Solve X Equation for Sum of Forces in X/Y Direction

[Rebel_Yell]

How do you set up equations for sum of forces in the x and y directions? I can do the y when it is only moving in the x direction, but I don't know how to set up the x equation.

 

[HallsofIvy]

Yes, of course, you can do the "y direction when it is only moving in the x direction" because that's trivial! I'm not sure what you really meant.

Normally one starts with "movement in a line" where we only have to worry about one direction. If something is moving on the x-axis with a force in the x-direction then v_x= at+ v₀ and x= (1/2)at²+ v₀t+ x₀.

Now you have motion in the plane (or 3 dimensions with x, y, z) so you have to separate everything into x, y, z components. If you are given "initial velocity" or "force vector" as length and direction, then you need to use trigonometry to decompose into x and y components. For example if the problem says "the initial speed is 20m/s at a direction 30 degrees above the x-axis, draw a picture showing a line at 30 degrees above the x-axis. Mark off a "length" of 20 on that line and draw a perependicular from that point to the x-axis. You see a right triangle with hypotenuse of length 20, "near side" along the x-axis (so its length is x), and "opposite side" parallel to the y-axis (so its length is y). Remembering that "sine" is defined as "opposite over hypotenus", sin(30)= y/20 or y= 20sin(30). Remembering that "cosine" is defined as "near side over hypotenuse", cos(30)= x/20 or x= 30 sin (30).
Once you have those components, you can just work with "x" and "y" separately- that's the nice thing about using components.

 

[M98Ranger]

To solve for the sum of forces in the x direction, you will need to use the concept of vector components. This involves breaking down each force into its x and y components, and then adding them together to find the net force in the x direction.

To set up the x equation, you will first need to identify all the forces acting in the x direction. These forces could include applied forces, frictional forces, and any other forces acting in the x direction.

Next, you will need to determine the x component of each force. This can be done by using trigonometric functions such as cosine, which relates the adjacent side of a right triangle to the hypotenuse.

Once you have determined the x components of all the forces, you can add them together to find the net force in the x direction. This can be represented as:

ΣFx = F1x + F2x + F3x + ...

Where ΣFx represents the sum of all the forces in the x direction, and F1x, F2x, F3x, etc. represent the x components of each individual force.

Using this equation, you can solve for the net force in the x direction and use it to determine the acceleration or motion of the object in that direction.

In summary, to set up equations for sum of forces in the x direction, you will need to identify all the forces acting in that direction, determine their x components, and then add them together to find the net force. This approach can also be used for the y direction, but with the y components of the forces instead.