Tension in the string in a moveable mass-pulley system In the following diagram, what's the force that's acting on $m$, the Tension in the string? There is no friction between $M$ and the ground but there is between $m$ and $M$ and I need to find the minimum value of $F$ for which $m$ slides on $M$, but my problem is that I'm not sure what the Tension that acts on $m$ is... Is it $T=F$ or $T=F*sin(angle)$?

 A: The tension is just $T$, because the tension at every point of the inextensible string is the same.
think about it : if the tension acting on $m$ was $T \sin(\alpha)$ it would be $0$ when $\alpha = 0$, which isn't the case.
A: I think the problem consists three assumptions:


*

*The rope is mass-less.

*The rope is inelastic.

*There is no friction between rope and pulleys.
So, you can determine the tension of the rope by considering to its free body diagram:

If we isolate an arbitrary infinitesimal element of the rope, then the forces those acting along the element length must cancel each other. Because the rope is mass-less ($m=0\;\Rightarrow\;\Sigma F=ma=0$)
I.e. for any arbitrary infinitesimal element of the rope, we have $T(s)=T(s+\mathrm ds)$
So, we can obviously see that $F=T_0$ because
$$F=T(l-\mathrm ds)=T(l-2\mathrm ds)=\cdots =T(0+\mathrm ds)=T(0)=T_0$$
Not that $\color{blue}{\mathrm dN}$s are perpendicular to the rope and there is no friction between rope and pulleys.
A: As this seems to be a homework question I will just give a couple of hints.
The wheels which guide the string are probably assumed to be without any friction. Therefore the tension within the string is just the external force $F$.
The force on the big mass $M$ will depend on the angle. Imagine pulling very softly at $\alpha = 0^\circ$. Then you would not overcome the friction $\mu$ but slightly accelerate $M$ and also $m$. If you pull at $\alpha = 90^\circ$, then you are not accelerating the mass $M$ forward at all. You have to pull until you overcome the stick-friction $\mu$ and then you can pull the mass $m$. The tension in the rope will just be the caused by the friction $\mu$ and the acceleration of $m$.
Now if $\alpha$ is somewhere in between, you have to think what happens and what depends on the angel in what way.
