Chapter 6.3, Problem 76E

To determine

To find:The angle between the forces given the magnitude of their resultant.

Answer to Problem 76E

Theangle between the forces is $47.4°$ .

Explanation of Solution

Given information:

The magnitude of force 1 is 3000 pounds and it is in the positive x-axis direction, and that of force 2 is 1000 pouds and it is making an angle $\theta$ with the positive x-axis. The magnitude of resultant force is 3750 pounds.

Calculations:

Given forces are

  $\begin{array}{l}{F}_1=3000i\\ F_2=1000\cos \theta i+1000\sin \theta j\end{array}$

So the resultant force will be

  $\begin{array}{l}{F}_1+F_2=(3000i)+(1000\cos \theta i+1000\sin \theta j)\\ =(3000+1000\cos \theta )i+(1000\sin \theta )j\end{array}$

Using the given magnitude of resultant force,

  $\begin{array}{l}$ {\left(\Vert F_1+F_2\Vert \right)}^2={3750}^2={(3000+1000\cos \theta )}^2+{(1000\sin \theta )}^2$14062500=9000000+6000000\cos \theta +1000000{\cos }^2\theta +1000000{\sin }^2\theta \\ 14062500-9000000=6000000\cos \theta +1000000({\cos }^2\theta +{\sin }^2\theta )\\ 5062500=6000000\cos \theta +1000000\\ 5062500-1000000=6000000\cos \theta \\ 4062500=6000000\cos \theta \\ \Rightarrow \cos \theta =\frac{4062500}{6000000}=\frac{65}{96}\\ \Rightarrow \theta ={\cos }^{-1}\left(\frac{65}{96}\right)\\ \Rightarrow \theta =47.4°\end{array}$

Conclusion:

Therefore, the angle between the forces is $47.4°$ .