Two forces act on an object at 20 degree to each other. One force has a magnitude of 200N, and the resultant has a magnitude of 340N. Determine the magnitude of the second force and the direction it makes with the resultant.

To determine the magnitude of the second force and the direction it makes with the resultant, we can use vector addition.

Step 1: Split each force into its x and y components.
Let the first force (F1) be F1x and F1y, and the second force (F2) be F2x and F2y.

F1 = 200N
F2 = ?
Resultant (R) = 340N

Step 2: Find the x and y components of F1 using trigonometry.
F1x = F1 * cosθ = 200N * cos(20°)
F1y = F1 * sinθ = 200N * sin(20°)

Step 3: Use vector addition to find the x and y components of the resultant force (R).
R = F1 + F2
Rx = F1x + F2x
Ry = F1y + F2y

Step 4: Equate the magnitudes of R and its x and y components.
we know R = 340N, Rx = F1x + F2x, and Ry = F1y + F2y.

340N = sqrt(Rx^2 + Ry^2)
340^2 = Rx^2 + Ry^2 ... (Equation 1)

Step 5: Use trigonometry to find F2x and F2y.
Rx = F2x + F1x = F2x + 200N * cos(20°)
Ry = F2y + F1y = F2y + 200N * sin(20°)

Step 6: Substitute F2x and F2y into Equation 1 and solve for R.
340^2 = (F2x + 200N * cos(20°))^2 + (F2y + 200N * sin(20°))^2

Solve this equation to find the magnitude of the second force (F2).

Step 7: Find the direction of F2.
To find the direction, use the inverse tangent (tan^-1) function to find the angle:
tanθ = F2y / F2x
θ = tan^-1(F2y / F2x)

This will give you the direction that F2 makes with the resultant.

To solve this problem, we can use the concept of vector addition. We'll start by breaking down the given information:

1. The first force has a magnitude of 200N.
2. The resultant force has a magnitude of 340N.

Now, let's determine the magnitude of the second force:

Since forces are vectors, we can represent them graphically as arrows. Construct a diagram with the first force as one arrow and the resultant as another arrow. Label the first force as F1 and the resultant as R.

Now, we'll use vector addition to find the second force. To do this, we need to add F1 and the unknown second force to get the resultant R.

Since we know the magnitude of F1 is 200N and the magnitude of R is 340N, we can write down the equations:

|F1| + |F2| = |R|
200N + |F2| = 340N

Simplifying the equation, we get:

|F2| = 340N - 200N
|F2| = 140N

Therefore, the magnitude of the second force is 140N.

Now, let's determine the direction of the second force:

To find the direction, we need to calculate the angle between the second force and the resultant. We can do this using trigonometry.

Let's assume the angle between the second force (F2) and the resultant (R) is θ.

We can use the equation:

tan(θ) = (|F1| sin(α)) / (|F1| cos(α) + |F2|)
tan(θ) = (200N sin(20°)) / (200N cos(20°) + 140N)

Now, we can solve for θ:

θ = tan^(-1)((200N sin(20°)) / (200N cos(20°) + 140N))

Evaluating this expression, we find:
θ ≈ 31.16°

Thus, the magnitude of the second force is 140N, and it makes an angle of approximately 31.16° with the resultant.

200 in + x direction

F at angle 20 above x axis
R = 340 at angle T above x axis

340 cos T =200 + F cos 20
340 sin T = F sin 20

solve for F and T

F is magnitude we want
The angle we want is 20 - T