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

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 \( 2\hat{i} + \hat{j} + \hat{k} \).