The magnitudes of the four displacement vectors shown in the drawing are A = 15.0 m, B = 11.0 m, C = 11.0 m, and D = 27.0 m. Determine the (a) magnitude and (b) direction for the resultant that occurs when these vectors are added together. Specify the direction as a positive (counterclockwise) angle from the +x axis.
To determine the magnitude and direction of the resultant vector, you need to add all the displacement vectors together.
(a) Magnitude of the resultant vector:
To find the magnitude of the resultant vector, you can use the Pythagorean theorem. The Pythagorean theorem states that the square of the hypotenuse (in this case, the resultant vector) is equal to the sum of the squares of the other two sides.
Let's find the magnitude:
R = √(A^2 + B^2 + C^2 + D^2)
Substituting the given values:
R = √(15.0^2 + 11.0^2 + 11.0^2 + 27.0^2)
R = √(225 + 121 + 121 + 729)
R = √(1196)
R ≈ 34.6 m
Therefore, the magnitude of the resultant vector is approximately 34.6 m.
(b) Direction of the resultant vector:
To determine the direction of the resultant vector, you can use trigonometry. The direction is given as a positive angle from the +x axis.
Let's find the direction:
We can use the tangent function (tan) to find the angle (θ). Use the inverse tangent (arctan) function to find the angle.
θ = arctan((∑y) / (∑x))
Where (∑y) is the sum of all the vertical components and (∑x) is the sum of all the horizontal components.
In this case, we have to find the x and y components of each vector:
Ax = A * cos(α) Ay = A * sin(α)
Bx = B * cos(β) By = B * sin(β)
Cx = C * cos(γ) Cy = C * sin(γ)
Dx = D * cos(δ) Dy = D * sin(δ)
α, β, γ, and δ are the angles made by vectors A, B, C, and D with the +x axis, respectively.
Substituting these values:
∑x = Ax + Bx + Cx + Dx
∑y = Ay + By + Cy + Dy
Upon calculating the sums:
∑x = (15.0 * cos(α)) + (11.0 * cos(β)) + (11.0 * cos(γ)) + (27.0 * cos(δ))
∑y = (15.0 * sin(α)) + (11.0 * sin(β)) + (11.0 * sin(γ)) + (27.0 * sin(δ))
Once you find the values of (∑x) and (∑y), you can substitute them into the equation to find the angle:
θ = arctan(∑y / ∑x)
By substituting the values, you can find the direction of the resultant vector in degrees or radians.