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

Answer: (C)

$2\hat{i} + \hat{j} + \hat{k}$

Answer 2

Given: - Mass of the body: $m = 4 \, \text{kg}$ - Forces acting on the body:

$\vec{F_1} = 5\hat{i} + 8\hat{j} + 7\hat{k}$ $\vec{F_2} = 3\hat{i} - 4\hat{j} - 3\hat{k}$

Step 1: Calculating the Net Force

The net force acting on the body is given by the vector sum of $\vec{F_1}$ and $\vec{F_2}$:

$\vec{F_{\text{net}}} = \vec{F_1} + \vec{F_2}$

Substituting the given values:

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

Combining like terms:

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

Step 2: Calculating the Acceleration

The acceleration $\vec{a}$ is given by Newton’s second law:

$\vec{a} = \frac{\vec{F_{\text{net}}}}{m}$

Substituting the values:

$\vec{a} = \frac{1}{4} (8\hat{i} + 4\hat{j} + 4\hat{k})$ $\vec{a} = 2\hat{i} + \hat{j} + \hat{k}$

Conclusion:

The acceleration acting on the body is $2\hat{i} + \hat{j} + \hat{k}$.