How Can Angular Momentum Principles Solve Ice Skater and Rotating Disk Problems?

[jcumby]

I am having a lot of trouble understanding this! I'm not even sure how to begin these :(

First problem:
Two ice skaters, both of mass 60 kg, approach on parallel paths 1.4 m apart. Both are moving at 3.2 m/s with their arms outstretched. They join hands as they pass, still maintaining their 1.4 m separation, and begin rotating about one another. What is their angular speed?

I think that this has to do with moment of inertia and the distance from the axis of rotation, but I am confused.

Second problem:
Two disks, one above the other, are on a frictionless shaft. The lower disk, of mass 440 g and radius 3.4 cm is rotating at 180 rpm. The upper disk, of mass 270 g and radius 2.3 cm, is initially not rotating. It drops freely down the shaft onto the lower disk, and frictional forces act to bring the two disks to a common rotational speed. a.) what is that speed? b.) what fraction of the initial kinetic energy is lost to friction?

I think that I should be using energy considerations here, but I'm not sure how I should set this up.

Third problem:
Two small beads of mass m are free to slide on a frictionless rod of mass M and length l. Initially, the beads are held together at the rod center, and the rod is spinning freely with initial angular speed $$\omega$$₀ about a vertical axis. The beads are released, and they slide to the ends of the rod and then off. Find the expressions for the angular speed of the rod a.) when the beads are halfway to the ends of the rod b.) when they're at the ends, and c.) after the beads are gone.

I believe that I should treat the beads as point masses, but I am confused about where to go from here.

 

[tiny-tim]

Hi jcumby!

(have an omega: ω )

i] Don't bother about moment of inertia … you can treat them both as point masses.

Use conservation of angular momentum … the angular momentum before they touch is the same as the angular momentum when they're turning.

ii] Part a): You don't need energy, only conservation of angular momentum, = ∑ Iω.

Part b): Rotational KE = ∑ (1/2)Iω².

iii] Yes, point masses again.

And use conservation of angular momentum again (the radial speed of the beads doesn't matter, since it doesn't contribute to angular momentum )

 

[thomate1]

Dear student,

I understand that you are struggling with these angular momentum problems. Let me try to provide some guidance and explanation to help you understand them better.

First, let's start with the concept of angular momentum. Angular momentum is a measure of how much an object is rotating around a certain axis. It is calculated by multiplying the moment of inertia, which is a measure of how spread out the mass is from the axis of rotation, by the angular velocity, which is the rate at which the object is rotating.

In the first problem, we have two ice skaters rotating around each other after joining hands. To solve this problem, we need to use the conservation of angular momentum, which states that the total angular momentum of a system remains constant unless acted upon by external torque. In this case, we can assume that there is no external torque acting on the system, so the initial angular momentum of the two skaters must be equal to the final angular momentum after they join hands.

To calculate the initial and final angular momentum, we need to consider the moment of inertia and the angular velocity of the skaters. The moment of inertia for a point mass is given by mr^2, where m is the mass and r is the distance from the axis of rotation. Since the skaters are rotating at a distance of 1.4 m, we can calculate their initial and final moments of inertia using their masses. The initial angular velocity is given as 3.2 m/s, but after they join hands, they will have a new angular velocity, which we can calculate using the conservation of angular momentum equation.

In the second problem, we are dealing with the transfer of kinetic energy due to friction. To solve this problem, we need to use the conservation of energy principle, which states that energy cannot be created or destroyed, only transferred from one form to another. In this case, we can assume that the initial kinetic energy of the lower disk is transferred to the upper disk when it drops down and rotates at the same speed. To calculate the final rotational speed, we need to use the moment of inertia and the conservation of energy equation.

Lastly, in the third problem, we have a rod with two beads sliding on it. To solve this problem, we need to use the conservation of angular momentum again. The initial angular momentum of the system is equal to the final angular momentum after the beads slide off the rod. We can calculate the initial and final moments of inertia using the masses