1. A certain playground merry-go-round is essentially an 90-kg iron disk 2.0 m in radius. Two children, each with a mass of 46 kg, are originally standing on opposite sides of the disk near its outer edge. The merry-go-round is turning essentially without friction about a vertical axis once every 5 s. The children then clamber toward each other so that they are only 1.2 m from the center. What is its angular velocity now? (You do not need to specify its direction.)
Hint: Angular momentum is conserved.
The initial angular velocity is
w1 = (2*pi)/5 = 1.26 rad/s
Moment of inertia of the platform is
I = (1/2) M R^2 = 180 kg*m^2
There is additional angular momentum due to the children. Conservation of angular momentum requires that:
I*w1 + 2 m*5*2^2*w1 = I*w2 + 2*m*1^2*w2
m is each child's mass. Solve for w2
jacky
yo momma's a ho
To find the angular velocity of the merry-go-round after the children move towards the center, we can use the conservation of angular momentum.
Angular momentum, L, is defined as the product of the moment of inertia, I, and the angular velocity, ω:
L = I * ω
Initially, when the children are standing near the outer edge, the moment of inertia can be calculated as the sum of the moment of inertia of the disk and the two children:
I_initial = I_disk + 2 * I_child
Where:
- I_disk = 0.5 * m_disk * r_disk^2 (moment of inertia of a disk)
- I_child = m_child * r_child^2 (moment of inertia of a point mass)
Given:
- m_disk = 90 kg (mass of the disk)
- r_disk = 2.0 m (radius of the disk)
- m_child = 46 kg (mass of each child)
- r_child = 1.2 m (distance from the center)
Substituting the values into the formula, we get:
I_initial = (0.5 * 90 kg) * (2.0 m)^2 + 2 * (46 kg) * (1.2 m)^2
Now, we need to calculate the final moment of inertia, I_final, when the children are positioned 1.2 m from the center. Since the children have moved towards the center, their distance from the center is now equal to the radius of the disk.
Therefore, the final moment of inertia is:
I_final = I_disk + 2 * I_child
Substituting the values, we get:
I_final = (0.5 * 90 kg) * (2.0 m)^2 + 2 * (46 kg) * (1.2 m)^2
Now, we can use the conservation of angular momentum to find the final angular velocity, ω_final. The angular momentum should remain the same before and after the children move, so:
I_initial * ω_initial = I_final * ω_final
We are given that the initial angular velocity, ω_initial, is 1 revolution every 5 seconds. Since 1 revolution is equal to 2π radians, we can convert it to radians per second:
ω_initial = (1 revolution / 5 seconds) * (2π radians / 1 revolution) = 2π/5 radians/second
Substituting the values, we can solve the equation for ω_final:
(I_initial * ω_initial) / I_final = ω_final
Plug in the values for I_initial, I_final, and ω_initial, and calculate ω_final to get the final angular velocity of the merry-go-round.