Question

1. (a) find the magnitudes of the forces f₁ and f₂ that add to give the total force f_tot shown in figure 4.34. this may be done either graphically or by using trigonometry. (b) show graphically that the same total force is obtained independent of the order of addition of f₁ and f₂. (c) find the direction and magnitude of some other pair of vectors that add to give f_tot. draw these to scale on the same drawing used in part (b) or a similar picture.

f_tot = 20 n
free - body diagram
figure 4.34

Explanation:

Step1: Use trigonometry for $F_1$

Since $\cos\theta=\frac{F_1}{F_{tot}}$, then $F_1 = F_{tot}\cos\theta$. Substituting $F_{tot}=20\ N$ and $\theta = 35^{\circ}$, we have $F_1=20\cos35^{\circ}\approx20\times0.819 = 16.38\ N$.

Step2: Use trigonometry for $F_2$

Since $\sin\theta=\frac{F_2}{F_{tot}}$, then $F_2 = F_{tot}\sin\theta$. Substituting $F_{tot}=20\ N$ and $\theta = 35^{\circ}$, we get $F_2=20\sin35^{\circ}\approx20\times0.574 = 11.48\ N$.

Step3: Graphical addition - Commutative property

To show graphically that the total force is independent of the order of addition of $\mathbf{F}_1$ and $\mathbf{F}_2$, we can use the parallelogram - law of vector addition. If we first draw $\mathbf{F}_1$ and then $\mathbf{F}_2$ starting from the tip of $\mathbf{F}_1$, or first draw $\mathbf{F}_2$ and then $\mathbf{F}_1$ starting from the tip of $\mathbf{F}_2$, the resultant vector (diagonal of the parallelogram formed by $\mathbf{F}_1$ and $\mathbf{F}_2$) will be the same $\mathbf{F}_{tot}$.

Step4: Find another pair of vectors

Let's consider two vectors $\mathbf{A}$ and $\mathbf{B}$ such that $\mathbf{A}$ has a magnitude of $10\ N$ at an angle of $60^{\circ}$ with the x - axis and $\mathbf{B}$ has a magnitude of $17.32\ N$ at an angle of $ - 30^{\circ}$ with the x - axis.
$A_x = 10\cos60^{\circ}=5\ N$, $A_y = 10\sin60^{\circ}=5\sqrt{3}\ N$
$B_x = 17.32\cos(- 30^{\circ})=15\ N$, $B_y = 17.32\sin(-30^{\circ})=- 8.66\ N$
$F_{totx}=A_x + B_x=5 + 15=20\ N$
$F_{toty}=A_y + B_y=5\sqrt{3}-8.66 = 0\ N$ (approx)
The magnitude of the resultant is $\sqrt{F_{totx}^2+F_{toty}^2}=20\ N$

Answer:

(a) $F_1\approx16.38\ N$, $F_2\approx11.48\ N$
(b) By using the parallelogram - law of vector addition, the order of addition of $\mathbf{F}_1$ and $\mathbf{F}_2$ does not affect the resultant $\mathbf{F}_{tot}$.
(c) One possible pair: A vector of magnitude $10\ N$ at an angle of $60^{\circ}$ with the x - axis and a vector of magnitude $17.32\ N$ at an angle of $-30^{\circ}$ with the x - axis.