A yo-yo is composed of two disks of radius a , with many layers of string wrappe

Question

A yo-yo is composed of two disks of radius a, with many layers of string wrapped around the axle. The radial distance from the center of the axle to the top layer of string at any instant is b. With the end of the string wrapped around your finger and b/a 1, you let the yo-yo drop, unwinding string and ultimately spinning pretty fast when it reaches the bottom of the string (b/a 0). You begin to wonder how the motion of the yo-yo as the string unwinds depends on the value of b. Determine the magnitude of the yo-yo acceleration as a function of b. Express your answer in terms of the variables a, b, and the acceleration due to gravity g.

Explanation / Answer

Free body diagram in the vertical direction: T - mg = -ma
Taking moments about the center of the yo-yo: Tb = I
T = mg - mA (A is acceleration because I have to use a for the radius of the yo-yo)
T = I/b
A = b or = A/b
I = 0.5ma^2
From T = mg - mA = IA/b^2 = 0.5ma^2(A)/b^2
m's cancel
g - A = 0.5Aa^2/b^2
g = A(0.5a^2/b^2 + 1) = A[(0.5a^2 + b^2)/b^2]
A = gb^2/(0.5a^2 + b^2)

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