Question

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

at 02 = 30°. Find their resultant force on the box.

(a) the magnitude of vector sum Fi + F2

(b) the direction of of the vector sum Fi + F2

Answer 1

a) F = 64.30 N, b) θ = 121.4º

Step-by-step 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) (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