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blalien
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[SOLVED] Playing with a yoyo (Angular momentum problem)
This problem is from Gregory. This is only the first part of the problem, since I understand how the solution to the first part implies the second part.
A uniform circular cylinder (a yo-yo) has a light inextensible string wrapped around it so that it does not slip. The free end of the string is fastened to a support and the yo-yo moves in a vertical straight line with the straight part of the string also vertical. At the same time the support is made to move vertically having upward displacement Z(t) at time t. Find the acceleration of the yo-yo.
a: the radius of the yo-yo
m: the mass of the yo-yo
I: moment of inertia of the yo-yo
v: velocity of the center of mass of the yo-yo
\omega: angular velocity of the yo-yo
g: acceleration of gravity
Z(t): the displacement of the support
T: the tension force the string exerts on the yo-yo
\tau: torque of the yo-yo
Since the yo-yo is a cylinder, I = ma^2/2
\tau = T*a = I \omega'
m v' = -mg + T + mZ''(t)
v = -a\omega
Work a little substitution magic:
m v' = -mg + I \omega'/a + mZ''(t)
m v' = -mg - ma^2/2 v'/a^2 + mZ''(t)
m v' = -mg - m v'/2 + mZ''(t)
3/2m v' = -mg + mZ''(t)
v' = -2/3g + 2/3Z''(t)
The solution given in the book is v' = -2/3g + 1/3Z''(t). So, somewhere along the line I missed something with Z''(t). Can anybody guess where I went wrong?
Homework Statement
This problem is from Gregory. This is only the first part of the problem, since I understand how the solution to the first part implies the second part.
A uniform circular cylinder (a yo-yo) has a light inextensible string wrapped around it so that it does not slip. The free end of the string is fastened to a support and the yo-yo moves in a vertical straight line with the straight part of the string also vertical. At the same time the support is made to move vertically having upward displacement Z(t) at time t. Find the acceleration of the yo-yo.
a: the radius of the yo-yo
m: the mass of the yo-yo
I: moment of inertia of the yo-yo
v: velocity of the center of mass of the yo-yo
\omega: angular velocity of the yo-yo
g: acceleration of gravity
Z(t): the displacement of the support
T: the tension force the string exerts on the yo-yo
\tau: torque of the yo-yo
Homework Equations
Since the yo-yo is a cylinder, I = ma^2/2
\tau = T*a = I \omega'
m v' = -mg + T + mZ''(t)
v = -a\omega
The Attempt at a Solution
Work a little substitution magic:
m v' = -mg + I \omega'/a + mZ''(t)
m v' = -mg - ma^2/2 v'/a^2 + mZ''(t)
m v' = -mg - m v'/2 + mZ''(t)
3/2m v' = -mg + mZ''(t)
v' = -2/3g + 2/3Z''(t)
The solution given in the book is v' = -2/3g + 1/3Z''(t). So, somewhere along the line I missed something with Z''(t). Can anybody guess where I went wrong?
