Angular momentum conservation help

In summary, the bola is a native South American weapon consisting of three massive spheres connected by sturdy string. To launch it, one rotates the spheres horizontally around a common point by holding one sphere overhead and rotating the hand. The ratio of the angular speeds before and after launching is f/i, and the ratio of the rotational kinetic energies is Kf/Ki. The conservation of angular momentum can be used to find the relationship between f and i.
  • #1
snoggerT
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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).
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(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?
 
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  • #2
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...
 
  • #3



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
 

What is angular momentum conservation?

Angular momentum conservation is a fundamental principle in physics that states that the total angular momentum of a system remains constant unless acted upon by an external torque. This means that the amount of angular momentum in a system cannot be created or destroyed, but can only be transferred between objects or converted into other forms of energy.

Why is angular momentum conservation important?

Angular momentum conservation is important because it helps us understand and predict the behavior of objects in motion. It allows us to analyze rotational motion and determine how forces and torques affect the motion of objects. This principle is also used in practical applications such as spacecraft navigation, gyroscopes, and spinning tops.

How is angular momentum conserved?

Angular momentum is conserved through Newton's third law of motion, which states that for every action, there is an equal and opposite reaction. In rotational motion, this means that the torque exerted on one object will result in an equal and opposite torque on another object, thus keeping the total angular momentum of the system constant.

What factors affect angular momentum conservation?

The factors that affect angular momentum conservation include the mass, velocity, and moment of inertia of the objects in the system. The greater the mass, velocity, or moment of inertia of an object, the greater its angular momentum will be. External torques, such as friction or air resistance, can also affect angular momentum conservation.

How is angular momentum conservation different from linear momentum conservation?

Angular momentum conservation is similar to linear momentum conservation in that they both state that the total amount of their respective quantities remains constant in a closed system. The main difference is that angular momentum deals with rotational motion, while linear momentum deals with straight-line motion. Additionally, angular momentum takes into account the distribution of mass and velocity in an object, while linear momentum only considers the overall velocity of an object.

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