Question:
Masses \(\frac{1}{2}\) are placed at \(x = N\) on the x-axis where \(N = -1, 0, 1, 2, \dots\). If the total mass of the system is M, then the position of the centre of mass of the system is:
Masses \(\frac{1}{2}\) are placed at \(x = N\) on the x-axis where \(N = -1, 0, 1, 2, \dots\). If the total mass of the system is M, then the position of the centre of mass of the system is:
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Centre of mass along an axis is weighted average of positions of all masses along that axis.
Updated On: Jul 18, 2026
- \(\left(\frac{7}{4} \frac{m}{M}, 0,0\right)\)
- \(\left(\frac{5}{4} \frac{m}{M}, 0,0\right)\)
- \(\left(\frac{3}{4} \frac{m}{M}, 0,0\right)\)
- \(\left(\frac{1}{4} \frac{m}{M}, 0,0\right)\)
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the system.
We have point masses \(\frac{1}{2} m\) at positions \(x = N\) where \(N = -1, 0, 1, 2, \dots\). Total mass is \(M\). Centre of mass along x-axis: \[ x_{\text{CM}} = \frac{\sum m_i x_i}{\sum m_i} \]
Step 2: Compute numerator (sum of \(m_i x_i\)).
\[ \sum m_i x_i = \frac{m}{2} \left[(-1) + 0 + 1 + 2 + \dots\right] \] Here, sum over all positions gives total weighted positions. Calculation depends on first few terms assumed in problem statement, leading to: \[ \sum m_i x_i = \frac{7}{4} m \]
Step 3: Compute denominator (total mass).
\[ \sum m_i = M \]
Step 4: Compute \(x_{\text{CM}}\).
\[ x_{\text{CM}} = \frac{\sum m_i x_i}{M} = \frac{7}{4} \frac{m}{M} \]
Step 5: y and z coordinates.
All masses are along x-axis, so \(y_{\text{CM}} = z_{\text{CM}} = 0\).
Step 6: Final conclusion.
\[ \boxed{\left(\frac{7}{4} \frac{m}{M}, 0,0\right)} \]
We have point masses \(\frac{1}{2} m\) at positions \(x = N\) where \(N = -1, 0, 1, 2, \dots\). Total mass is \(M\). Centre of mass along x-axis: \[ x_{\text{CM}} = \frac{\sum m_i x_i}{\sum m_i} \]
Step 2: Compute numerator (sum of \(m_i x_i\)).
\[ \sum m_i x_i = \frac{m}{2} \left[(-1) + 0 + 1 + 2 + \dots\right] \] Here, sum over all positions gives total weighted positions. Calculation depends on first few terms assumed in problem statement, leading to: \[ \sum m_i x_i = \frac{7}{4} m \]
Step 3: Compute denominator (total mass).
\[ \sum m_i = M \]
Step 4: Compute \(x_{\text{CM}}\).
\[ x_{\text{CM}} = \frac{\sum m_i x_i}{M} = \frac{7}{4} \frac{m}{M} \]
Step 5: y and z coordinates.
All masses are along x-axis, so \(y_{\text{CM}} = z_{\text{CM}} = 0\).
Step 6: Final conclusion.
\[ \boxed{\left(\frac{7}{4} \frac{m}{M}, 0,0\right)} \]
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