- #1
nicnic344
- 1
- 0
Homework Statement
A large turnable shaped like a wooden board with sizes A and B and mass M rotates. A drop of water with mass m drops on 80cm from the axis of rotation. Find the final angular velocity of the system if the initial angular speed is ω.
Homework Equations
I know how to work this question out if it were to land on the edge of the turntable. However, I am not suree what to do if it lands 80cm away from the axis of rotation. I am not really looking for the answer, just the theory to work it out. I don't have any values... I just made the question up. Thanks!
The Attempt at a Solution
he angular momentum before the drop drops equals the angular momentum after the drop drops, or
Lb = La (b, a refer to before and after)
since angular momentum, L, = I w where I is the moment of inertia and w the angular velocity,we need to find the moment of inertia of a rectangle of sides, A, B
the moment of inertia of a rectangle around an axis perpendicular to the plane and passing throught the middle of the plane is
I=1/2 M(A^2+B^2)
so we have:
1/2 M(A^2+B^2) w = Ia wa
the moment of inertia after is the original moment of inertia + the moment due to the drop of water; for point masses (and we consider a drop a point mass), this contribution is mr^2 where r is the distance from the rotation axis
the pythagorean theorem tells us that the distance of m from the rotation axis is
r^2=(A/2)^2 +(B/2)^2 = 1/4(A^2+B^2)
so we have:
1/12 M(A^2+B^2) wb = [1/12M(A^2+B^2)+1/4 m(A^2+B^2)]wa
collect terms and solve for wa (notice, interestingly, that the (A^2+B^2) terms drop out)