A large turntable rotates about a fixed vertical axis, making one revolution in 6.00 s. The moment of inertia of the turntable about this axis is 1200 kg * mg^2. A childe of mass 40.0 kg, initually staning at the center of the turntable, runs out along a radius. What is the angular speed of the turntable when the child is 2.00 m from the center, assuming that you can treat the child as a particle?
dont answer this again i didn't mean to post it over... that dang copy button :)
To find the angular speed of the turntable when the child is 2.00 m from the center, we can use the principle of conservation of angular momentum.
The angular momentum of an object can be calculated as the product of its moment of inertia and its angular velocity. Mathematically, angular momentum (L) is given by:
L = I * ω
Where:
L is the angular momentum of the object
I is the moment of inertia of the object
ω is the angular velocity of the object
In this case, the moment of inertia of the turntable about its vertical axis is given as 1200 kg * m^2, and the initial angular velocity (ω) of the turntable is the unknown we want to find.
When the child runs out along a radius, the moment of inertia of the system changes because the child is no longer at the center. However, we can treat the child as a point mass, which means we can ignore the change in moment of inertia of the turntable and treat it as constant.
Since the child is treated as a particle, its angular momentum can be calculated as the product of its mass (m) and its linear velocity (v), multiplied by the distance (r) from the center of rotation. Mathematically, the angular momentum of the child (L_child) is given by:
L_child = m * v * r
The angular momentum of the system before and after the child moves can be assumed to be conserved. Therefore, the initial angular momentum of the turntable-child system is equal to the final angular momentum of the turntable-child system.
Initially, the child is at the center of the turntable, so its distance from the center of rotation is 0. Therefore, the initial angular momentum is:
L_initial = I * ω_initial
Finally, when the child moves to a distance of 2.00 m from the center, the final angular momentum is:
L_final = (I + m * r^2) * ω_final
Since we assumed the angular momentum is conserved, we can equate the initial and final angular momenta and solve for ω_final.
I * ω_initial = (I + m * r^2) * ω_final
Substituting the given values:
I = 1200 kg * m^2
m = 40.0 kg
r = 2.00 m
We can now solve for ω_final:
I * ω_initial = (I + m * r^2) * ω_final
(1200 kg * m^2) * ω_initial = (1200 kg * m^2 + (40.0 kg * (2.00 m)^2)) * ω_final
To find ω_final, simply rearrange the equation and solve for ω_final:
ω_final = (I * ω_initial) / (I + m * r^2)
Substitute the given values to find the angular speed of the turntable when the child is 2.00 m from the center.