Two point masses of 0.3 kg and 0.7kg are fixed at the ends of a rod of length 1.4 m and of negligible mass. The rod is set rotating about an axis perpendicular to its length with a uniform angular speed. The point on the rod through which the axis should pass in order that the work required for rotation of the rod is minimum, is located at a distance of

Options

A. `0.42 m` from the mass of `0.3 kg`
B. `0.70 m` from the mass of `0.7 kg`
C. `0.98 m` from the mass of `0.3 kg`
D. `0.98 m` from the mass of `0.7 kg`

Explanation

To solve the problem of finding the point on the rod through which the axis should pass to minimize the work required for rotation, we will follow these steps:

### Step 1: Understand the System
We have two point masses, \( m_1 = 0.3 \, \text{kg} \) and \( m_2 = 0.7 \, \text{kg} \), fixed at the ends of a rod of length \( L = 1.4 \, \text{m} \). The goal is to find the position along the rod where the axis of rotation should be placed to minimize the work done for rotation.

### Step 2: Define the Center of Mass
The center of mass (CM) of a system of point masses can be calculated using the formula:
\[ x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]
where \( x_1 \) and \( x_2 \) are the positions of the masses. We can set \( x_1 = 0 \) (position of \( m_1 \)) and \( x_2 = 1.4 \, \text{m} \) (position of \( m_2 \)).

### Step 3: Substitute the Values
Substituting the values into the center of mass formula:
\[ x_{cm} = \frac{(0.3 \, \text{kg} \cdot 0) + (0.7 \, \text{kg} \cdot 1.4 \, \text{m})}{0.3 \, \text{kg} + 0.7 \, \text{kg}} \]
\[ x_{cm} = \frac{0 + 0.98 \, \text{kg m}}{1.0 \, \text{kg}} = 0.98 \, \text{m} \]

### Step 4: Interpret the Result
The center of mass is located at a distance of \( 0.98 \, \text{m} \) from the \( 0.3 \, \text{kg} \) mass. This is the point on the rod through which the axis of rotation should pass to minimize the work required for rotation.

### Step 5: Conclusion
Thus, the point on the rod through which the axis should pass is located at a distance of \( 0.98 \, \text{m} \) from the \( 0.3 \, \text{kg} \) mass.

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