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Homework Statement
Hi everybody! I know that is a classical problem, but I haven't been able to find an answer to my questions in the other threads so here we go:
We have a yoyo made of 3 aluminium disks (density ρ). The two side disks have a radius R and thickness D, and the middle disk has a radius R_{0} and a thickness D_{0}. The string has length l (see picture).
a) Calculate the moment of inertia of the yoyo. Derive then the moment of inertia of a disk of mass M and radius r about its axis of symmetry.
b) Calculate the torque when the yoyo is "falling". In which direction does it point?
c) What is the angular frequency ω when the string is half unrolled?
Homework Equations
Moment of inertia, torque, tangential acceleration and so on.
The Attempt at a Solution
a) My first issue attempting to solve this problem is that I am unsure whether the radius R_{0} should be considered when calculating the moment of inertia of the yoyo. I am confused, because the string is responsible for the rotation but yet the yoyo is rotating about its axis of symmetry. Therefore I would assume the moment of inertia of the yoyo is:
I_{yoyo} = 2⋅I_{big disk} + I_{small disk}
= 2⋅(ρπDR^{4}) + ½ρπD_{0}R_{0}^{4}
= ρπ⋅(DR^{4} + ½⋅D_{0}R_{0}^{4})
I quickly add the calculation for the moment of inertia of a disk:
I_{disk} = ρ∫ r^{2}dV = ρDπr^{4}/2 = ½ ⋅Mr^{2}
Is that correct, or is there any need to use the parallel axis theorem?
b) Here it becomes complicated. I think the only force playing a role on the rotation is the tension force of the string, therefore:
Στ = I⋅α = F⋅R_{0} where α is of course the angular acceleration ⇒ α = a_{T}/R_{0}.
I also think a_{T} = a_{y} and M_{yoyo}⋅g  I⋅α/R_{0} = M_{yoyo}⋅g  I⋅a_{y}/R_{0}^{2} = M_{yoyo}⋅a_{y}
⇒ a_{y} = M_{yoyo}⋅g / (M_{yoyo} + I/R_{0})
But when I insert that in my torque equation, it gets crazy:
Στ = I⋅α = (I/r_{0})⋅(M_{yoyo}⋅g / M + I/R_{0})
When I plug my moment of inertia in, nothing good comes out and it gets very messy. Have I made a mistake or should I also put up an equation of energy?
Thank you very much in advance.
EDIT: I just saw a similar problem on internet where the guy takes the point of contact of the string with the disk as the torque axis. Is that really allowed?? If so, would I have to use the parallel axis theorem because we don't take the center of the yoyo as axis of rotation anymore?
Julien.
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