Resolve a 500-N force into two rectangular vector components such that the ratio of their magnitudes is 2:1. Calculate the angle between the greater component and the 500-N force.
To resolve a force into two rectangular vector components, we can use trigonometry. Let's denote the magnitude of the first component as twice the magnitude of the second component.
Let F be the magnitude of the 500-N force, and let F1 and F2 be the magnitudes of the two components.
Given:
F = 500 N
F1 = 2F2
We can start by finding the magnitudes of the components:
F = F1 + F2
Substituting the given values:
500 N = 2F2 + F2
Combining like terms:
500 N = 3F2
Solving for F2:
F2 = 500 N / 3
F2 ≈ 166.67 N
Since F1 = 2F2:
F1 = 2 * 166.67 N = 333.34 N
Now that we have the magnitudes of the two components, let's find the angle θ between the greater component (F1) and the 500-N force.
Using trigonometry, we can use the inverse tangent function:
tan(θ) = opposite/adjacent
In this case, the opposite side is F2 and the adjacent side is F1.
tan(θ) = F2 / F1
Substituting the values:
tan(θ) = (166.67 N) / (333.34 N)
Taking the inverse tangent of both sides:
θ = tan^(-1)((166.67 N) / (333.34 N))
Using a calculator:
θ ≈ 26.5°
Therefore, the angle between the greater component and the 500-N force is approximately 26.5 degrees.
To resolve a force into rectangular components, we can use trigonometry.
Let's assume that the magnitude of the greater component is 2x, and the magnitude of the smaller component is x. So, we have the ratio of their magnitudes as 2:1.
To find the angle between the greater component and the 500-N force, we need to determine the angles of each component with the horizontal axis.
Let's consider the horizontal component first. Since it is along the x-axis, its angle with the horizontal axis is 0 degrees (or π/2 radians).
Now, let's find the angle for the vertical component. Since it is perpendicular to the horizontal component, the angle between the vertical component and the horizontal axis is 90 degrees (or π/2 radians).
To find the angle between the greater component and the 500-N force, we can use the inverse tangent (arctan) function. We'll divide the magnitude of the vertical component (x) by the magnitude of the horizontal component (2x) to get the tangent of the angle:
tan(angle) = (x) / (2x)
tan(angle) = 1 / 2
Take the inverse tangent of both sides to solve for the angle:
angle = arctan(1 / 2)
angle ≈ 26.57 degrees
So, the angle between the greater component and the 500-N force is approximately 26.57 degrees.