The system of small object is rotating at 2rad/s. If the object are connected by small light spokes that can be lengthened or shortened. What is the new angular speed if the spokes are shortened to 0.5m
The angular speed of the rotating system will increase when the spokes are shortened. This is because the moment of inertia of the system decreases, which means that the same angular momentum can be achieved with a smaller angular speed.
To calculate the new angular speed, we need to use the conservation of angular momentum:
I₁ω₁ = I₂ω₂
where I₁ and ω₁ are the moment of inertia and angular speed of the system before the spokes are shortened, and I₂ and ω₂ are the moment of inertia and angular speed after the spokes are shortened.
Since the object are small, we can approximate them as point masses and use the formula for the moment of inertia of a point mass:
I = mr²
where m is the mass of each object and r is the distance from the axis of rotation.
Let's assume that each object has a mass of 1 kg and the original distance from the axis of rotation is 1 m. Therefore, the moment of inertia of the system before the spokes are shortened is:
I₁ = 4(1 kg)(1 m)² = 4 kg·m²
When the spokes are shortened to 0.5 m, the new distance from the axis of rotation is halved, which means that the moment of inertia is also halved:
I₂ = 4(1 kg)(0.5 m)² = 1 kg·m²
Now we can solve for ω₂:
I₁ω₁ = I₂ω₂
(4 kg·m²)(2 rad/s) = (1 kg·m²)ω₂
ω₂ = (4 kg·m²)(2 rad/s) / (1 kg·m²)
ω₂ = 8 rad/s
Therefore, the new angular speed of the system is 8 rad/s when the spokes are shortened to 0.5 m.