How Fast Must a Bullet Travel to Tip a Rotating Cube?

[MickeyGoh]

Homework Statement

A solid cube of wood with side 2a and mass M is resting on a horizontal surface. The cube is constraint to rotate about a fixed axis AB. A bullet of mass m and speed v is shot at the face opposite ABCD at a height of 4a/3. The bullet becomes embedded in the cube. Find the minimum value of speed v required to tip the cube so that it falls on the face ABCD. Assuming m << M.


2. Homework Equations
I = MR² (the integral one)
KE = 0.5 mv²
Force*height = Moment of Inertia*angular acceleration
(i am not sure)

The Attempt at a Solution

Actually, I have just learned moment of inertia and rotational motion. And teacher just give me this. I have difficulties in finding moment of inertia of a cube. I just can't get the result something like ms²/6 if the axis passes through center.

I am confused... I know that for a sphere we can make it into a very thin shell that dV = 4 πr²dr
But how can I do it when it comes to cube? The r seems different when it touches the side and the corner?

https://fbcdn-sphotos-h-a.akamaihd.net/hphotos-ak-xpa1/v/t34.0-12/10681646_10203072127269453_1684323490_n.jpg?oh=38647e54fa690c109ae2d505e3bff890&oe=5434F8D3&__gda__=1412758462_e1726fd93896039363a30fbcd178af5e

 

[SteamKing]

You are having difficulty calculating the MOI of the block because the definition of MOI you are using is not quite general enough.

The MOI is calculated as defined in the following article:

http://www.efunda.com/math/solids/massmomentofinertia.cfm

This article contains a list of MOI for common shapes:

http://en.wikipedia.org/wiki/List_of_moments_of_inertia

Be aware that these MOI are the values about axes whose origin is at the centroid of the body. If the body is not being rotated about its centroid, then you must use the parallel axis theorem to find the MOI about the axis of rotation.

 

[MickeyGoh]

So for a cube, x, y and z must be used?
How can I apply parallel-axis theorem in this?
I know it states that

I = I_cm + MD²
But in this case, how do I measure D? From where to the axis? Corner?

 

[SteamKing]

So for a cube, x, y and z must be used?

Yes, if you wish to derive the MOI for the block. However, you can also use the dimensions of the block
and the appropriate formula in the table attached to my previous post.

How can I apply parallel-axis theorem in this?
I know it states that

I = I_cm + MD²
But in this case, how do I measure D? From where to the axis? Corner?

Presumably, you'll start with the MOI of the block about an axis running thru its centroid. D must be the distance between the centroidal axis and the axis of rotation. The axis through the centroid and the axis of rotation must be parallel to one another, in order to apply the parallel axis theorem, so there is only one distance which can be used.

 

[MickeyGoh]

thank you very much
I'll try my best