A children's merry-go-round in a park consists of a uniform 200kg disk rotating about a vertical axis. the radius of the disk is 6 m and a 100kg man is standing on the outer edge when the disk is rotating at a speed of 0.20rev/s. how fast will the disk be rotating if the man walks 3m in toward the center along a radial line?
MomentofINertia= 1/2 massdisk*r^2+massman*distanceheis
So figure the moment of inertia at each point he stands.
Conservation of angular momentum applies
originalmomentum=finalmomentum
Iorig*worig=Ifinal*wfinal
To solve this problem, we can use the principle of conservation of angular momentum. The angular momentum of the system remains constant unless acted upon by an external torque.
The angular momentum (L) of a rotating object is given by the formula: L = Iω, where I is the moment of inertia and ω is the angular velocity.
In this case, the moment of inertia (I) of the disk is given by the equation: I = (1/2)mr^2, where m is the mass of the disk and r is the radius of the disk.
Given:
Mass of the disk (m) = 200 kg
Radius of the disk (r) = 6 m
Mass of the man (m_man) = 100 kg
Initial angular velocity (ω_initial) = 0.20 rev/s
First, let's calculate the initial angular momentum (L_initial) of the system with the man at the outer edge of the disk:
L_initial = I_initial * ω_initial
The moment of inertia of the disk can be calculated using the formula mentioned above:
I_initial = (1/2) * m * r^2
Substituting the given values, we have:
I_initial = (1/2) * 200 kg * (6 m)^2
Now, calculate I_initial.
Next, we can calculate L_initial and set it equal to the final angular momentum (L_final) when the man walks 3 m towards the center along a radial line. Let's assume the final angular velocity to be ω_final.
L_initial = L_final
I_initial * ω_initial = I_final * ω_final
Now, we need to find the new moment of inertia (I_final) and solve for ω_final.
The mass of the system after the man moves towards the center is the sum of the disk mass and the man's mass:
m_final = m_disk + m_man
Now, we can calculate I_final using the formula mentioned above:
I_final = (1/2) * m_final * r_final^2
Here, r_final is the new distance of the man from the axis of rotation after walking 3 m towards the center along the radial line. It can be calculated as:
r_final = r_initial - d
Where d is the distance the man walked towards the center, which is 3 m in this case.
Now, substitute the values into the equation to calculate r_final.
Now, we have I_final in terms of ω_final, and we can solve the equation:
I_initial * ω_initial = I_final * ω_final
Substitute the values into the equation.
Finally, solve the equation for ω_final to find the new angular velocity of the disk.
This will give you the answer to how fast the disk will be rotating if the man walks 3 m toward the center along a radial line.