Question

Three people pull simultaneously on a stubborn donkey. Jack pulls eastward with a force of 80.5 N, Jill pulls with 81.7 N in the northeast direction, and Jane pulls to the southeast with 131 N. (Since the donkey is involved with such uncoordinated people, who can blame it for being stubborn

Answer 1

F = 233.52 N, θ' = 351.41º

Step-by-step explanation:

In this exercise we must find the net force applied on the donkey.

For this we use Newton's second law, where we create a reference frame with the horizontal x axis

let's decompose the forces

Jack

= 80.5 N

Jill

cos 45 = F_{2x} / F₂2

sin 45 = F_{2y} / F₂2

F_{2x} = F₂ cos 45

F_{2y} = F₂ sin 45

F_{2x} = 81.7 cos 45 = 57.77 N

F_{2y} = 81.7 sin 45 = 57.77 N

Jane

cos (270 + 45) = F_{3x} / F₃3

sin 315 = F_{3y} / F₃

F_{3x} = 131 cos 315 = 92.63 N

F_{3y} = 131 sin 315 = -92.63 N

the force can be found in each axis

X axis

F_{x} = F_{1x} + F_{2x} + F_{3x}

F_{x} = 80.5 +57.77 + 92.63

F_{x} = 230.9 N

Axis y

F_{y} = F_{1y} + F_{2y} + F_{3y}

F_{y} = 0 + 57.77 -92.63

F_{y} = -34.86 N

we can give the result in two ways

a) F = (230.9 i ^ - 34.86 j ^) N

b) in the form of module and angle

we use the Pythagorean theorem

F = √(Fₓ² + F_{y}²

F = √(230.9² + 34.86²)

F = 233.52 N

let's use trigonometry for the angle

tan θ =
`(F_y)/(F_x) }`

θ = tan⁻¹ (\frac{F_y}{F_x} })

θ = tan⁻¹ (-34.86 / 230.9)

θ = -8.59º

if we measure this angle from the positive side of the x-axis counterclockwise

θ' = 360 -θ

θ‘= 360- 8.59

θ' = 351.41º