Why Is My Calculation of Rotational Inertia Incorrect?

[sskk221]

Homework Statement

Three identical balls, with masses of M, 2M, and 3M, are fastened to a massless rod of length
L as shown. The rotational inertia about the left end of the rod is:

http://img13.imageshack.us/img13/1424/1234sjy.jpg

Homework Equations

I = MR^2

The Attempt at a Solution

I total = (3M)(0)^2 + (2M)(L/2)^2 + (M)(L)^2

I total = 3ML^2/2

It says the answer is 3ML^2/4 though. Thanks for the help

 

[Haths]

Hmm...

[tex]
I = mr^{2}
[/tex]

[tex]
I = (3M)0^{2} + 2M(\frac{L}{2})^{2}) + ML^{2}
[/tex]

Therefore;

[tex]
I = \frac{1}{2}ML^{2} + ML^{2}
[/tex]

[tex]
I = \frac{3}{2} ML^{2}
[/tex]

I get the same as you.

The definition of the moment of inertia is;

SUM( m_ir_i² ) So I'm preaty certian that we are following the correct method. So perhaps it is some arithmetical mistake we are both making.

2/4 is 1/2 yes and 1/2mr² + mr² is mr²(1/2 + 1)
(1/2 + 1) = 3/2. Nope. I think your answer book might be flawed.

UNLESS you are meant to use the distribution of mass in a sphere for the first mass (3M). Mindyou, because they havn't given you a radius for the sphere I would assume not.

Haths

 

[nlightner]

Your solution is incorrect because you have not taken into account the distance of each ball from the point of rotation. The correct equation for rotational inertia is I = Σmr^2, where m is the mass of each object and r is the distance of the object from the point of rotation. In this problem, the distance from the left end of the rod to the first ball (with mass M) is L/2, the distance to the second ball (with mass 2M) is L, and the distance to the third ball (with mass 3M) is 3L/2. Therefore, the correct equation for rotational inertia is:

I total = (M)(L/2)^2 + (2M)(L)^2 + (3M)(3L/2)^2

I total = ML^2/4 + 4ML^2 + 9ML^2/4

I total = 14ML^2/4

I total = 7ML^2/2

Therefore, the correct answer is 7ML^2/2. This can also be simplified to 3.5ML^2, which is equivalent to 3ML^2/2. So, both the given answer and your attempt at a solution are correct.