Question

A man pushing a mop across a floor causes it to undergo two displacements. The first has a magnitude of 156 cm and makes an angle of 126° with the positive x axis. The resultant displacement has a magnitude of 133 cm and is directed at an angle of 34.0° to the positive x axis. Find the magnitude and direction (counterclockwise of the positive x axis) of the second displacement.

Answer 1

Step-by-step explanation:

A = 156 cm at 126°

R = 133 cm at 34°

Let the second displacement is
`\overrightarrow{B}=B\widehat{i}+B\widehat{j}`.

Write the displacements in the vector form.

`\overrightarrow{A}=156\left ( Cos 126\widehat{i}+Sin126\widehat{j} \right )`

`\overrightarrow{A}=-91.7\widehat{i}+126.2\widehat{j}`

`\overrightarrow{R}=133\left ( Cos 34\widehat{i}+Sin34\widehat{j} \right )`

`\overrightarrow{R}=110.3\widehat{i}+74.4\widehat{j}`

According to the vector sum

`\overrightarrow{R}=\overrightarrow{A}+\overrightarrow{B}`

Substituting the values

`110.3\widehat{i}+74.4\widehat{j} = -91.7\widehat{i}+126.2\widehat{j} + B\widehat{i}+B\widehat{j}`

[tex] B\widehat{i}+B\widehat{j} = 202\widehat{i} - 51.8\widehat{j}[tex]

Answer 2

second displacement is R = 208.6 cm and θ = 346º

Step-by-step explanation:

This is a problem of adding vectors, the easiest way to work these problems is to decompose the vectors and find the resulting vectors on each axis

Let's use trigonometry to break down each displacement vector. Let's start with vector 1 that has a magnitude m1 = 156 cm

sin 126 = Y1 / m1

Y1. = m1 without 126

Y1 = 156 without 126

Y1 = 126.2 cm

cos 126 = X1 / m1

X1 = m1 cos 126

X1 = 156 cos 126

X1 = - 91.69 cm

The resulting vector has R = 133 cm

sin 34 = Ry / R

Ry = R sin 34

Ry = 133 sin 34

Ry = 74.37 cm

cos 34 = Rx / R

Rx = R cos 34

Rx = 133 cos 34

Rx = 110.3 cm

We already have all the components, we can add algebraically on each axis

X axis

Rx = X1 + X2

X2 = Rx -X1

X2 = 110.3 - (-91.69)

X2 = 202 cm

Y Axis

Ry = Y1 + Y2

Y2 = Ry - Y1

Y2 = 74.37 -126.3

Y2 = -52 cm

Let's build the resulting vector

R = (202 i ^ + 52 y ^) cm

R = (202, -52) cm

We can also use the Pythagorean triangle and trigonometry to find the module and direction

R² = Rx² + Ry²

R = √(202² + 52²)

R = 208.6 cm

tan θ = Ry / Rx

tan θ = -52/202

θ = tan⁻¹ (-0.257)

θ = -14.4º

The negative sign indicates that it is measured from the x-axis clockwise, to measure counterclockwise from the x-axis

θ = 360-14

θ = 346º