Q1E - University Physics with Modern Physics

Question

Two dogs pull horizontally on ropes attached to a post; the angle between the ropes is  $60°$. If Rover exerts a force of 270 N and Fido exerts a force of 300 N, find the magnitude of the resultant force and the angle it makes with Rover’s rope.

Step-by-Step Solution

Answer

If two dogs Rover and Fido pull their ropes attached to a post with 270 N and 300 N forces respectively having an angle $60°$ between them, the resultant force is $493.86 N$  and makes an angle  $31.74°$ with Rover's rope.

Step 1: Given data

The magnitude of the force applied by Rover is

  $\overrightarrow{A}=270 \text{N}$

The magnitude of the force applied by Fido is

  $\overrightarrow{B}=300 \text{N}$

The angle between the two forces is

  $\theta =60°$

Step 2: Vector addition

The magnitude of vector sum of two vectors $\overrightarrow{A}$  and $\overrightarrow{B}$  with an angle  $\theta$ between them is

  $R=\sqrt{A^2+B^2+2ABcos\theta } .....\left(1\right)$

The angle made by the resultant vector with  $\overrightarrow{A}$ is

    $f=ta{n}^{-1}\left(\frac{Bsin\theta }{A+Bcos\theta }\right) .....\left(2\right)$

Step 3: Sum of two forces

From equation (1), the magnitude of sum of the two forces is

  $\begin{array}{rcl}R& =& \sqrt{{270}^2+{300}^2+2\times 270\times 300\cos 60°} \text{N}\\ & =& 493.86 \text{N}\\ & & \end{array}$

From equation (2), the angle made by the resultant force with Rover's rope is

 $\begin{array}{rcl}\varphi & =& {\tan }^{-1}\left(\frac{300\sin 60°}{270+300\cos 60°}\right)\\ & =& {\tan }^{-1}\left(0.619\right)\\ & =& 31.74°\\ & & \end{array}$

Thus, the resultant force is $493.86 \text{N}$  and makes an angle  $31.74°$ with Rover's rope.