The system of small objects shown in the figure below is rotating at an angular speed of 2.0 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 to 0.15 m? 1.0m x 1.0m distance.

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To determine the new angular speed of the system when the spokes are shortened to 0.15 m, we need to use the principle of conservation of angular momentum.

The angular momentum of a rotating system is given by the formula:

L = Iω

where L is the angular momentum, I is the moment of inertia, and ω is the angular speed.

In this case, the moment of inertia of the system remains constant since neither the masses nor their distribution change. Therefore, we can set up the equation for conservation of angular momentum as follows:

I1ω1 = I2ω2

where I1 and ω1 are the initial moment of inertia and angular speed, and I2 and ω2 are the final values.

Since the entire system is rotating as a single object, the moment of inertia I is given by:

I = mr²

where m is the mass of each object and r is the distance from the axis of rotation.

In the initial configuration, the distance r is given as 1.0 m, so we have I1 = m(1.0)² = m.

In the final configuration, the distance r is given as 0.15 m, so we have I2 = m(0.15)² = 0.0225m.

The initial angular speed ω1 is given as 2.0 rev/s. To convert it to rad/s, we multiply by 2π since there are 2π radians in one revolution. Therefore, ω1 = 2.0 rev/s * 2π rad/rev = 12.57 rad/s.

Now we can rearrange the equation for conservation of angular momentum to solve for ω2:

(12.57 rad/s)(m) = (0.0225m)(ω2)

By canceling out the mass term m, the equation simplifies to:

12.57 rad/s = 0.0225ω2

To find ω2, we can isolate it by dividing both sides of the equation by 0.0225:

ω2 = 12.57 rad/s / 0.0225 = 557.33 rad/s

Therefore, the new angular speed is approximately 557.33 rad/s when the spokes are shortened to 0.15 m.