The problem is the following:

The figure shows a block on a slope. Find the minimum force that should be applied to the block so that the body of mass $ m = 2 , kg $ as that body moves with constant speed up the slope. It is known that the coefficient of friction between surfaces is $ mu = 0.3 $ and the angle of inclination is $ alpha = 30 ^ { circ} $.

The alternatives given in my book are the following:

$ begin {array} {ll}

1. & 21 , N \

2. & 23 , N \

3. & 18 , N \

4. and 20 , N \

5. and 2.2 , N \

end {array} $

I really need help with this problem. Initially I thought I should break down strength and weight. Which I assumed that the figure of the force is parallel to the floor, which is the base of the inclination.

By doing this and considering the coefficient of friction (which I assumed to be static), this would translate as follows:

$ F cos alpha – mu N = 0 $

The normal or the tilt reaction I found using this logic:

$ N- mg cos alpha – F sin alpha = 0 $

$ N = mg cos alpha + F sin alpha $

Inserting this in the previous equation:

$ F cos alpha – mu left (mg cos alpha + F sin alpha right) = 0 $

$ F cos alpha – mu mg cos alpha – mu F sin alpha = 0 $

$ F left ( cos alpha – mu sin alpha right) = mu mg cos alpha $

$ F = frac { mu mg cos alpha} { cos alpha – mu sin alpha} $

Therefore, inserting there the given information would become:

$ F = frac { frac {3} {10} (2 times 10) cos 30 ^ { circ}} { cos 30 ^ { circ} – frac {3} {10} sen 30 ^ { circ}} $

$ F = frac { frac {6 sqrt {3}} {2}} { cos 30 ^ { circ} – frac {3} {10} without 30 ^ { circ}} $

$ F = frac { frac {6 sqrt {3}} {2}} { frac { sqrt {3}} {2} – frac {3} {10} times frac {1} { 2}} $

This is where simplification becomes ugly:

$ F = frac {3 sqrt {3}} { frac {10 sqrt {3} -3} {20}} $

$ F = frac {60 sqrt {3}} {10 sqrt {3} -3} approx 27.25 $

Therefore, in the end I get that value for strength. But it is not close to the answers. Can anyone help me with this? What could I have done wrong? How could I simplify this? Can anyone offer an FBD help for this problem?