now is time to see what happens if ##L=R\sqrt{2}##.

- #53

member 731016

##I_{side}= \frac {2mR^2}{4} + \frac{2mR^2}{12} ##

##I_{side} = \frac {mR^2}{2} + \frac{mR^2}{6} ##

##I_{square} = 4[\frac {mR^2}{2} + \frac{mR^2}{6}]##

##I_{square} = 2mR^2 + \frac{2mR^2}{3}##

##I_{square} = \frac{8mR^2}{3}##

Many thanks!

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