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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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