Two forces of 55N and 85N act on an object simultaneously and the resultant force is 125N. What is the measurement of the angle between the two forces?

Question

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Answer 1

Denote the angle by θ.

Start with a parallelogram rule. The sum vector is along the diagonal of the parallelogram, splitting the parallelogram into two triangle with sides of lengths 55, 85, and 125. The angle opposite to the side of length 125 is π - θ radians. Now apply the law of cosines to finish.

125² = 55² + 85² - 2*55*85 cos(π - θ)

15625 = 3025 + 7225 - 9350 cos(π - θ)

cos(π - θ) = 5375/9350 = -215/374

cos θ = 215/374

θ = 0.95835 radians ( 54.90972 degrees )

Answer 2

Use law of cosine,

125² = 55² + 85² - 2*55*85cos(θ)

Solve for θ,

θ = 125.09 ^o

Answer 3

Let us indicate the magnitude of the two forces with A and B for simplicity, and C their resultant, and let θ be the angle between the two of them. Let us introduce a frame of reference such that the line of action of A is along x.

Let us write the components of A and B:

A_x = A, A_y = 0

B_x = B cosθ, B_y = B sinθ

so that the components of the resultant read:

C_x = A_x + B_x = A + B cosθ

C_y = A_y + B_y = Bsinθ

The magnitude of the resultant is C^2 = C_x^2 + C_y^2 = A^2 + B^2 + 2ABcosθ. Solving this for cosθ gives:

cosθ = 0.575

so that θ = 55 deg

Answer 4

125^2 = 85^2 + 55^2 -2(85)(55)CosR where R=angle opposite the resultant length= about 125.1

angle between the two forces = 180-R= 180-125.1 = about 54.9 degrees