A particle undergoes three displacements. The ?rst has a magnitude of 15 m and m

Question

A particle undergoes three displacements.
The ?rst has a magnitude of 15 m and makes
an angle of 22?with the positive x axis. The
second has a magnitude of 8.4 m and makes
an angle of 131?with the positive x axis.
After the third displacement the particle
returns to its initial position.Find the magnitude of the third displacement.
Answer in units of m

Find the angle of the third displacement (measured from the positive x axis, with counterclockwise positive within the limits of -180?to +180?).
Answer in units of? Degree

Explanation / Answer

from the question, we know that the D1 + D2 = - D3
find the displacement of D1 and D2 in x y component

D1 = (15cos22i + 15sin22j) = (13.91i + 5.62j)
D2 = (8.4cos131i + 8.4sin131j) = (-5.51i + 6.34j)

D3 = - D1 - D2
D3 = -(13.91i + 5.62j) - (-5.51i + 6.34j) = (-8.4i - 11.96j)
magnitude of D3 = ((-8.4)2 + (-11.96)2) = 14.62 m

angle = tan-1 (11.96/8.4) = 54.92 degree

since both x and y components are negative , the angle will be from below the negative x-axis.

180 - 54.92 = 125.08 degree

the direction is -125.08 degree counterclockwise from the positive x-axis.

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