The system of nine small 3 kg objects is rotating at an angular speed of 6 rev/s. The objects are connected by light, flexible spokes that can be lengthened or shortened.
What is the new angular speed if the spokes are shortened from 6 m to 2 m?
To calculate the new angular speed, we can use the principle of conservation of angular momentum.
Angular momentum (L) is given by the formula L = Iω, where I is the moment of inertia and ω is the angular speed.
The moment of inertia (I) for a system of point masses rotating around a common axis can be calculated by summing the individual moments of inertia of each mass.
For a point mass rotating around an axis at a distance r, the moment of inertia is given by I = m*r^2, where m is the mass of the object.
In this case, the system consists of nine objects with a mass of 3 kg each. Before the spokes are shortened, the moment of inertia (I_initial) can be calculated as:
I_initial = m*r_initial^2 = 3 kg * (6 m)^2 = 108 kg*m^2
After the spokes are shortened to 2 m, the moment of inertia (I_final) can be calculated as:
I_final = m*r_final^2 = 3 kg * (2 m)^2 = 12 kg*m^2
The conservation of angular momentum states that the initial angular momentum (L_initial) is equal to the final angular momentum (L_final). Therefore, we can write:
I_initial * ω_initial = I_final * ω_final
Solving for ω_final:
ω_final = (I_initial * ω_initial) / I_final
Plugging in the given values:
ω_final = (108 kg*m^2 * 6 rev/s) / 12 kg*m^2
Calculating:
ω_final = 54 rev/s
Therefore, the new angular speed is 54 rev/s when the spokes are shortened from 6 m to 2 m.
To find the new angular speed when the spokes are shortened, 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.
Since the system consists of nine small 3 kg objects, the total mass is given by:
mass = 9 * 3 kg = 27 kg
To find the moment of inertia, we need to consider the distribution of mass. In this case, since the objects are connected by light, flexible spokes, we can treat the system as a disc.
The moment of inertia of a disc is given by the equation:
I = (1/2) * m * r^2
Where m is the mass and r is the radius.
Initially, when the spokes are 6 m long, the radius of the disc is 6 m. Therefore, the moment of inertia is:
I_initial = (1/2) * mass * (6 m)^2
Now, when the spokes are shortened to 2 m, the radius of the disc becomes 2 m. Therefore, the moment of inertia is:
I_new = (1/2) * mass * (2 m)^2
Since the angular momentum is conserved, we can set the initial angular momentum equal to the new angular momentum:
I_initial * ω_initial = I_new * ω_new
Substituting the formulas for I_initial, I_new and the given angular velocity ω_initial, we can solve for ω_new:
(1/2) * mass * (6 m)^2 * 6 rev/s = (1/2) * mass * (2 m)^2 * ω_new
Simplifying the equation, we get:
36 * 36 * 6 rev/s = 4 * ω_new
2592 rev/s = 4 * ω_new
ω_new = 2592 rev/s / 4
ω_new = 648 rev/s
Therefore, the new angular speed when the spokes are shortened to 2 m is 648 rev/s.
conservation of angular momentum applies
Itotal*w=I'total*w'
but I' is changed by deltar^2, or (1/3)^2
w'=worig*9