Question

The magnitudes of the vectors are F = 84 N and P = 77 N. They act at angles theta = 47 deg and phi = 52 deg. Find the angle between the resultant of the two forces and the x-axis in deg. (Note: these values may be different from above!)

Answer 1

Answer:

Explanation:

Force, F = 84 N at 47°

Force, P = 77 N at 52°

First write the forces in vector form

$\overrightarrow{F}=84\left ( Cos47\widehat{i}+Sin47\widehat{j} \right )$

$\overrightarrow{F}=57.3\widehat{i}+61.4\widehat{j}$

$\overrightarrow{P}=77\left ( Cos52\widehat{i}+Sin52\widehat{j} \right )$

$\overrightarrow{P}=47.4\widehat{i}+60.7\widehat{j}$

Let R be the resultant of two forces.

$\overrightarrow{R} = \overrightarrow{F} + \overrightarrow{P}$

$\overrightarrow{R}=(57.3+47.4)\widehat{i}+(61.4+60.7)\widehat{j}$

$\overrightarrow{R}=104.7\widehat{i}+122.1\widehat{j}$

Let it makes an angle θ from X axis

$tan \theta =\frac{122.1}{104.7}$

θ = 49.4°

Answer 2

Answer:

The angle between the resultant of the two forces and the x-axis is 56.93°.

Explanation:

Given that,

Magnitude of the vector F = 84 N

Magnitude of the vector P = 77 N

Angle for F= 47°

Angle for P = 52°

We need to calculate the resultant vector

Using formula of resultant vector

$\vec{R}=\vec{F}+\vec{P}$

$\vec{R}=85(\cos47i+\sin47j)+77(\cos52i+\sin52j)$

$\vec{R}=85\cos47+77\cos52+85\sin47+77\sin52$

$\vec{R}=105.35i+122.84j$

We need to calculate the magnitude

$R=\sqrt{(105.35)^2+(122.84)^2}$

$R=161.82\ N$

We need to calculate the angle between the resultant of the two forces and the x-axis

Using formula of angle

$\tan\theta=\dfrac{R}{105.34}$

$\theta=\tan^{-1}(\dfrac{R}{105.34})$

Put the value into the formula

$\theta=\tan^{-1}(\dfrac{161.82}{105.34})$

$\theta=56.93^{\circ}$

Hence, The angle between the resultant of the two forces and the x-axis is 56.93°.