force and motion

Question

Consider two displacements, one of magnitude 3 m and another of magnitude 4 m. What angle between the directions of this two displacements give a resultant displacement of magnitude (a) 7 m, (b) 1 m, and (c) 5 m.

Answer 1

R(θ) = magnitude of resultant

 

Without loss of generality, align the 4m displacement with the x-axis.

 

Rx = 4 + 3cosθ

 

Ry = 3sinθ 

 

R^2 = Rx^2 + Ry^2 = (4 + 3cosθ)^2 + (3sinθ)^2

 

R^2 = 16 + 9(cosθ)^2 + 24cosθ + 9(sinθ)^2 = 16 + 9[(cosθ)^2 + (sinθ)^2]+ 24cosθ

 

R^2 = 16 + 9 + 24cosθ = 25 + 24cosθ

 

θ = cos^-1[(R^2 - 25) / 24]

 

(a)  R = 7

 

θ = cos^-1[(7^2 - 25) / 24] = 0 degrees

 

(b)  R = 1

 

θ = cos^-1[(1^2 - 25) / 24] = 180 degrees

 

(c)  R = 5

 

θ = cos^-1[(5^2 - 25) / 24] = 90 degrees or 270 degrees