Answer:
0.14442 N
Step-by-step explanation:
To find the magnitude and direction of the force needed to cause equilibrium in the system, we can use vector addition.
First, we need to convert the forces given in polar coordinates (direction and magnitude) to Cartesian coordinates (x and y components):
Force A: (60 grams) cos(60°) = 30 grams in the x direction, (60 grams) sin(60°) = 51.96 grams in the y direction.
Force B: (100 grams) cos(142°) = -36.07 grams in the x direction, (100 grams) sin(142°) = 94.78 grams in the y direction.
Force C: (130 grams) cos(255°) = -109.57 grams in the x direction, (130 grams) sin(255°) = -62.79 grams in the y direction.
Next, we can add up the x and y components of the three forces to find the net force acting on the system:
Net force in x direction = 30 grams - 36.07 grams - 109.57 grams = -115.64 grams
Net force in y direction = 51.96 grams + 94.78 grams - 62.79 grams = 83.95 grams
Since the system is in equilibrium, the net force in both the x and y directions must be zero. Therefore, we can set up the following equations:
-115.64 grams + F_Dx = 0
83.95 grams + F_Dy = 0
Solving for F_Dx and F_Dy, we get:
F_Dx = 115.64 grams
F_Dy = -83.95 grams
The magnitude of Force D can be found using the Pythagorean theorem:
|F_D| = sqrt(F_Dx^2 + F_Dy^2) = sqrt((115.64 grams)^2 + (-83.95 grams)^2) = 144.42 grams
To convert this to Newtons, we can divide by the conversion factor of 1000 grams per Newton:
|F_D| = 144.42 grams / (1000 grams per Newton) = 0.14442 Newtons
Finally, the direction of Force D can be found using the inverse tangent function:
theta = atan(F_Dy / F_Dx) = atan(-83.95 grams / 115.64 grams) = -37.21 degrees
Therefore, Force D needs to be 144.42 grams (0.14442 Newtons) in magnitude and 37.21 degrees in direction to cause equilibrium in the system.