Answer to Question #255767 in Classical Mechanics for Henry

Question

Three vectors

𝒗1,𝒗2, 𝒗3 originate from a single point and are coplanar on the x-y plane. The vector 𝒗1 is at an angle of 30deg clockwise from the positive y axis. The vector 𝒗2 is along the positive x axis and has a magnitude of 200 units. The vector 𝒗3 has a magnitude of 260 units and is at an angle 𝜃 counter clockwise from the negative y axis, such than tan𝜃= 5/12.

A) Determine the magnitude of v1 and the resultant vector, such that the magnitude of resultant vector is minimized. Report results in three significant figures.

B) What is the angle that the resultant vector makes with x-axis?

Answer 1

$F_x=v_1\sin 30°+v_2+v_3\sin\arcsin \frac 5{13}=0.5v_1+300,$

$F_y=v_2\cos30°-v_3\cos\arccos \frac{ 12}{13}=\frac{\sqrt 3}2v_1-240,$

a)

$F^2=F_x^2+F_y^2=v_1^2+v_1(300-240\sqrt 3)+147600,$

$v_1=120\sqrt3-150=57.8,$

$F=379.8,$

b)

$F_x=328.9,$

$F_y=-189.9,$

$\theta=\arctan\frac{F_y}{F_x}=-30°.$