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:
- $-2\hat{i} - \hat{j} - \hat{k}$
- $4\hat{i} + 2\hat{j} + 2\hat{k}$
- $2\hat{i} + \hat{j} + \hat{k}$
- $2\hat{i} + 3\hat{j} + 3\hat{k}$
The Correct Option is C
Approach Solution - 1
We are given a body of mass \(4 \, \text{kg}\) which experiences two forces: \(\vec{F}_1 = 5\hat{i} + 8\hat{j} + 7\hat{k}\) and \(\vec{F}_2 = 3\hat{i} - 4\hat{j} - 3\hat{k}\). We are asked to find the acceleration acting on the body.
To find this, we need to first calculate the net force acting on the body using the formula:
\(\vec{F}_{\text{net}} = \vec{F}_1 + \vec{F}_2\)
Substitute the given forces into the formula:
\(\vec{F}_{\text{net}} = (5\hat{i} + 8\hat{j} + 7\hat{k}) + (3\hat{i} - 4\hat{j} - 3\hat{k})\)
Combine the like terms:
\(\vec{F}_{\text{net}} = (5 + 3)\hat{i} + (8 - 4)\hat{j} + (7 - 3)\hat{k}\) \(\vec{F}_{\text{net}} = 8\hat{i} + 4\hat{j} + 4\hat{k}\)
Now, we use Newton’s second law of motion to find the acceleration: \(\vec{F}_{\text{net}} = m \vec{a}\), where \(m\) is the mass of the body.
Rearranging for the acceleration \(\vec{a}\):
\(\vec{a} = \frac{\vec{F}_{\text{net}}}{m}\)
Substitute the values:
\(\vec{a} = \frac{8\hat{i} + 4\hat{j} + 4\hat{k}}{4}\)
Simplify by dividing each component of the vector by the mass:
\(\vec{a} = 2\hat{i} + \hat{j} + \hat{k}\)
Thus, the acceleration acting on the body is \(2\hat{i} + \hat{j} + \hat{k}\)
Approach Solution -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 \( 2\hat{i} + \hat{j} + \hat{k} \).
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