00:02
We're given this disk with an angular velocity omega, and we need to find the angle between the angular momentum about the point g of this disc and the axle of the given diagram.
00:19
So we're essentially going to have to find the equation for the angular momentum about this point g, and use that to find the angle that we need.
00:29
So we'll start the problem by writing the equations for the different components of the moment of inertia.
00:41
And keep in mind that in this free body diagram, there's going to be no z component of the angular momentum due to the direction that this angular velocity is given.
00:54
So for the x prime component of the moment of inertia of this disk, this is going to be equal to one -half mr squared and for the y prime direction we know that this is equal to 1 over 4 times mr squared for the angular velocity in the x prime direction this is equal to omega times the cosine of beta and for the angular velocity in the y prime direction this is equal to negative omega times the sign of beta so now for for the x component of the angular momentum, this is just the moment of inertia times the angular velocity, which we can simplify as mr squared times the cosine of beta.
02:18
So mr squared times omega times the cosine of beta divided by 2.
02:28
Now for the y prime component, so it's the moment of inertia times the angular velocity.
02:41
And we can simplify this as negative m r squared times omega times the sine of beta divided by 4.
03:04
So now the equation for the angular momentum about the point g, that's equal to the x component.
03:19
So i'll write it in black.
03:27
So angular momentum about the point g, that's equal to the x component, the x component i, minus the y component j so next what we're going to do is we're going to find the angle between this axis x prime and the angular momentum about point g so i'll call that angle phi so angle phi that's going to be equal to the tangent inverse of the y component of the angular momentum divided by the x component so rewriting that angle phi and plugging in those terms, you have the inverse tangent of the y component divided by the x component.
05:05
So divided by 4 and divided by 2...
