Two dogs pull horizontally on ropes attached to a post; the angle between the ropes is 51.0 deg....

Question:

Two dogs pull horizontally on ropes attached to a post; the angle between the ropes is 51.0°. Dog A exerts a force of 290 N, and dog B exerts a force of 328 N.

a) Find the magnitude of the resultant force?

b) Find the angle the resultant force makes with the rope of dog A?

Vector Addition:

Force is known as the push or pulls on an object with mass and this interaction results in a change in velocity of an object. Force is a vector, which means it has both magnitude and direction.

Answer and Explanation:

Given:

  • Force exerted by Dog A : {eq}F_A = 290\ N{/eq}
  • Force exerted by Dog B : {eq}F_B = 328\ N{/eq}
  • Angle between the ropes : {eq}\theta = 51^{\circ}{/eq}

Find:

  • Magnitude and angle of the resultant force : {eq}F_R =?{/eq} and {eq}\theta= ?{/eq}


To further understand the problem we need to visualize it first using the figure as shown below,

This image was generated using draw.io


(Part a: Magnitude)

We will use the parallelogram Law of vector addition to solve for the resultant force

$$\begin{align} F_R^2 &= F_A^2 + 2F_A F_B \cos \theta +F_B^2 \\[0.3cm] &= \sqrt {(290\ \rm N)^2 + 2 (290\ \rm N)(328\ \rm N)(cos(51.0^{\circ})) + (328\ \rm N)^2) } \\[0.3cm] &= \sqrt{311405.9 \ \rm N^2} \\[0.3cm] &\approx \boxed{558.0\ \rm N} \end{align} $$


(Part b: Angle)

To solve for the angle of the resultant force,we will use the formula,

$$\begin{align} \theta &= \tan^{-1}\left(\frac{\sum y}{\sum x} \right) \\[0.3cm] &= \tan^{-1} \left(\frac{F_B \sin \theta}{F_A +F_B \cos \theta} \right) \\[0.3cm] &= \tan^{-1} \left[ \frac{(328\ \rm N) \sin(51^{\circ})}{(290\ \rm N)+(328\ \rm N) \cos(51^{\circ})} \right] \\[0.3cm] &= \tan^{-1} \left(\frac{258.0}{496.4} \right) \\[0.3cm] &= \tan^{-1}(0.52^{\circ}) \\[0.3cm] &\approx \boxed{27.4^{\circ}} \end{align} $$


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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