Two forces of 40 pounds and 28 pounds act on an object. The angle between the two forces is 65 degrees. Find the magnitude of the resultant force to the nearest pound. Using this answer, find the measure of the angle formed between the resultant and the smaller force, to the nearest degree.
I saw this problem in a text while I was reading and was wondering how to do this problem. I was trying to teach myself about this topic and I would like to see how a problem like this would be solved. Can someone please show me.
Best Answer
Hint 1: Using properties of the dot product, we get $$ \begin{align} (f_1+f_2)\cdot(f_1+f_2) &=f_1\cdot f_1+2f_1\cdot f_2+f_2\cdot f_2\\ &=40^2+2\cdot40\cdot28\cos(65^\circ)+28^2 \end{align} $$ Hint 2: The Law of Sines says $$ \frac{\sin{\theta}}{40}=\frac{\sin(65^\circ-\theta)}{28} $$ Hint 3: if we let $f$ be the magnitude of the resulting force, then we can also use the Law of Cosines to get $$ 40^2=28^2+f^2-2\cdot28f\cos(\theta) $$
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