A body, under the action of a force $\vec{F} = 6\hat{i} - 8\hat{j} + 10\hat{k}$, acquires an acceleration of $

Question

A body, under the action of a force $\vec{F} = 6\hat{i} - 8\hat{j} + 10\hat{k}$, acquires an acceleration of $1 	ext{ m/s}^2$. The mass of this body must be:

Accepted Answer

1. **Concept:** According to Newton's Second Law of Motion, the force vector is the product of mass and the acceleration vector ($\vec{F} = m\vec{a}$). In terms of magnitude, $|\vec{F}| = m |\vec{a}|$ [NCERT Class 11, Physics Part I, Section 5.5].
2. **Calculate Magnitude of Force:** The force is given by $\vec{F} = 6\hat{i} - 8\hat{j} + 10\hat{k}$. The magnitude is calculated as:
   $$ |\vec{F}| = \sqrt{F_x^2 + F_y^2 + F_z^2} $$
   $$ |\vec{F}| = \sqrt{6^2 + (-8)^2 + 10^2} = \sqrt{36 + 64 + 100} $$
   $$ |\vec{F}| = \sqrt{200} = \sqrt{100 	imes 2} = 10\sqrt{2} 	ext{ N} $$
3. **Calculate Mass:** Given the magnitude of acceleration $|\vec{a}| = 1 	ext{ m/s}^2$.
   $$ m = \frac{|\vec{F}|}{|\vec{a}|} = \frac{10\sqrt{2}}{1} = 10\sqrt{2} 	ext{ kg} $$
   (Reference: NCERT Class 11, Physics Part I, Chapter 4: Motion in a Plane, Section 4.4 for vector magnitude, and Chapter 5: Laws of Motion, Section 5.5).

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