Two forces of 90 N act on an object. The forces make an angle of 48 degrees to each other. Calculate the resultant force and the force that must be applied to the object to create equilibrium.

My work, using geometric vectors:

| r |^2 = 90^2 + 90^2 - 2(90)(90)cos132
| r | = 164.4 N

sin132 / 164.4 = sinx / 90
x = 24 degrees

Therefore the resultant force is 164.4 N, 24 degrees from 90 N.

*************MY QUESTION*********
-------- How do I find the equilibrium to this problem? Can you please provide a step by step solution?

Thanks

I assume you did the law of cos and of sin right, not checking

To hold this force in equilibrium, you need a force equal and magnitude and opposite in direction to the resultant.
so 164.4 N at 180+24 = 204 degrees from one of the original 90 N forces.

To find the force that must be applied to the object to create equilibrium, we need to consider the vector sum of the two given forces and find its magnitude and direction.

Step 1: Resolve the forces into horizontal and vertical components.

Force 1 (90 N at an angle of 48 degrees):
Horizontal component: F1x = 90 N * cos(48) = 59.38 N
Vertical component: F1y = 90 N * sin(48) = 66.31 N

Force 2 (90 N at an angle of 48 degrees, but in the opposite direction):
Horizontal component: F2x = -90 N * cos(48) = -59.38 N
Vertical component: F2y = -90 N * sin(48) = -66.31 N

Step 2: Find the sum of the horizontal and vertical components separately.

Horizontal component: Fx = F1x + F2x = 59.38 N + (-59.38 N) = 0 N
Vertical component: Fy = F1y + F2y = 66.31 N + (-66.31 N) = 0 N

Step 3: Determine the magnitude and direction of the force required for equilibrium.

Since the sum of the horizontal and vertical components is zero, it means there is no net force in either direction. Hence, the force required for equilibrium is zero.

In conclusion, to create equilibrium, no additional force needs to be applied to the object.

To find the force that must be applied to the object to create equilibrium, we need to consider that the object is currently experiencing two forces, each with a magnitude of 90 N. These forces are making an angle of 48 degrees with respect to each other.

In equilibrium, the net force acting on the object must be zero. Since the two given forces are not in the same direction, we can use the concept of vector addition to find the equilibrium force.

Here's a step-by-step solution:

1. Draw a diagram: Draw a vector diagram representing the two forces acting on the object. Label the forces as F1 and F2. The angle between them is 48 degrees, so you can sketch them accordingly.

2. Resolve the forces: Resolve each force into its horizontal and vertical components. To do this, you can use trigonometry. The vertical component of each force is given by F * sin(angle) and the horizontal component is given by F * cos(angle).

Let's call the vertical components V1 and V2, and the horizontal components H1 and H2.

V1 = F1 * sin(angle)
= 90 N * sin(48 degrees)
≈ 64.81 N

V2 = F2 * sin(angle)
= 90 N * sin(48 degrees)
≈ 64.81 N

H1 = F1 * cos(angle)
= 90 N * cos(48 degrees)
≈ 59.99 N

H2 = F2 * cos(angle)
= 90 N * cos(48 degrees)
≈ 59.99 N

3. Add the horizontal components: The horizontal components of the forces must cancel out for equilibrium. Add H1 and H2 together to find the net horizontal force.

H = H1 + H2
≈ 59.99 N + 59.99 N
≈ 119.98 N

4. Add the vertical components: The vertical components of the forces must also cancel out for equilibrium. Add V1 and V2 to find the net vertical force.

V = V1 + V2
≈ 64.81 N + 64.81 N
≈ 129.62 N

5. Calculate the magnitude and direction of the equilibrium force: Now, we have the horizontal and vertical components of the equilibrium force. We can use these components to find the magnitude and direction of the force.

Equilibrium force magnitude:
|E| = √(H² + V²)
= √(119.98 N² + 129.62 N²)
≈ 169.41 N

Equilibrium force direction:
θ = arctan(V / H)
= arctan(129.62 N / 119.98 N)
≈ 48.87 degrees

Therefore, the force that must be applied to the object to create equilibrium is approximately 169.41 N, at an angle of approximately 48.87 degrees with respect to the horizontal.