Could Someone Check This Answer?

kitz

Hello!!

I did this problem and have gotten it wrong-- but my math is right. Maybe I'm missing something? Here's the question:

We want to support a thin hoop by a horizontal nail and have the hoop make one complete small-angle oscillation each 2.0 s. What must the hoop's radius be?

So this is a physical pendulum, and in order to find the radius, I need to know the period. The period is found by:

$$T=2\pi\sqrt{\displaystyle{\frac{I}{mgd}}}$$
Where I is the moment of inertia, g is the acceleration due to gravity, and d is the distance from the pivot point to the center of gravity.

For a hoop, the moment of inertia is: $$I=MR^2$$
And the center of gravity is in the center of the hoop, so d is the radius.

So with that information, I set up the following:

$$T=2\pi\sqrt{\displaystyle{\frac{MR^2}{MgR}}}$$
$$T=2\pi\sqrt{\displaystyle{\frac{R}{g}}}$$

And in plugging in the values, I get:

$$2=2\pi\sqrt{\displaystyle{\frac{R}{9.8}}}$$
Solving for R, I get .9929475

Which, when I plug into my equation, I get very close to 2 for my period. However, this is incorrect...

Any ideas as to why?

Thanks!!!!

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kitz

That makes sense. How would I go about finding the moment of inertia about the nail?

...This is a longshot, but would the moment of inertia be 2MR^2??
(using the parallel-axis theorem...)

kitz

...There was a reply here a minute ago... ...That suddenly disappeared...

StatusX

Homework Helper
Sorry, that was me. I wasn't sure about my answer because I haven't done this stuff in a while, but you can try it if you want. And yes, that is the correct moment of inertia around the nail.

CarlB

Homework Helper
Yes, you're right, you have to adjust the MI by adding the appropriate amount because you're spinning the hoop around a spot on the rim instead of its center.

If I recall, the formula for adjusting the MI is to take the formula you look up, and add to it $$md^2$$, where "m" is the mass of the object, and "d" is the distance away from the center of mass. So you get $$MR^2+MR^2 = 2MR^2$$, like you suspected.

Carl

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