Answer:
[tex]\vec{F_3}=(-34.0N)\hat{i}+(-12.0N)\hat{j}[/tex]
Explanation:
You have three forces F1, F2 an F3 that produce the following acceleration:
a = −(8.00m/s^2)ˆi + (6.00m/s^2)ˆj
you know that force F1 and F2 are:
F1 = (30.0N)ˆi + (16.0N)ˆj
F2 = −(12.0N)ˆi + (8.00N)ˆj
and the force F3 is unknown:
F3 = F3x ˆi + F3y ˆj
The second Newton law is given by the following equation:
[tex]\vec{F}=m\vec{a}[/tex]
F: the total force = F1 +F2 + F3
m: mass of the object = 2 kg
By the properties of vectors you have:
[tex]\vec{F_1}+\vec{F_2}+\vec{F_3}=m\vec{a}\\\\(30.0-12.0+F_{3x})N\hat{i}+(16.0+8.00+F_{3y})N\hat{j}=(2.0kg)[(-8.00m/s^2)\hat{i}+(6.00m/s^2)\hat{j}]\\\\(18.0+F_{3x})N\hat{i}+(24.0+F_{3y})\hat{j}=(-16.00N)\hat{i}+(12.0N)\hat{j}[/tex]
Both x and y component must be equal in the previous equality, then you have:
[tex]18.0N+F_{3x}=-16.00N\\\\F_{3x}=-34.00N\\\\24.0N+F_{3y}=12.0N\\\\F_{3y}=-12.00N[/tex]
Hence, the vector F3 is:
[tex]\vec{F_3}=(-34.0N)\hat{i}+(-12.0N)\hat{j}[/tex]
