Analyzing the Vertical Acceleration of a Yo-yo Using Moment of Inertia

[Cepterus]

Homework Statement

A yo-yo consists of two cylinders with radius $R$, mass $M$ and height $H$ connected by a smaller cylinder of radius $r$, mass $m$ and height $h$.

Determine the yo-yo's acceleration in $z$-direction, using its moment of inertia.

Homework Equations

Moment of inertia of a solid cylinder: $I=\frac12mr^2$

The Attempt at a Solution

The yo-yo's total moment of inertia is $I_{\text{tot}}=\frac12mr^2+MR^2$, using the formula for solid cylinders. However, I don't know how to proceed from here.

 

[TSny]

Free body diagram.
$\sum F = ma$.
$\sum \tau = I \alpha$.

 

[BvU]

Hi Cep,

Could gravitational acceleration appear somewhere ?

 

[Cepterus]

Could gravitational acceleration appear somewhere ?

I guess so. The acceleration probably consists of both gravitational acceleration downwards and the acceleration upwards caused by the "user" of the yo-yo.
Is that what you mean?

 

[BvU]

Yes. Make a sketch where the driving force acts and how it results in a torque.