Conservation of Angular Momentum / Kinematics.

In summary, the conversation discusses a problem involving a tetherball with a mass of 2 kg, initial velocity of 20 m/sec, and initial length of 1.5 m. The cord is slowly drawn in until it has a length of 1/3 of its initial length. The problem involves finding the final velocity and angle of the cord. The conversation also mentions using the conservation of angular momentum and the relationship V = rdØ/dt to solve the problem, but the person is struggling to find a solution. They also mention that they have calculated the final velocity in symbolic form and are looking for hints or suggestions to solve the problem.
  • #1
holdenthefuries
5
0

Homework Statement



Consider the tetherball shown:
231975007.jpg


At t=0, the ball, of mass m, is moving in a horizontal circle with velocity Vo and has length Lo. The cord is now slowly drawn in until it has length Lo/3.

Take

Vo = 20 m/sec.
Lo = 1.5 m
m = 2 kg.

What is the final velocity of the ball, and what angle does the cord make with the verticle?

Homework Equations



This is a conservation of angular momentum problem... so IiWi = IfWf...

I am treating the tetherball as a single particle so therefore its moment of inertia = mr^2.

(=)

2((LoSin(Ø)^2))Wi = 2((LfSin(Ø+∆Ø)^2))Wf

R is given at all times by LxSin(Ø+∆Ø)... for our purposes where Lx is either Lo or Lo/3. Of course at Lo, ∆Ø=0.

Also, Tangential velocity = rw...

or:

V = r dØ/dT

The Attempt at a Solution



So, I have been working this problem for so long that I am starting to go a bit crazy, and seem to find myself sputtering around in a rut!

It is clear to me that I have been looking at this problem a bit poorly...

I am trying to set Vdt/r = dØ and integrating both sides... but I'm not sure that is a valid operation due to the sinØ inside the radius. If it is a valid operation, then theta = 20 do after completing the IVP... But from there, how do I find the final time?

I am completely lost on this one... I think.

ANY help would be vastly appreciated.

You don't want to see the pages of failed attempts... this is about where I am right now.

Thanks BUNDLES!
Sean
 
Last edited:
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  • #2
And as a side... I have calculated Vf in symbolic form to be:

60sinØ/(sin(Ø+∆Ø)

If I could just figure out what theta and time were, I'd be gold! Any hints to this would be perfect. Thanks!
 
  • #3
I think in this case, it would be easier if you assumed the final angle to be theta, and find the final speed (vf) in terms of theta, and using conservation of energy calculate vf (the loss in T=gain in Ug). You'll have two relations in theta and vf to solve for as everything else is a constant.
 

1. What is conservation of angular momentum?

Conservation of angular momentum is a fundamental principle in physics that states that the total angular momentum of a system remains constant as long as there are no external torques acting on the system. This means that the angular momentum of an object will not change unless a force acts upon it.

2. How is angular momentum defined?

Angular momentum is defined as the product of an object's moment of inertia (a measure of its rotational inertia) and its angular velocity (the rate at which it rotates around a fixed point). It is a vector quantity, meaning it has both magnitude and direction.

3. What is the difference between angular momentum and linear momentum?

Linear momentum is a measure of an object's motion in a straight line, while angular momentum is a measure of an object's motion around a fixed point. Linear momentum is a vector quantity, while angular momentum is a vector quantity.

4. How is conservation of angular momentum applied in real-world situations?

Conservation of angular momentum is applied in many real-world situations, such as the motion of planets and satellites in orbit, the spinning of a top, and the motion of a gyroscope. It is also used in sports, such as figure skating and diving, where athletes use their body's angular momentum to perform tricks and maneuvers.

5. How does conservation of angular momentum relate to rotational kinematics?

Conservation of angular momentum is closely related to rotational kinematics, as both involve the motion of objects in circular or rotational motion. Rotational kinematics deals with the relationships between angular position, velocity, and acceleration, while conservation of angular momentum focuses on the constancy of angular momentum in a system.

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