Okay so I have a 1/2 in thick ellipsoidal steel plate where (p=493 lb/ft^3) and it has a major axis diameter of 4ft and a minor of 2ft. the plate rolls w/out slipping (friction=0.15) and in a vertical position has an angular velocity of of w=4 rad/s(i.e. the 4 ft diameter is along the y axis with the 2ft parallel to the x axis. I need to determine the angular velocity when the plate has rotated such that the major 4ft axis is horizontal.
To determine the angular velocity when the plate has rotated such that the major 4ft axis is horizontal, we can use the principle of conservation of angular momentum.
Angular momentum is given by the equation:
L = I * ω
Where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
The moment of inertia for an ellipsoid is given by:
I = (1/5) * m * (a^2 + b^2)
Where m is the mass of the plate, a is the semi-major axis, and b is the semi-minor axis.
First, we need to calculate the mass of the plate:
m = V * p
Where V is the volume of the plate and p is the density.
The volume of an ellipsoid is given by:
V = (4/3) * π * a * b^2
Next, we need to calculate the moment of inertia for the initial position where the plate is vertical. In this case, the semi-major axis (a) is 4ft and the semi-minor axis (b) is 2ft.
Now we can calculate the initial angular momentum (L_initial) using the given angular velocity (w) and moment of inertia (I_initial) for the vertical position.
Next, we need to calculate the final moment of inertia (I_final) for the plate when the major 4ft axis is horizontal. In this case, the semi-major axis (a) is 2ft and the semi-minor axis (b) is 4ft.
Finally, we can use the principle of conservation of angular momentum to calculate the final angular velocity (w_final) when the major axis is horizontal. Since angular momentum is conserved, we can set L_initial equal to L_final and solve for w_final.
I hope this explanation helps you understand the process of determining the angular velocity when the plate has rotated such that the major 4ft axis is horizontal.