How can I determine the instant the block overcomes static friction?

[hutchphd]

So finish the problem please. The proof is in the solution.

 

[Thermofox]

So finish the problem please. The proof is in the solution.

$\frac 1 2$ $k x_{max}^2$ + $\frac 1 2$ $I_c ω^2$ + $\frac 1 2$ $m_s ω^2 R^2$ = $F x_{max}$;

; $ω^2(m_s R^2 + I_c)$ + $k x_{max}^2$ = $2F x_{max}$;

; $ω^2$ = $\frac {2F x_{max} - k x_{max}^2} {m_s R^2+ \frac 2 5 m_s R^2}$ => $ω$ = $\sqrt {\frac {2F x_{max} - k x_{max}^2} { \frac 7 5 m_s R^2}}$ = $\sqrt {\frac {2~20~0.0491 - 200~(0.0491)^2} { \frac {21} 5 (0.3)^2}}$ $≈ 1,252~rad/s$

Then I was wondering that to find the remaning informations of the sphere (friction force on sphere $:= f_s$ and acceleration of center of mass $:= a_c$) I could just use the force balance equation of the sphere:
$$\begin{cases}
F + f_s - F_s = m_s a_c \\
N_s = m_s g \\
τ_{fs} = I_c α
\end{cases}$$
=> $f_s = \frac {I_c α} R$ and $a_c = \frac {F + f_s - F_s} {m_s}$ , since I know that $α = \frac {a_c} R$ I have 2 equations and 2 unknowns.

Edit: adjusted moment of inertia of the sphere

 

[haruspex]

$\frac 1 2 m_s R^2$

It's a sphere, presumably solid, not a cylinder.

Then I was wondering that to find the remaning informations of the sphere (friction force on sphere $:= f_s$ and acceleration of center of mass $:= a_c$) I could just use the force balance equation of the sphere:
$$F + f_s - F_s = m_s a_c $$

Which direction are you taking as positive for $f_s$?
$F_s$ is not constant.

 

[Thermofox]

It's a sphere, presumably solid, not a cylinder.

Which direction are you taking as positive for $f_s$?
$F_s$ is not constant.

Yeah I didn't think it through, I've always done problems with cylinders so in my mind the sphere transformed into a cylinder.
$f_s$ has the same direction of $F$, so I'm considering the right direction as positive
I should've said that I'm finding this two values the moment the block starts moving, so that I can use $F_{s,max}$ which is constant

 

[haruspex]

Yeah I didn't think it through, I've always done problems with cylinders so in my mind the sphere transformed into a cylinder.
$f_s$ has the same direction of $F$, so I'm considering the right direction as positive
I should've said that I'm finding this two values the moment the block starts moving, so that I can use $F_{s,max}$ which is constant

Ok. There is another equation relating $f_s$ and $a_c$.

 

[Thermofox]

Ok. There is another equation relating $f_s$ and $a_c$.

But I don't need it, I think.
$\begin{cases} f_s = \frac {I_c a_c} {R^2} \\ a_c = \frac {F + f_s - F_s} {m_s} \end{cases}$
If I substitute $f_s$ or $a_c$ into the other equation don't I get that value?

 

[haruspex]

But I don't need it, I think.
$f_s = \frac {I_c a_c} {R^2}$

That's the one.

 

[Thermofox]

That's the one.

Ok, so am I all good?

 

[haruspex]

Ok, so am I all good?

Yes.

 

[Thermofox]

Yes.

Thanks for helping!

 

[hutchphd]

Thanks, well done.

 

[TSny]

$\sqrt {\frac {2~20~0.0491 - 200~(0.0491)^2} { \frac {21} 5 (0.3)^2}}$ $≈ 1,252~rad/s$

Check the evaluation. (Maybe you already caught this.)