Question

two forces, $f_t = 3n$ and $f_s = 4n$, exerted in opposite directions as shown in the figure, cause an object to move with an acceleration of magnitude $a_0$. if these forces are instead exerted perpendicular to each other, what is the new magnitude of the acceleration of the object?
(a) $\frac{1}{5}a_0$
(b) $a_0$
(c) $5a_0$
(d) $7a_0$

Explanation:

Step1: Find initial net - force

When the forces are in opposite directions, the net - force $F_1=\vert F_S - F_T\vert$. Given $F_T = 3N$ and $F_S=4N$, then $F_1=\vert4 - 3\vert=1N$. According to Newton's second law $F = ma$, so $1N=ma_0$, where $m$ is the mass of the object.

Step2: Find new net - force

When the forces are perpendicular to each other, use the Pythagorean theorem to find the net - force $F_2$. $F_2=\sqrt{F_T^{2}+F_S^{2}}$. Substitute $F_T = 3N$ and $F_S = 4N$ into the formula, we get $F_2=\sqrt{3^{2}+4^{2}}=\sqrt{9 + 16}=\sqrt{25}=5N$.

Step3: Find new acceleration

Again, according to Newton's second law $F=ma$. Let the new acceleration be $a$. Since $F_2=ma$ and $F_1 = ma_0$ with $F_1 = 1N$ and $F_2 = 5N$, we have $5N=ma$ and $1N=ma_0$. Then $a = 5a_0$.

Answer:

C. $5a_0$