A force vector applied on a mass is represented as $\vec{F} = 6\hat{i} - 8\hat{j} + 10\hat{k}$ and accelerates

Question

A force vector applied on a mass is represented as $\vec{F} = 6\hat{i} - 8\hat{j} + 10\hat{k}$ and accelerates with $1 	ext{ m/s}^2$. What will be the mass of the body?

Accepted Answer

1. **Newton's Second Law:** The relationship between force, mass, and acceleration is given by $\vec{F} = m\vec{a}$. In terms of magnitude, $|\vec{F}| = m|\vec{a}|$ [Source 75, 77].
2. **Magnitude of Force Vector:** The magnitude of the force vector $\vec{F} = F_x\hat{i} + F_y\hat{j} + F_z\hat{k}$ is calculated using the formula $|\vec{F}| = \sqrt{F_x^2 + F_y^2 + F_z^2}$ [Source 53, 54].
   Substituting the given components ($F_x=6, F_y=-8, F_z=10$):
   $$ |\vec{F}| = \sqrt{(6)^2 + (-8)^2 + (10)^2} $$
   $$ |\vec{F}| = \sqrt{36 + 64 + 100} = \sqrt{200} = 10\sqrt{2} 	ext{ N} $$
3. **Calculation of Mass:** Given the acceleration magnitude $a = 1 	ext{ m/s}^2$:
   $$ m = \frac{|\vec{F}|}{a} = \frac{10\sqrt{2}}{1} = 10\sqrt{2} 	ext{ kg} $$

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