Question

shows three ropes tied together in a knot. One of your friends pulls on a rope with 3.0 units of force and another pulls on a second rope with 5.0 units of force. How hard and in what direction must you pull on the third rope to keep the knot from moving?

Short answer

$\overrightarrow{F}$makes an angle of ${83.4}^{°}$below the negative x-axis.

Step-by-step solution

Step 1. Given infomation

Consider that the knot in the ropes as particle in static equilibrium. The below figure shows the three ropes tied together in a knot.

From the above figure the magnitude of the force $F_1$is 3 units and the magnitude of the force$F_2$ is 5 units. Let the magnitude of the third vector be $F_3$.

Angle made by ${\overrightarrow{F}}_1$with the positive x-axis,${\theta }_1=0^{°}$

Angle made by ${\overrightarrow{F}}_2$with the positive x-axis, ${\theta }_2={120}^{°}$

Angle made by ${\overrightarrow{F}}_3$with the negative x-axis,$\theta$

Step 2. Explanation

Now express the vectors using the unit vectors.

$F_1=3i\text{and}{\overrightarrow{F}}_2=-5\sin {30}^{°}i+5\cos {30}^{°}j$Since,${\overrightarrow{F}}_1+{\overrightarrow{F}}_2+{\overrightarrow{F}}_3=0$

Rearrange the equation in terms of vector${\overrightarrow{F}}_3$

$\begin{array}{l}{\overrightarrow{F}}_3=-\left({\overrightarrow{F}}_1+{\overrightarrow{F}}_2\right)\\ {\overrightarrow{F}}_1=\left(3\right)\cos 0^{°}\widehat{i}+\left(3\right)\sin 0^{°}\widehat{j}\\ {\overrightarrow{F}}_2=-5\sin {30}^{°}\widehat{i}+5\cos {30}^{°}\widehat{j}\\ =-2.5\widehat{i}+4.33\widehat{j}\end{array}$

Therefore, the vector,$\overrightarrow{F_3}=-[3\widehat{i}-2.5\widehat{i}+4.33\widehat{j}]=-0.5\widehat{i}-4.33\widehat{j}$

Magnitude of the third force,

$F_3=\sqrt{{(-0.5)}^2+{(-4.33)}^2}$$=4.36$units

Therefore, the required force you must pull on the third rope to keep the knot from moving is 4.36 units

Direction of ${\overrightarrow{F}}_3$is,

$\theta ={\tan }^{-1}\left(\frac{-4.36}{-0.5}\right)={83.4}^{°}\ln 3^{\text{rd}}$quadrant

Hence, makes an angle of ${83.4}^{°}$below the negative x-axis.