Angular momentum conservation help

[snoggerT]

A bola consists of three massive, identical spheres conected to a common point by identical lengths of sturdy string (Fig. 11-51a). To launch this native South American weapon, you hold one of the spheres overhead and then rotate that hand about its wrist so as to rotate the other two spheres in a horizontal path about the hand. Once you manage sufficient rotation, you cast the weapon at a target. Initially the bola rotates around the previously held sphere at angular speed i but then quickly changes so that the spheres rotate around the commonconnection point at angular speed f (Fig. 11-51b).

(a) What is the ratio f /i?

(b) What is the ratio Kf /Ki of the corresponding rotational kinetic energies?



Li=Lf

The Attempt at a Solution

- I'm absolutely stumped on this problem. Mainly because I'm given no numbers and don't know how to get a ratio with just the equations. Where would I begin on a problem like this?

 

[learningphysics]

I believe the idea is that the angular momentum about the center of mass is conserved... gravity is the only force acting once the bola is released... and gravity doesn't exerts 0 torque about the center of mass of the system...

given the initial angular speed i... what is the angular momentum about the center of mass initially... you'll have to find the center of mass...

and what's the angular momentum about the center of mass afterwards...

finally set the initial angular momentum = final angular momentum... this will give a relationship between f and i...

 

[ogb p]

I understand your confusion and can provide some guidance on how to approach this problem. First, let's define some key terms and equations:

- Angular momentum (L): This is a measure of an object's rotational motion, calculated by multiplying its moment of inertia (I) by its angular velocity (ω). The equation is L = Iω.

- Moment of inertia (I): This is a measure of an object's resistance to rotational motion, based on its mass and distribution around its axis of rotation. For a solid sphere, the moment of inertia is given by I = (2/5)mr^2, where m is the mass and r is the radius.

- Angular velocity (ω): This is the rate of change of an object's angular displacement, measured in radians per second.

Now, let's apply these concepts to the given scenario. We are dealing with a bola, which consists of three identical spheres connected by strings. At the beginning, the bola is rotating around the sphere held overhead at a certain angular speed, let's call it ωi. When the bola is released, it quickly changes its rotation so that the spheres rotate around the common connection point at a different angular speed, ωf.

(a) To find the ratio f/i, we can use the equation L = Iω. Since the bola is rotating around a fixed point (either the held sphere or the common connection point), its angular momentum must be conserved. This means that Li = Lf, or Iiωi = Ifωf. Since the moment of inertia is the same for all three spheres and the strings are identical, we can cancel them out and rearrange the equation to get ωf/ωi = Ii/If. This gives us the ratio of the final and initial angular speeds.

(b) To find the ratio of the corresponding rotational kinetic energies, we can use the equation K = (1/2)Iω^2. Again, since the moment of inertia is the same for all three spheres, we can cancel it out and use the same ratio we found in part (a) to get Kf/Ki = (ωf/ωi)^2. This gives us the ratio of the final and initial kinetic energies.

I hope this helps to clarify the problem and guide you in finding a solution. Remember to always start by defining the key terms and equations, and then applying them to the given scenario