A uniform, 0.0300-kg rod of length 0.400 m rotates in a horizontal plane about a fixed axis through its center and perpendicular to the rod. Two small rings, each with mass 0.0200 kg, are mounted so that they can slide along the rod. They are initially held by catches at positions 0.0500 m on each side of the center of the rod, and the system is rotating at 48.0 rev/min. With no other changes in the system, the catches are released, and the rings slide outward along the rod and fly off at the ends. What is the angular speed (a) of the system at the instant when the rings reach the ends of the rod; (b) of the rod after the rings leave it?

The conservation of angular momentum allows finding the results for the angular velocity of the system are:

     a) With the rings at the ends of the rod. w = 12 rpm

     b) When the rings leave the rod. w = 15 rpm.

Given parameters.

To find.

    a) Angular velocity with the rings at the ends.

    b) Angular velocity without rings.

The angular momentum is equivalent to the linear momentum for rotation, it is a vector magnitude, which is conserved if the system is isolated.

        L = I w

where L is the angular momentum, I the moment of inertia and w the angular velocity.

The symmetrical bodies moment of inertia is tabulated.

a) We define the system to be formed by the rod and the rings, therefore the system is isolated and the angular momentum is conserved.

Initial instant. Before releasing the rings

       L₀ = (I₁ + 2 I₂₀) w₀

The two comes from having two rings.

Final instant. When the rings reach the end of the rod.

Angular momentum is conserved.

We calculate

       I₁ =  0.03 0.4² = 4 10-4 kg m²

       I₂₀ = 0.02 0.05² = 5 10-5 kg m²

        = 0.02 0.2² = 8 10-4 kg m²

We substitute

        w = 12 rpm

b) when the rings have left the rod  

Initial instant. With the rings at the ends of the rod.

       L₀ = (I₁ +2 )  

Final moment. Without the rings

Let's calculate.

       w = 15 rpm

In conclusion using the conservation of angular momentum we can find the results for the angular velocity of the system are:

     a) With the rings at the ends of the rod, w = 12 rpm

     b) When the rings leave the rod,  w = 15 rpm.

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A) ω₂ = 28 rev/min

B) ω_3 = 28 rev/min

Explanation:

A) Initial moment of inertia (I₁) is the rod's own ((1/12)ML²) plus that of the two rings (two lots of mr² if we treat each ring as a point mass).

Thus;

I₁ = ((1/12)ML²) + 2(mr²)

Where;

M is mass of rod = 0.03 kg

L is length of rod = 0.4m

m is mass of small rings = 0.02 kg

r is radius of small rings = 0.05 m

Thus;

I₁ = ((1/12) x 0.03 x 0.4²) + 2(0.02 x 0.05²)

I₁ = 0.0021 kg.m²

Now, when the rings slide and reach the ends of the rod i.e at r = L/2 = 0.4/2 = 0.2m from axis, the new Moment of inertia is;

I₂ = ((1/12) x 0.03 x 0.4²) + 2(0.02 x 0.2²)

I₂ = 0.0036 kg.m²

From conservation of angular momentum, we know that:

I₁ω₁ = I₂ω₂

We are given ω₁ = 48 rev/min.

Thus; plugging in the relevant values;

0.0021 x 48 = 0.0036ω₂

0.1008 = 0.0036ω₂

ω₂ = 0.1008/0.0036

ω₂ = 28 rev/min

b) For this, we have the same scenario as the case above where the ring just reaches the ends of the rods.

Thus,

I₂ω₂ = I_3•ω_3

So,0.0021 x 48 = 0.0036ω_3

ω_3 = 28 rev/min