To find the rotational inertia of the board plus the children about a vertical axis through the center of the board, we can use the parallel axis theorem. The parallel axis theorem states that the rotational inertia about an axis parallel to an axis through the center of mass is given by the sum of the rotational inertia about the center of mass and the product of the mass and the square of the distance between the two axes.
In this case, we have two children with mass m = 11.0 kg each, and a board with mass M = 9.4 kg. The length of the board is L = 4.5 m. The rotational inertia of each child is given by I_child = (1/3) * m * W^2, where W is the width of the board (0.30 m).
The rotational inertia of the board about its center of mass is given by I_board = (1/12) * M * (L^2 + W^2).
Using the parallel axis theorem, the total rotational inertia of the board plus the children about a vertical axis through the center of the board is:
I_total = I_board + 2 * I_child
Now we can substitute the values into the formula to find the rotational inertia.
I_total = (1/12) * 9.4 * (4.5^2 + 0.3^2) + 2 * (1/3) * 11.0 * 0.3^2
Now we can calculate the value of I_total.
Next, to find the magnitude of the angular momentum of the system, we can use the formula:
L = I_total * ω
where ω is the angular speed, given as 2.89 rad/s.
Substituting the values into the formula, we can calculate the magnitude of the angular momentum.
Finally, when the children pull themselves towards the center, the moment of inertia changes. According to the conservation of angular momentum, the angular momentum of the system remains constant.
So, we can write:
I_total * ω_before = I_total * ω_after
The children are now half as far from the center, so the new distance is L/2. We can calculate the new moment of inertia as described earlier using this new distance, and the angular speed is what we need to find.
Finally, to calculate the change in kinetic energy of the system as a result of the children changing their positions, we can use the formula:
ΔKE = KE_final - KE_initial
where KE_final is the final kinetic energy of the system and KE_initial is the initial kinetic energy of the system.
The kinetic energy of a rotating system is given by:
KE = (1/2) * I_total * ω^2
Using this formula, we can calculate the initial and final kinetic energy and then find the change in kinetic energy.
By following these steps, you should be able to find the rotational inertia, magnitude of the angular momentum, resulting angular speed, and change in kinetic energy for the given system.