Answer :
Answer
given,
F₁ = 15 lb
F₂ = 8 lb
θ₁ = 45°
θ₂ = 25°
Assuming the question's diagram is attached below.
now,
computing the horizontal component of the forces.
F_h = F₁ cos θ₁ - F₂ cos θ₂
F_h = 15 cos 45° - 8 cos 25°
F_h = 3.36 lb
now, vertical component of the forces
F_v = F₁ sin θ₁ + F₂ sin θ₂
F_v = 15 sin 45° + 8 sin 25°
F_v = 13.98 lb
resultant force would be equal to
[tex]F = \sqrt{F_h^2+F_v^2}[/tex]
[tex]F = \sqrt{3.36^2+13.98^2}[/tex]
F = 14.38 lb
the magnitude of resultant force is equal to 14.38 lb
direction of forces
[tex]\theta =tan^{-1}(\dfrac{F_v}{F_h})[/tex]
[tex]\theta =tan^{-1}(\dfrac{13.98}{3.36})[/tex]
θ = 76.48°
The resultant force will be 14.3844 N making an angle of 76.5° from the horizontal line.
Given to us
- F1 = 15 lb
- F2 = 8 lb
- [tex]\theta_1 = 45^o[/tex]
- [tex]\theta_2 = 25^o[/tex]
We know that a force is a vector quantity and can be divided into two component a vertical and a horizontal component. As shown below in the image.
The horizontal component is the cosine component while the vertical component is the sine component.
Resultant of the horizontal components of the forces,
As the forces are pointing in different directions, therefore, the force on the left will be taken as positive while the force on the right is taken as negative.
[tex]F_{H} = F_1 Cos \theta_1 + F_2 Cos \theta_2\\F_{H} = [15\times Cos(45^o)]+[-8\times Cos(25^o)]\\F_H = 3.3561\ N[/tex]
Resultant of the Vertical components of the forces,
As the forces are pointing in the same direction, therefore, the net force will be in the same direction,
[tex]F_{V} = F_1 Cos \theta_1 + F_2 Cos \theta_2\\F_{V} = [15\times Sin(45^o)]+[8\times Sin(25^o)]\\F_V = 13.9875\ N[/tex]
Resultant Force
[tex]F_R = \sqrt{F_H^2+ F_V^2}[/tex]
[tex]F_R = \sqrt{3.3561^2+13.9875^2}\\F_R = 14.3844\ N[/tex]
Angle of Resultant Force
[tex]Tan(\theta_R) = \dfrac{F_V}{F_H} = \dfrac{13.9875}{3.3561}= 4.167784\\\\(\theta_R) = Tan^{-1}(4.167784)\\(\theta_R) = 76.5^o[/tex]
As both the forces are positive the resultant force will be in the first quadrant, making an angle of 76.5° from the horizontal line.
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