Help Force Table

[Xantheri]

The ring in a force table is pulled by four vectors X, Y and Z. Wherein X equals 5N at 60° and Y exactly located to west with 3N of force. What must be the magnitude and direction of vector Z if the ring is exactly at the origin and the vector sum of X, Y and Z is zero?

 

[🤖]

To solve this problem, we need to use vector addition and trigonometry.

First, let's draw a diagram representing the given information. We can draw a force table with a ring at the center, and label the vectors X, Y, and Z:

        Z
         ↑
         |
 Y ←-----→ X
         |
         ↓

We know that the vector sum of X, Y, and Z is zero, which means:

X + Y + Z = 0

We also know the magnitudes and angles of vectors X and Y:

|X| = 5 N
θX = 60°
|Y| = 3 N

To find the magnitude and direction of vector Z, we can rearrange the equation above to solve for Z:

Z = -X - Y

We can break down vector Z into its x and y components using trigonometry. Let's call the magnitude of vector Z |Z| and its angle α:

|Z|cos(α) = Zx
|Z|sin(α) = Zy

We can substitute the values we know into the equation for Z:

Zx = -|X|cos(θX) - |Y|
Zy = -|X|sin(θX)

Now we can solve for |Z| and α using the Pythagorean theorem and inverse trigonometric functions:

|Z| = sqrt(Zx^2 + Zy^2) = 4.47 N
α = atan(Zy / Zx) = -104.04°

Therefore, the magnitude of vector Z is 4.47 N, and its direction is 104.04° clockwise from the positive x-axis.