Respuesta :
Answer:
The magnitude of the resultant force [tex]F_{net}[/tex], is 676.643 N
The angle it makes with David;s rope is 77.19°
Explanation:
Here we have
David's force F₁ = 400 N and direction of F₁ = horizontal θ₁ = 0
Stephanie's force F₂ = 300.0 N direction of F₂ = θ₂ = 30°
The angle between the ropes θ = 30°
Therefore we resolve the forces applied by the two teenagers as follows
[tex]F_1 = F_{1x} +F_{1y}[/tex]
F₁ₓ = F₁×cos (0) = F₁ = 400.0 N
[tex]F_{1y}[/tex] = F₁×sin (0) = 0 N
Also
[tex]F_2 = F_{2x} +F_{2y}[/tex]
F₂ₓ = F₂×cos (θ₂) = 300 ×cos (30)= 259.81 N
[tex]F_{2y}[/tex] = F₂×sin (θ₂) = 300 ×sin (30)= 150 N
Therefore, the combined force of both teenagers is;
∑Fₓ = F₁ₓ + F₂ₓ = 400.0 N + 259.81 N = 659.81 N and
∑[tex]F_{y}[/tex] = [tex]F_{1y}[/tex] + [tex]F_{2y}[/tex] = 0 N + 150 N =150 N
The resultant force [tex]\overrightarrow{\rm F}[/tex] is given by
[tex]\overrightarrow{\rm F}[/tex] = [tex]659.81\hat {i} + 150\hat {j}[/tex]
The magnitude of the resultant force is, [tex]F_{net}[/tex];
[tex]F_{net}=\sqrt{\Sigma F^2_x+\Sigma F^2_y} =\sqrt{658.81^2+150^2} = \sqrt{457846.1} = 676.643\hspace {0.09cm} N[/tex]
The angle it makes with David's rope is
[tex]tan \hspace {0.09cm} \alpha = \frac{F_y}{F_x} = \frac{659.81}{150} =4.399\\\alpha = tan^{-1}4.399 =77.19 \textdegree[/tex]
