Question

The following forces pull on a ring along the same plane (i.e. coplanar): 200 N at 30° E of N, 500 N at 10° N of E, 300 N at 60° W of S, and an unknown force that keeps the ring in equilibrium. Find the magnitude and direction of this unknown force.

Answer 1

To maintain equilibrium, the unknown force must have equal magnitude and opposite direction to the resultant of the given forces. The magnitude is approximately 442.4 N and the direction is almost directly south.

Step-by-step explanation:

To find the magnitude and direction of the unknown force that keeps the ring in equilibrium, we must first resolve each of the given forces into its x (east-west) and y (north-south) components and then find the resultant force that will balance these components to maintain equilibrium. The forces given are:

200 N at 30° East of North.

500 N at 10° North of East.

300 N at 60° West of South.

We will calculate the x and y components of these three forces using trigonometry:

Force 1:
Force 2:

Force 3:

Next, we add the components from all forces:

Total x-component = 173.2 N + 87.2 N - 259.8 N = 0.6 N east

Total y-component = 100 N + 492.4 N - 150 N = 442.4 N north

To keep the ring in equilibrium, the unknown force must equal but opposite to the resultant of these components. Thus, the unknown force's components are:

x-component = -0.6 N (west)

y-component = -442.4 N (south)

The magnitude of the unknown force (F) is calculated using the Pythagorean theorem:

F = √((-0.6 N)^2 + (-442.4 N)^2) = approximately 442.4 N

Since the x-component is very small relative to the y-component, the direction of the unknown force is almost directly south.