How Does Angular Velocity Change as String Length Reduces in a Rotating System?

In summary: I_{orig} ω_{orig} = I_{new} ω_{new}I_{orig} = mL^2I_{new} = m(L/2)^2 = (1/4)mL^2Hence, mL^2 ω_{orig} = (1/4)mL^2 ω_{new}ω_{new} = 4ω_{orig}To find the tension in the string:Consider a small element of the string of length dr at a distance r from the hole.The force on this element is given by dF = (mω^2r)dr (centripetal force)Integrating from r=0 to r=L/2,T
  • #1
Simfish
Gold Member
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This is a problem from Kibble Classical Mechanics, so it may be harder than it looks.

Homework Statement



10. A particle of mass m is attached to the end of a light string of length
L. The other end of the string is passed through a small hole and is slowly pulled through it. Gravity is negligible. The particle is originally spinning round the hole with angular velocity ω. Find the angular velocity when the string length has been reduced to L/2. Find also the
tension in the string when its length is r, and verify that the increase in
kinetic energy is equal to the work done by the force pulling the string
through the hole.

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Attempt:

[tex]PE_{orig} = (1/2) k L^2[/tex]

[tex]PE_{new} = (1/2) k (L/2)^2 = (1/8) k L^2[/tex]

So [tex]PE_{new} / PE_{old}[/tex] is 4.

Now, I know that rotational kinetic energy is E = 1/2 I ω^2. And the answer says that the new angular velocity is 4 times that of the old angular velocity. But how do I get that?

Also, how do I get the tension in the string? Tension comes from a force opposing an applied force. Also, I know that W = Fd = F*L/2. So... maybe [tex]F = W/d = \Delta PE/d = (7/8) k L^2 / (L/2) = (7kL/16)[/tex]. But that's still nowhere near the answer
 
Last edited:
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  • #2
just use angular momentum conservation
 
  • #3
Let particle and string be a system
Since no external torques act, angular momentum is conserved.
Hence,
 

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FAQ: How Does Angular Velocity Change as String Length Reduces in a Rotating System?

What is the tension of a spinning string?

The tension of a spinning string refers to the amount of force or pull that is being applied to the string as it spins.

How is the tension of a spinning string measured?

The tension of a spinning string can be measured using a variety of methods, such as using a tension meter or by calculating the force of the string based on its mass and speed.

What factors affect the tension of a spinning string?

The tension of a spinning string can be affected by factors such as the mass of the string, the speed at which it is spinning, and any external forces acting on the string.

Why is tension important in a spinning string?

Tension is important in a spinning string because it determines the stability and accuracy of the string's movement. Too much or too little tension can cause the string to break or produce incorrect results.

How can the tension of a spinning string be adjusted?

The tension of a spinning string can be adjusted by changing the speed at which it is spinning, changing the mass of the string, or by using external devices such as tensioners to control the amount of tension on the string.

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