Falling yo-yo attached to string

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SUMMARY

The discussion revolves around calculating the velocity of a giant yo-yo with a mass of 410 kg and a radius of 1.2 m, dropped from a height of 63 m, while unwinding from a string. The relevant equations include the conservation of energy equation \( mgh = 0.5Iw^2 + 0.5mv^2 \) and the moment of inertia \( I = 0.5MR^2 \). The user struggles with incorporating rotational energy and torque into their calculations, leading to an incorrect formula for velocity. The correct approach requires a clear understanding of the relationship between translational and rotational kinetic energy.

PREREQUISITES
  • Understanding of conservation of energy principles in physics
  • Familiarity with rotational motion concepts, including moment of inertia
  • Knowledge of angular velocity and its relationship to linear velocity
  • Basic algebra skills for manipulating equations
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  • Study the derivation of the conservation of energy equation in rotational motion
  • Learn about the relationship between torque, angular acceleration, and moment of inertia
  • Explore examples of similar problems involving rolling objects and energy conversion
  • Review the concept of angular momentum and its application in rotational dynamics
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Students studying physics, particularly those focusing on mechanics and rotational motion, as well as educators seeking to clarify concepts related to energy conservation in dynamic systems.

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1.Homework Statement

A giant yo-yo of mass 410 kg measuring about
1.2 m in radius was dropped from a platform
63 m high. One end of the string was tied
to the platform, so the yo-yo unwinds as it
descended. Assuming that the axle of the yo-yo has a
radius of 0.3 m, find the velocity of descent at
the end of the fall. The acceleration of gravity
is 9.81 m/s
Answer in units of m/s.


Homework Equations



M=410 kg
R=1.2 m
h=63 m
r=0.3 m
g=9.81 m/s2

mgh = 0.5Iw2 +0.5mv2

I = 0.5MR2

w = v/R

The Attempt at a Solution



So I've been trying to figure this problem out since last night. Please be patient with me. I've always struggled with problems involving rotational motion for some reason. If I'm understanding this right, the potential energy will be split up into the translational kinetic energy and the rotational kinetic energy. My problem is that I'm having a REALLY hard time visualizing how to break up the rotational energy. I'm assuming that my equation for I (the moment of inertia) is where my problem is because there's torque from gravity and tension from the string, but I don't really understand how I'm supposed to incorporate that.

In my first attempt at a solution, I substituted I and w into my equation for mgh, and after doing some algebra, I came up with v = 2*sqrt(gh/3) and that obviously gave me a wrong answer.

Can someone explain to me what I'm doing wrong conceptually?
 
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I really don't understand why my equation for I isn't working. And how should I properly break up the rotational energy in this problem? Thanks
 

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