Answer:
F = 45 (2.4142 i ^ + 4.414 j ^)
F = 226.40 N, θ= 61.3
Step-by-step explanation:
For this exercise we will use that the forces are vectors and we will add them, the force due to being electric charges must comply, go Coulomb's law
F₁₂ = k q₁ q₂ / r₁₂²
To apply this equation to our case, they indicate that all charges are of the same sign and their value, also the charge Q is located in the upper left corner, unfortunately the diagram of the other charges is not loaded, but the most general is that is in sequence, see attached, for the sum of the forces let's add its components
x-axis (horizontal)
Fₓ = F₂₁ + F₂₄ₓ
Let's use trigonometry for the force component, as the weathering figure is a square the angle between the force and the x axis is 45
cos 45 = F₂₄ₓ / F₂₄
F₂₄ₓ = F₂₄ cos 45
y-axis (vertical)
= F₂₃ + F_{24y}
sin 45 = F_{24y} / F₂₄
F_{24y} = F₂₄ sin 45
let's search every distance
The side of the square is worth l = 5cm = 0.05 m
we can find the diagonal with the Pythagorean theorem
d = √(l² + l²) = l √2
now we can search every force
F₂₁ = k q₁ q₂ / r₁₂²
F₂₁ = k Q 2Q / l²
F₂₁ = k 2Q² / l²
this force points in the positive x direction
F₂₃ = k q₂ q₃ / r₂₃²
F₂₃ = k 2Q 3Q / l²
this force points in the direction of the positive y
F₂₄ = k q₄q₂ / d²
F₂₄ = k 4Q 2Q / 2 l²
let's find the resultant in each x
X axis
Fₓ = k 2Q² / l² + k 8Q² / 2 l² cos 45
Fₓ = k 2Q² / l² (1 + 4/2 cos 45)
Fₓ = k 2Q²/ l² 2.4142 i ^
Axis y
F_{y} = k 6Q² / l² + k 8Q² / 2 l² sin 45
F_{y} = k 2Q² / l² (3 + 4/2 sin45)
F_{y} = k 2Q² / l² (4,414)
the resultant force
F = Fₓ i ^ + F_{y} j ^
F = k 2Q² / l² ( 2.4142 i ^ + 4.414 j ^)
let's substitute the values
F = 9 10⁹ 2 (2.5 10⁻⁶) 2 / 0.05² (2.4142 i ^ + 4.414 j ^)
F = 45 (2.4142 i ^ + 4.414 j ^)
The result we can give is this form and in the form of module and angle
let's use the Pythagorean theorem to find the modulus
F =√ (Fₓ² +
²)
F = 45 √ (2.4142² + 4.414²)
F = 226.40 N
we use trigonometry for the angle, measured from the x-axis
tan θθ = Fy / Fx
θ = tan⁻¹ (4.414 / 2.4142)
θ= 61.3