I am getting two ways to find F. One is Fx0.2=30 Nm which gives F=150N Other is Fx0.2-640x0.13=0 which gives F=416N I am confused which one is incorrect. I know problem is too basic but still not clear where I am wrong. Question is shown in picture. PS: Sorry for such picture, don't have better one.
My interpretation is "the force causing the rod to pressure the hydraulic cylinder that results in 640 N compressive force in the rod/cylinder, and a net moment/torque of 30 Nm about the pivot point. If so, the force should be:
$0.2F + 30 - 640*0.13 = 0$
$F = 266 N$
Other than the 30 Nm moment, the pivot point has a horizontal force that is equal to $640 - F_x$, and a vertical force that is equal to $F_y$. $F_x$ & $F_y$ are component forces of $F$ in $x$ & $y$ axes respectively.
Note, the 640 N force was drawn as the applied force in the direction of action. For equilibrium consideration, its direction needs to be reversed.
Unless there are other clarifying statements on the text not being shown here, you are right. There is one extra piece of date here that crashes the equation. They have already supplied the wrong answer.
The question is wrong. If you start from the moment on the pedal you absolutely get 150 N force at the piston.
And if you accept640N force at the piston you must have F=416N.
They have clearly shown in the figure that both forces are measured perpendicular to the pivot and the direction of the force is parallel to the line of displacement.