Question

A body of mass $4 \, \text{kg}$ experiences two forces \[\vec{F}_1 = 5\hat{i} + 8\hat{j} + 7\hat{k} \quad \text{and} \quad \vec{F}_2 = 3\hat{i} - 4\hat{j} - 3\hat{k}.\]The acceleration acting on the body is:

• A. $-2\hat{i} - \hat{j} - \hat{k}$
• B. $4\hat{i} + 2\hat{j} + 2\hat{k}$
• C. $2\hat{i} + \hat{j} + \hat{k}$
• D. $2\hat{i} + 3\hat{j} + 3\hat{k}$

Answer 1

The Correct Option is C

A body with a mass of \(4 \, \text{kg}\) is subjected to two forces: \(\vec{F}_1 = 5\hat{i} + 8\hat{j} + 7\hat{k}\) and \(\vec{F}_2 = 3\hat{i} - 4\hat{j} - 3\hat{k}\). The objective is to determine the acceleration acting on the body.

The net force acting on the body is calculated as the vector sum of the applied forces:

\(\vec{F}_{\text{net}} = \vec{F}_1 + \vec{F}_2\)

Substituting the given force vectors:

\(\vec{F}_{\text{net}} = (5\hat{i} + 8\hat{j} + 7\hat{k}) + (3\hat{i} - 4\hat{j} - 3\hat{k})\)

Combining the components of the vectors:

\(\vec{F}_{\text{net}} = (5 + 3)\hat{i} + (8 - 4)\hat{j} + (7 - 3)\hat{k}\) \(\vec{F}_{\text{net}} = 8\hat{i} + 4\hat{j} + 4\hat{k}\)

Newton's second law of motion, \(\vec{F}_{\text{net}} = m \vec{a}\), is applied to find the acceleration \(\vec{a}\), where \(m\) represents the mass.

The equation for acceleration is rearranged as:

\(\vec{a} = \frac{\vec{F}_{\text{net}}}{m}\)

Inputting the calculated net force and the given mass:

\(\vec{a} = \frac{8\hat{i} + 4\hat{j} + 4\hat{k}}{4}\)

The acceleration vector is obtained by dividing each component by the mass:

\(\vec{a} = 2\hat{i} + \hat{j} + \hat{k}\)

Consequently, the acceleration of the body is \(2\hat{i} + \hat{j} + \hat{k}\).