physics help! argg

[Omegachi]

i have this problem that i have to solve, and i can't get the answer right.

A uniform rectangle plate has mass m and sides a and b. (a) show by integration that the moment of inertia of the plate about an axis that is perpendicular to the plate and passes through one corner is 1/3m(a^2+b^2). (b) what is the moment of inertia about an axis that is perpendicular to the plate and passes through its center?

for part a, i can only get 1/3(a^2+b^2)^2. what ever happened to the power of 2?

cause to find the moment of inertia of that thing, the formula is integral of r^2dm.

 

[Omegachi]

bump

 

[Omegachi]

bump again

 

[MereMortal]

Here's a hint:

Take the plate to be in the x-y plane with a corner at (0,0). It is relatively easy to find the moment of inertia about the x and y axes through integration.

Then to find the moment of inertia about the z-axis, which is what you want, use the perpendicular axes theorem.

 

[Omegachi]

still can't find it. i keep on getting 1/3 m (a^2+b^2)^2.

the center of mass of the plate should be right in the center of the plate right? then the circlewill have r of (a/2)^2+(b/2)^2?

 

[MereMortal]

Okay, if you insist on doing a double integration (the method I explained above is two single integrations) 😉


You have a plate of mass M = s*a*b, where s=surface density, and {a, b} are the lengths of the sides. Thinking about this in terms of the CoM isn't really all that helpful here. Instead think of a mass element dM, being revolved about the z-axis.

Then dM = s*dx*dy. The mass element is a distance r=sqrt(x^2+y^2) from the axis of revolution, so the moment of inertia for dM is given by dI=r^2*dM = s*(x^2 + y^2)* dx*dy. To get the I for the plate, you just need to integrate dI over the surface of the plate.

Is this any clearer?

 

[MereMortal]

I don't know if I ended up helping you in any way, but here are a few things to keep in mind.

1) If you are going to use a center of mass approach, like you seemed to want to, then your axis of rotation must be at the CoM. You could solve part (a) of the problem with this approach, but I don't think that is what they are after. Plus it's extra work. Either of the ways that I suggested give trivial integrations with simple limits of integration (the CoM way isn't any harder, just messier algebraically--then you must use the parallel axis theorem to boot!).

2) I have no idea why you get (a^2+b^2)^2 without seeing how you set up the problem. 😉

3) The CoM becomes important in part (b), where you can use the parallel axis theorem to solve for I_cm. Then you need to use the fact that the axes of rotation are (a/2)^2+(b/2)^2 apart.

 

[Omegachi]

hmm.... so the radius, or the "c" of the plate is a+b? no wonder, i put a^2+b^2