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To solve this problem it is necessary to use the conservation equations of both kinetic, rotational and potential energy.
By definition we know that
Where,
KE =Kinetic Energy
KR = Rotational Kinetic Energy
PE = Potential Energy
In this way
Where,
m = mass
v= Velocity
I = Moment of Inertia
Angular velocity
g = Gravity
h = Height
We know as well that for velocity (v) and Radius (r)
Therefore replacing we have
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Therefore the height must be 0.3915 for the yo-yo fall has a linear speed of 0.75m/s