Question

A man pushing a mop across a floor causes it to undergo two displacements. The first has a magnitude of 154 cm and makes an angle of 122° with the positive x axis. The resultant displacement has a magnitude of 147 cm and is directed at an angle of 38.0° to the positive x axis. Find the magnitude and direction of the second displacement.

Answer 1

Total displacement will be given by:

d = d1 + d2

d1 = first displacement = 154 cm at angle of 122 deg with +ve x-axis

d2 = 2nd displacement = ?

d = total displacement = 147 cm at an angle of 38 deg with +ve x-axis

So,

d2 = d - d1

Now Suppose given that Vector is R and it makes angle A with +x-axis, then it's components are given by:

Rx = R*cos A

Ry = R*sin A

Using above rule: d1 = d1x i + d1y j, where

d1x = 154*cos 122 deg = -81.6 cm

d1y = 154*sin 122 deg = 130.6 cm

Similarly

dx = 147*cos 38 deg = 115.8 cm

dy = 147*sin 38 deg = 90.5 cm

So Using these values:

d2 = (dx i + dy j) - (d1x i + d1y j)

d2 = (115.8 - (-81.6)) i + (90.5 - 130.6) j

d2 = 197.4 i - 40.1 j

Magnitude of d2 = sqrt (d2x^2 + d2y^2)

|d2| = sqrt (197.4^2 + (-40.1)^2) = 201.4 cm

direction of d2 = arctan (d2y/d2x)

direction = arctan (-40.1/197.4) = -11.5 deg = 360 - 11.5

Direction = 348.5 deg to the positive x-axis

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