A 33-N force acing due north and a 44-N force acting at 30 degrees act concurrently on point P. What is the magnitude and direction of a third force that produces equilibrium at point P?

"..acting at 30 degrees" is not clear.

Which way does the 30° go, east of north ?, west of north ?,

I will solve it as if it was 30° east of north. If not just make the necessary adjustments.

the resultant of our two forces, I will consider north to be 90°
= (33cos90, 33sin90) + (44cos60, 44sin60)
= (0, 33) + (22, 22√3)
= (22, 33+22√3)

magnitude = √(22^2 + (33+22√3)^2
= √( 484 + 1089 + 1452√3 + 1452)
= √(3025 + 1452√3)
= appr 74.43 N

angle of resultant:
tanØ = 1452√3/3025
Ø = appr 39.74°
So the force keeping it in equilibrium has to go in the opposite direction, which is 219.74° with a magnitude of 74.32 N
I will leave it to you to express the direction in the form needed. Make a sketch to determine it.

30 degrees which way? I assume to the East.

So you want the resultant, then the equilibrant will be opposite (read add 180 degrees).

Ok, forces: N direction
33 + 44cos30= you do it, N component

E direction:
44 sin30= you do it, E component

Now, the resultant:
magnitude: sqrt(N^2 + E^2)
direction: arctan (E/N) E of N

Equilibrate: same magnitude as resultant. Direction, add 180
so direction=arctan(E/N) W of S

To find the magnitude and direction of the third force that produces equilibrium at point P, we can use vector addition. Since the two given forces are acting concurrently, we can find the resultant force by adding them together.

Let's break down the given forces into horizontal and vertical components:

Force 1 (acting due north):
- Horizontal component: 0 N
- Vertical component: 33 N

Force 2 (acting at 30 degrees):
- Horizontal component: 44 N * cos(30°)
- Vertical component: 44 N * sin(30°)

To achieve equilibrium, the sum of the horizontal components and the sum of the vertical components should both be equal to zero.

Horizontal component sum:
0 N + 44 N * cos(30°) = 0

Vertical component sum:
33 N + 44 N * sin(30°) = 0

Solving these equations will give us the magnitude and direction of the third force.

From the first equation, we find:
44 N * cos(30°) = 0
44 N * cos(30°) = 0
44 N * 0.866 = 0
38.024 N ≈ 0

From the second equation, we find:
33 N + 44 N * sin(30°) = 0
33 N + 44 N * 0.5 = 0
33 N +22 N = 0
55 N = 0

Based on these calculations, we find that the sum of the horizontal components is zero, and the sum of the vertical components is zero. This means that a force of magnitude 55 N acting due south (180 degrees) will produce equilibrium at point P.

To find the magnitude and direction of the third force, we can use the concept of vector addition. Here's how you can solve it step by step:

Step 1: Draw a diagram: Draw a coordinate system and label point P as the origin. Mark the 33-N force acting due north and the 44-N force acting at a 30-degree angle.

Step 2: Resolve the forces: Break down the forces into their horizontal and vertical components. The 33-N force acting due north will have no horizontal component and a vertical component of 33 N. The 44-N force acting at 30 degrees can be resolved into two components: a horizontal component of 44 N * cos(30°) and a vertical component of 44 N * sin(30°).

Step 3: Add the horizontal and vertical components separately: Add up all the horizontal components and separately add up all the vertical components. The sum of the horizontal components should be zero since the forces are in equilibrium. Similarly, the sum of the vertical components should also be zero.

Step 4: Solve for the magnitude and direction of the third force: Since the sum of the horizontal components is zero, it means that the horizontal component of the third force is equal to the negative of the sum of the other two horizontal components. Similarly, the vertical component of the third force is equal to the negative of the sum of the other two vertical components.

Step 5: Use the Pythagorean theorem and trigonometry: The magnitude of the third force can be calculated using the Pythagorean theorem: magnitude = √(horizontal component^2 + vertical component^2). The direction of the third force can be found by calculating the inverse tangent of the vertical component divided by the horizontal component.

By following these steps, you should be able to find the magnitude and direction of the third force that produces equilibrium at point P.