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## Homework Statement

We have a rod (length

**L**, mass

**m**) suspended at a point whose distance from the center of mass is

**a**.

1) prove that (generally) there exist two values of a (a

2) derive and explain: [tex] T = 2\pi\sqrt{\frac{a_1+ a_2}{g}}[/tex]

3) with what value of a do we get the smallest period?

_{1}, a_{2}) for which the pendulum oscillates with the same period.2) derive and explain: [tex] T = 2\pi\sqrt{\frac{a_1+ a_2}{g}}[/tex]

3) with what value of a do we get the smallest period?

## Homework Equations

[tex] E = 1/2 I w^2 [/tex]

[tex]J= J^* + ma^2[/tex]

[tex]J= 1/12 mL^2 + ma^2 [/tex]

[tex] T = 2\pi\sqrt{\frac{J}{mga}}[/tex]

[tex] T = 2\pi\sqrt{\frac{L}{g}}[/tex]

## The Attempt at a Solution

**1)**I thought that two periods being the same would mean T

_{1}= T

_{2}

[tex] 2\pi\sqrt{\frac{J_1}{mga}}= 2\pi\sqrt{\frac{J_2}{mga}}[/tex]

[tex]J^* + ma_1^2 =J^* + ma_2^2[/tex]

[tex]a_1^2 =a_2^2[/tex]

[tex]a_1 = +/- a_2[/tex]

and since a

_{1}= a

_{2}would be the same location a

_{1}= - a

_{2}which would mean that the only way to get the same period for two different values of a, would be to have the a be the same but in different directions from the center of mass.

**2)**I noticed that [tex] T = 2\pi\sqrt{\frac{a_1+ a_2}{g}}[/tex] looks a lot like [tex] T = 2\pi\sqrt{\frac{L}{g}}[/tex]

So I thought that a

_{1}+ a

_{2}= L

and that that would mean that the rod would be suspended on one end and the mass would be concentrated on the other end of the rod? I guess it sounds silly but I just couldn't figure out anything else.

**3)**[tex] \lim_{x\rightarrow 0} T = 2\pi\sqrt{\frac{J}{mga}} = 2\pi\sqrt{\frac{1/12 L^2 + a^2}{ga}} [/tex]

since [tex] 2\pi\sqrt{\frac{1}{g}} [/tex] is a constant I'll derive the rest because a limit is just a special type of derivation (I think?)

[tex] \frac{1/2 L^2 + a^2}{a}[/tex] derived is:

[tex] La + a^2 + {\frac {L^2}{2}}[/tex]

and from that I can see that the smaller the a the smaller the period of the oscillation so we would get the shortest period if the pendulum was hung from its center of mass.

I was told my solution is wrong. I don't know which parts, I don't know why and I don't know how to correct it. If you could help me out somehow I'd be really grateful.

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