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 R0 and a thickness D0. 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?
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 R0 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:
Iyoyo = 2⋅Ibig disk + Ismall disk
= 2⋅(ρπDR4) + ½ρπD0R04
= ρπ⋅(DR4 + ½⋅D0R04)
I quickly add the calculation for the moment of inertia of a disk:
Idisk = ρ∫ r2dV = ρDπr4/2 = ½ ⋅Mr2
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⋅R0 where α is of course the angular acceleration ⇒ α = aT/R0.
I also think aT = ay and Myoyo⋅g - I⋅α/R0 = Myoyo⋅g - I⋅ay/R02 = Myoyo⋅ay
⇒ ay = Myoyo⋅g / (Myoyo + I/R0)
But when I insert that in my torque equation, it gets crazy:
Στ = I⋅α = (I/r0)⋅(Myoyo⋅g / M + I/R0)
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?