To solve forces in two dimensions using the Pythagorean Theorem, we need to understand that we can represent force vectors as arrows placed tail-to-tip on a grid system. Let's look at two forces, ( \vec{F_1} ) and ( \vec{F_2} ), that act at right angles (90 degrees) to each other.

Step 1: Break Down the Forces

First, we can split each force into horizontal (on the x-axis) and vertical (on the y-axis) parts. For example:

• For ( \vec{F_1} ):

  • The horizontal part is ( F_{1x} = F_1 \cos(\theta_1) )
  • The vertical part is ( F_{1y} = F_1 \sin(\theta_1) )

• For ( \vec{F_2} ):

  • The horizontal part is ( F_{2x} = F_2 \cos(\theta_2) )
  • The vertical part is ( F_{2y} = F_2 \sin(\theta_2) )

Step 2: Add Up the Forces

Next, we combine the parts from both forces to find the total (resultant) force in each direction:

• For the horizontal direction:
  • ( R_x = F_{1x} + F_{2x} )
• For the vertical direction:
  • ( R_y = F_{1y} + F_{2y} )

Step 3: Use the Pythagorean Theorem

Now that we have the total components, we can use the Pythagorean Theorem to find the overall strength of the resultant force, ( R ): $R = \sqrt{R_x^2 + R_y^2}$

Step 4: Find the Direction

To discover the direction (angle ( \phi )) of the resultant force compared to the x-axis, we can use a trigonometry function: $\phi = \tan^{-1}\left(\frac{R_y}{R_x}\right)$

Example

Let’s take an example where ( \vec{F_1} = 5\ \text{N} ) at ( 30^\circ ) and ( \vec{F_2} = 10\ \text{N} ) at ( 90^\circ ). Here’s how the math would work out:

• For ( F_{1x} \approx 4.33\ \text{N} ) and ( F_{1y} = 2.5\ \text{N} )
• For ( F_{2x} = 0\ \text{N} ) and ( F_{2y} = 10\ \text{N} )

Now we can add them up:

• ( R_x = 4.33\ \text{N} )
• ( R_y = 12.5\ \text{N} )

Now we calculate the overall force:

• ( R \approx 13.12\ \text{N} )

Finally, we find the direction:

• ( \phi \approx 71.57^\circ )

This method helps us break down and solve forces in two dimensions using the Pythagorean Theorem in physics.