Understanding Trig In Force Diagrams

[PhysicsReaper]

Hello,

I have the following problem part (b) which I already solved as you can see in the attached image. So I am not asking homework questions, I merely reviewing my homework for a better understanding for the test. I obtained the answer from a friend showing me his method. However, I am studying and a listed solution was the following:

F_y = 2*(k*(q^2/r^2))cos(30)


I've drawn out a force diagram but have no idea how, they have obtained the cosign. In the listed solution they state use oriented the y-axis such that it bisects charges q2 and q3.

 

[CWatters]

There are two forces acting on point 1. By symetry the result of these two forces will act along a line drawn through point 1 and a point mid way between 2 and 3. See diagram.

So work out the component of the two forces pointing in that direction.

Consider the triangle on the right.

Cos(30) = Fr/F

so

Fr = F Cos(30)

That's not the whole solution obviously, just where the cos(30) comes from. The angle doesn't come from the direction of the result per se, it comes from the direction of the result in relation to the forces. eg If you rotate the triangle/problem drawing 30 degrees so the resulting force is vertical (on the y-axis) the answer will still contain cos(30).

 

[PhysicsReaper]

Thank you very much! And because of symmetry is why it's multiplied by 2?

 

[CWatters]

Yes in this case.

 

[pmacias]

Hi there,

I am happy to provide some clarification on the use of trigonometry in force diagrams. In the solution provided, the cos(30) term comes from the use of trigonometric functions to find the components of the force in the y-direction.

In a force diagram, the forces are represented by arrows, with the length of the arrow representing the magnitude of the force and the direction of the arrow representing the direction of the force. In order to find the component of a force in a specific direction, we can use trigonometry.

In this case, the y-axis is oriented in such a way that it bisects the charges q2 and q3. This means that the angle between the y-axis and the line connecting q2 and q3 is 30 degrees. Using trigonometry, we can find the component of the force in the y-direction by multiplying the magnitude of the force by the cosine of this angle. This is where the cos(30) term comes from in the solution.

I hope this helps to clarify the use of trigonometry in force diagrams. It is a useful tool for finding the components of forces in different directions, and can be applied in many other areas of physics as well. Keep studying and good luck on your test!