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Moment of Inertia Problems

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The three problems attached having to do with moment of inertia of things like a yo-yo.

I need help setting them up. Please send explanations with the answer so I can see how your answer was achieved.

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1)
Let us assume the tension in the string be T, acceleration be a, angular acceleration be alpha, and time taken be t.
r = 5 in = 5/12 ft
R = 10 in = 10/12 ft
moment of inertia I = 0.24 slug.ft^2 = 0.24*32.2 lb.ft^2
mass m = 50 lb
gravitational acceleration g = 32.2 ft/s^2

Net force
m*g - T = m*a .........(1)

Moment about the center,
T*r = I*alpha = I*a/r
=> T = I*a/r^2 .........(2)

From Equations. (1) and (2),
m*g = (m + I/r^2)*a
=> a = m*g/(m + I/r^2)

Kinematics: s = 4 ft; u = 0 (rest)

Because,
s = u*t + (1/2)*a*t^2
=> s = (1/2)*t^2*m*g/(m + I/r^2)
=> t^2 = 2*s*(m + I/r^2)/(m*g)
=> t^2 = 2*4*(50 + ...

Solution Summary

The solution provides comprehensive workings-out of the three moment of inertia problems given, including step-by-step break-downs of the necessary calculations and snippets of written explanation where needed.

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