Need help in physics finding magnitude in word problem

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The 4.9 N weight is in equilibrium under the influence of the three forces acting on it. The F force acts from above on the left at an angle of α with the horizontal. The 5.9 N force acts from above on the right at an angle of 48◦ with the horizontal. The force 4.9 N acts straight down.

What is the magnitude of force F

Jun 2nd, 2015

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5.9 N ......... 4.9 N
...┌...... ......┐
.....\..... ...../
.....α\... .../ ß
-------- \ . /---------
...........V 4.9 N 

(It wasn't clear from your description whether the two angled forces act upward or downward, but if they're going to be in equilibrium with the weight, they should be upward.) 

So, we're in static equilibrium - the body is not accelerating. From the vector version of F = ma, that means that the net vector force is zero. If the vector force is zero, that means that orthogonal components of the net vector force are zero. So, we take the horizontal and vertical directions (which are orthogonal), find the net force components in those directions, and set them equal to zero. We need to define the positive and negative directions for those components, so let's say that the positive x direction is to the right, and the positive y direction is up.

In the x direction, the weight has no component. The 5.9 N force component acts in the -x direction, with magnitude -5.9 N * cos α. The 4.9 N force component acts in the +x direction, with magnitude +4.9 * cos ß. So, the sum of forces in the x direction (which is equal to zero) is 

ΣFx = -5.9 N * cos α + 4.9 N * cos ß = 0

In the y direction, the weight acts in the negative direction, with magnitude -9.6 N. The other forces act in the +y direction, with magnitude 5.5 N * sin α and 6.4 N * sin ß. So, the sum of forces in the y direction (which is equal to zero) is

ΣFy = -4.9 N + 4.9 N * sin α + 4.9 N * sin ß = 0

So, we have 2 equations, and 2 unknowns, and we can solve. Let's start with the first equation.

-5.9 N * cos α + 4.9 N * cos ß = 0
5.9 * cos α = 4.9 * cos ß
cos α = (4.9 / 5.9) * cos ß 

Please let me know if you need any clarification. I'm always happy to answer your questions.
Jun 2nd, 2015

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