# Solving the Angular Momentum & Kinetic Energy Equations: Find ωa & ωb

• mcheung4
In summary, the problem involves a ring of mass Mb and radius b mounted to a smaller ring of mass Ma and radius a, both rotating about a perpendicular axis. Dust of mass Ms is distributed on the inner surface of Ma and flies out at a constant rate λ, sticking to the outer ring. The subsequent angular velocities of the two rings, ωa and ωb, can be found using the equations for conservation of angular momentum and kinetic energy. However, in this case, both velocities turn out to be zero, indicating that something may be incorrect in the solution. To better understand the problem, it might be helpful to consider a related situation where a person standing on a rotating platform releases masses held in their hands.
mcheung4

## Homework Statement

A ring of mass Mb, radius b, is mounted to a smaller ring of mass Ma, radius a and with the same centre, and they are free to rotate about an axis which points through this centre and is perpendicular to the rings. Dust of mass Ms is distributed uniformly on the inner surface of Ma. At t=0, Ma rotates clock-wise at angular speed Ω while Mb is stationary. At t=0, small perforations in the inner ring are opend, and the dust start to fly out at a constant rate λ and sticks to the outer ring. Find the subsequent angular velocities of the 2 rings ωa and ωb. Ignore the transit time of the dust.

## Homework Equations

Conservation of angular momentum.
Conservation of kinetic energy.

## The Attempt at a Solution

(Ma + Ms)a2Ω = (Ma+Ms-λt)a2ωa + (Mb+λt)b2ωb

(Ma + Ms)a2Ω2 = (Ma+Ms-λt)a2ωa2 + (Mb+λt)b2ωb2

I tried to solve the system and both velocities turned out to be zero. Am I doing anyhting wrong?

mcheung4 said:
At t=0, small perforations in the inner ring are opend, and the dust start to fly out at a constant rate λ and sticks to the outer ring. Find the subsequent angular velocities of the 2 rings ωa and ωb.

Hello, mcheung4.
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When a piece of dust flies out and sticks to the outer ring, what type of collision is that?

Would you expect kinetic energy to be conserved in this type of collision?

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Think about how the angular velocity of the inner ring will change with time. It might help to consider a related situation. Suppose you're standing on a platform that is free to rotate. You are initially rotating at an angular speed Ω with your arms outstretched and a mass m held in each hand. You then release the masses in your hands (without "throwing" the masses). What happens to your angular speed?

## 1. What are the equations for solving angular momentum and kinetic energy?

The equations for solving angular momentum and kinetic energy are:Angular momentum (L) = Moment of inertia (I) x Angular velocity (ω)Kinetic energy (K) = 1/2 x Moment of inertia (I) x Angular velocity (ω)^2

## 2. How do you find the values of ωa and ωb?

To find the values of ωa and ωb, you need to first determine the moment of inertia (I) for both objects. Then, use the equations mentioned in question 1 to set up a system of equations with the given values of angular momentum and kinetic energy. Finally, solve for ωa and ωb using algebraic methods.

## 3. Can you solve for ωa and ωb if only one value is given?

No, in order to solve for both ωa and ωb, you need to have two known values for either angular momentum or kinetic energy. This is because the equations for solving these values involve both ωa and ωb, so having only one known value will not provide enough information to solve for both variables.

## 4. How does changing the moment of inertia affect ωa and ωb?

Changing the moment of inertia (I) will directly impact the values of ωa and ωb. A larger moment of inertia will result in a smaller angular velocity, and vice versa. This is because a larger moment of inertia requires more force to rotate the object, resulting in a slower angular velocity.

## 5. What are some real-life applications of these equations?

These equations are frequently used in engineering and physics to analyze the motion of rotating objects. They can also be applied in fields such as robotics, aerospace, and sports to understand the movement and stability of objects. In everyday life, these equations can help explain phenomena such as the motion of a spinning top or the rotation of planets around the sun.

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