While sliding a couch across a floor, Hannah and Andrea exerts forces FH and FA on the couch. Hannah's force is due north with a magnitude of FH = 59 N and Andrea's force is θ = 22° east of north with a magnitude of FA = 155 N. In this problem, use a coordinate system with y directed north and x directed east. Part (a) Find the net force in the y-direction in Newtons.
Part (b) Find the net force in the x-direction in Newtons.
Part (c) Calculate the angle in degrees north of east of the net force exerted on the couch by Hannah and Andrea, F HAF→HA.
Part (d) Hannah and Andrea's housemates, David and Stephanie disagree with the move and want to prevent its relocation. Their combined force F DSF→DS must be equal and opposite to that of FHAF→HA. What is the magnitude in Newtons of the force F DSF→DS which will prevent the relocation?

a) 202.7 N

b) 58.1 N

c) 74.1º N of E.

d) 210.9 N

Explanation:

a)

The net force exerted in the y-direction, will be the sum of FH (which is directed northwards) and the y-component of FA.Since the magnitude of FA is 155 N and the angle of FA with the y-axis, is 22º (E of N), we can find the N-S component of  FA, just applying the the definition of cosine, to the triangle defined by FA, the y- axis and a segment parallel to the x- axis between FA and the y-axis, as follows:

     

      ⇒ Fy = FH + FAy = 59 N + 143.7 N = 202.7 N (2)

b)

We can proceed exactly in the same way for the x-axis. Since FH is directed due North, it has no component along the x-axis.So, Fx is directly the component of FA along the x-axis, which can be found applying the definition of sine to the same triangle than in a) as follows:

     

c)

Taking the same triangle than in a) and b), we can apply the definition of tangent, in order to find the angle between F and the x-axis, as follows:

       

      ⇒ θ = tg⁻¹ (3.5) = 74.1º N of E. (5)

d)

In order to be equal and opposite to the combined force FH+FA, it must have the same magnitude.This magnitude can be found applying the Pythagorean Theorem to the same triangle that we used in a), b) and c):

     

Would like to . could you translate to english

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