The scale in Fig. P12 is being pulled on via three ropes. (The force on each rope is F = 180 N.) What net force does the scale read?
F 30 Degrees
The sum of the horizontal F comonents is 0.
The sum of the vertical F components is 180 N
We need to know the angle of inclination to vertical for each rope.
To determine the net force on the scale, we need to analyze the forces acting in different directions.
Given:
Force on each rope, F = 180 N
Angle between each force and the horizontal direction, θ = 30 degrees
Let's break down the forces into their horizontal and vertical components.
First, we need to find the horizontal and vertical components of a single force F acting at an angle of 30 degrees.
The horizontal component (F_h) can be calculated using the formula:
F_h = F * cos(θ)
Substituting the given values:
F_h = 180 N * cos(30 degrees)
Calculating the value:
F_h = 180 N * 0.866 (rounded to three decimal places)
F_h ≈ 155.88 N
The vertical component (F_v) can be calculated using the formula:
F_v = F * sin(θ)
Substituting the given values:
F_v = 180 N * sin(30 degrees)
Calculating the value:
F_v = 180 N * 0.5
F_v = 90 N
Now, let's analyze the forces acting on the scale.
There are three ropes pulling on the scale, each with the same magnitude and angle. Since each force is pulling at 30 degrees, the vertical components of the forces will cancel each other out, as they are equal in magnitude and opposite in direction.
Therefore, the net force acting on the scale will only include the horizontal components of the forces. Since there are three ropes, the net horizontal force will be three times the horizontal component (F_h) of a single rope.
Net horizontal force = 3 * F_h
Net horizontal force ≈ 3 * 155.88 N
Calculating the value:
Net horizontal force ≈ 467.64 N
Therefore, the scale will read a net force of approximately 467.64 N.
To find the net force that the scale reads, we need to consider the forces acting on it. In this case, there are three ropes pulling on the scale with a force of 180 N each, forming an angle of 30 degrees with the horizontal axis.
First, let's draw a diagram to visualize the forces. We have three ropes pulling on the scale, forming a triangle. The angle between two ropes is 30 degrees.
```
F F
_____________
\ /
\ /
\ /
\ /
\
\
\ F
```
Now, we can break down each force into its horizontal and vertical components. Since the angle between the ropes is 30 degrees, the vertical component of each force can be calculated using the sine function (sin(30)) and the horizontal component using the cosine function (cos(30)).
Vertical component of force = F * sin(angle)
Horizontal component of force = F * cos(angle)
Let's calculate the vertical and horizontal components of one force:
Vertical component = 180 N * sin(30 degrees) ≈ 90 N
Horizontal component = 180 N * cos(30 degrees) ≈ 155.88 N
Since there are three forces acting in the vertical direction, the net vertical force can be found by summing up the vertical components of each force:
Net vertical force = 3 * 90 N = 270 N
Similarly, the net horizontal force can be found by summing up the horizontal components of each force:
Net horizontal force = 3 * 155.88 N = 467.64 N
Finally, we can calculate the magnitude of the net force using the Pythagorean theorem:
Net force = √(Net vertical force^2 + Net horizontal force^2)
Net force = √(270 N^2 + 467.64 N^2) ≈ 543.35 N
Therefore, the scale would read a net force of approximately 543.35 N.