Question

1. Two forces act on a box as follows: F1 = 100 N at 01 = 170° and F2 = 75 N

Answer 1

Answer:

a)  F = 64.30 N,  b) θ = 121.4º

Explanation:

Forces are vector quantities so one of the best methods to add them is to decompose each force and add the components

let's use trigonometry

Force F1

          sin 170 = F_{1y} / F₁

          cos 170 = F₁ₓ / F₁

          F_{1y} = F₁ sin 170

          F₁ₓ = F₁ cos 170

          F_{1y} = 100 sin 170 = 17.36 N

          F₁ₓ = 100 cos 170 = -98.48 N

Force F2

          sin 30 = F_{2y} / F₂

          cos 30 = F₂ₓ / F₂

          F_{2y} = F₂ sin 30

          F₂ₓ = F₂ cos 30

          F_{2y} = 75 sin 30 = 37.5 N

          F₂ₓ = 75 cos 30 = 64.95 N

the resultant force is

X axis

          Fₓ = F₁ₓ + F₂ₓ

          Fₓ = -98.48 +64.95

          Fₓ = -33.53 N

Y axis

         F_y = F_{1y} + F_{2y}

         F_y = 17.36 + 37.5

         F_y = 54.86 N

a) the magnitude of the resultant vector

let's use Pythagoras' theorem

         F = Ra Fx ^ 2 + Fy²

         F = Ra 33.53² + 54.86²

         F = 64.30 N

b) the direction of the resultant

let's use trigonometry

        tan θ’= F_y / Fₓ

        θ'= $tan^{-1} \frac{F_y}{F_x}$

        θ'= tan⁻¹ (54.86 / (33.53)

        θ’= 58.6º

this angle is in the second quadrant

The angle measured from the positive side of the x-axis is

        θ = 180 -θ'

        θ = 180- 58.6

        θ = 121.4º