Find the magnitude and direction of the equilibrant of two forces, one being horizontal pull of 18 N and the other a pull of 23 N at 50 degrees to the 12 N force.

To find the magnitude and direction of the equilibrant of two forces, we need to understand the concept of vector addition and subtraction. The equilibrant is a single force that, when applied along with the given forces, brings the object to a state of equilibrium or zero net force.

First, let's find the resultant of the two given forces. We can break down the 23 N force into its horizontal and vertical components. The horizontal component can be found by multiplying the magnitude of the force (23 N) by the cosine of the angle (50 degrees), and the vertical component can be found by multiplying the magnitude of the force (23 N) by the sine of the angle (50 degrees).

Horizontal component: 23 N * cos(50°) ≈ 14.74 N
Vertical component: 23 N * sin(50°) ≈ 17.62 N

Now we can add the horizontal component of the 23 N force to the 18 N horizontal force to find the horizontal component of the resultant force.
Horizontal component of the resultant force = 18 N + 14.74 N ≈ 32.74 N

The vertical component of the resultant force is obtained by subtracting the vertical component of the 23 N force from the vertical component of the 12 N force.
Vertical component of the resultant force = 12 N - 17.62 N ≈ -5.62 N (negative because they are in opposite directions)

Using the Pythagorean theorem, we can find the magnitude (or resultant) of the resultant force:
Resultant force = sqrt((32.74 N)^2 + (-5.62 N)^2) ≈ 33 N

Now, let's determine the direction of the resultant force:
tan(theta) = Vertical component / Horizontal component
theta = arctan(Vertical component / Horizontal component)
theta = arctan(-5.62 N / 32.74 N) ≈ -9.96°

To obtain the direction of the equilibrant force, we add 180° to the calculated angle:
Equilibrant direction = -9.96° + 180° ≈ 170.04°

Therefore, the magnitude of the equilibrant force is approximately 33 N, and its direction is approximately 170.04°.