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The figure shows three ropes tied together in a knot. One of your friends pulls on a rope with...

Question:

The figure 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. What force (magnitude and direction) must you apply to the third rope to keep the knot from moving?

Vector Addition of Forces

Consider two vectors {eq}\displaystyle {\vec{a}} {/eq} and {eq}\displaystyle {\vec{b}} {/eq} making an angle {eq}\displaystyle {\theta} {/eq} with respect to each other. Then their resultant vector has a magnitude {eq}\displaystyle {\sqrt{a^2+b^2+2ab \cos \theta}} {/eq}. The resultant makes an angle {eq}\displaystyle {\alpha} {/eq} with the vector {eq}\displaystyle {\vec{a}} {/eq} given by, {eq}\displaystyle { \tan \alpha=\frac{b \sin \theta}{a+b \cos \theta}} {/eq}.

Answer and Explanation:

Here it is given that the three ropes are tied to form a know with two of your friends pulling it with forces of magnitudes say {eq}\displaystyle {a} {/eq} and {eq}\displaystyle {b} {/eq}, with {eq}\displaystyle {a=3} {/eq} and {eq}\displaystyle {b=5} {/eq} at an angle of {eq}\displaystyle {\theta=120^{\circ}} {/eq} with each other.

Clearly the resultant force has a magnitude,

{eq}\displaystyle { \sqrt{3^2+5^2+2\times 3\times 5 \cos 120}}=\sqrt{19} {/eq}.

The angle made by this resultant force with the horizontal vector {eq}\displaystyle {\vec{a}} {/eq} is,

{eq}\displaystyle { \tan \alpha=\frac{5 \sin 120}{3+5 \cos 120}}=8.66 {/eq}

This gives,

{eq}\displaystyle {\alpha=83.41^{\circ}} {/eq}.

Therefore to keep the knot in place you need to pull the third rope with a force of {eq}\displaystyle {\sqrt{19}\ N} {/eq} in the exactly opposite direction. This direction would make an angle of {eq}\displaystyle {180-83.41=96.59^{\circ}} {/eq} clockwise with the 3 N force.


Learn more about this topic:

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Vector Addition (Geometric Approach): Explanation & Examples

from UExcel Physics: Study Guide & Test Prep

Chapter 2 / Lesson 2
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