Force — Question 8

Question 8

Two prescribed force magnitudes have a feasibility interval

Two forces have magnitudes 7 N and 3 N; their angle can be selected.

1. Find the range of possible resultant magnitudes.

2. Find the angle needed for an 8 N resultant.

3. Can they produce a 2 N resultant or balance one another?

Question 8 - Solution

Two prescribed force magnitudes have a feasibility interval

(a) Use parallel and opposite limits.

$$|7-3|\le R\le7+3\ \Rightarrow\ \boxed{4\le R\le10\,\mathrm N}$$ The magnitude lies in $[|7-3|,7+3]=[4,10]$ N. The endpoints correspond to opposite and same directions.

(b) Use the cosine rule.

$$\cos\theta=\frac{8^2-7^2-3^2}{2(7)(3)}=\frac17,\quad\theta=81.787^\circ$$ $8^2=7^2+3^2+2(7)(3)\cos\theta$, so $\cos\theta=1/7$ and $\theta\approx81.787^\circ$.

(c) Apply the lower bound.

$$2\,\mathrm N<R_{\min}=4\,\mathrm N,\quad0<R_{\min}$$ Neither 2 N nor zero is reachable. Unequal magnitudes cannot cancel completely even when antiparallel.