A sled at rest is suddenly pulled in three horizontal directions at the same tim

Question

A sled at rest is suddenly pulled in three horizontal directions at the same time, but it does not move. Paul pulls to the northeast with a force of 31 lb. Johnny pulls at an angle of 41° south of due west with a force of 57 lb. Connie pulls with a force to be determined.

(a) Express the boys' two forces in terms of the usual unit vectors.

(b) Determine the third force (from Connie), expressing it first in component form and then as a magnitude and direction (angle). Neglect friction.
FConnie: lb î + lb

Explanation / Answer

a)

let,

force from Paul,

F=31lb

here,

F1=F*cos(45)i + F*sin(45)j

F1=31*cos(45)i + 31*sin(45)j

F1=(21.92)i + (21.92)j ----------->

and

froce from Johnny,

F=57lb

now,

F2=F*cos(180+41)i + F*sin(180+41)j

F2=57*cos(180+41)i + 57*sin(180+41)j

F2=(-43.02)i + (-37.39)j ----------->

b)

let,

force from Connie is F3=(Fx)i+(Fy)j

F3=(Fx)i+(Fy)j

F3=(21.92)i + (21.92)j + (-43.02)i + (-37.39)j

F3=(-21.09)i +(-16.29)j ------>

===>

magnitude of F3=sqrt((-21.09)^2 + (-16.29)^2)

F3=26.65 N

direction is, tan(theta)=(-16.29)/(-21.09)

==> theta=37.68 degrees

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