Find the moment fo inertia of the hydrogen molecules about an axis passing through its centre of mass and perpendicular to the internuclear axis. Given mass of H-atom `= 1.7 xx 10^(-27) kg`, interatomic distance `= 4xx 10^(-10)m`

Answer

To find the moment of inertia of a hydrogen molecule (H₂) about an axis passing through its center of mass and perpendicular to the internuclear axis, we can follow these steps:

### Step 1: Identify the mass of the hydrogen atoms
The mass of a single hydrogen atom is given as:
\[ m_H = 1.7 \times 10^{-27} \text{ kg} \]

### Step 2: Determine the interatomic distance
The interatomic distance (the distance between the two hydrogen atoms) is given as:
\[ d = 4 \times 10^{-10} \text{ m} \]

### Step 3: Calculate the distance from the center of mass to each atom
Since there are two hydrogen atoms, the center of mass (CM) will be located at the midpoint of the interatomic distance. Therefore, the distance from the center of mass to each atom (r) is:
\[ r = \frac{d}{2} = \frac{4 \times 10^{-10}}{2} = 2 \times 10^{-10} \text{ m} \]

### Step 4: Use the formula for moment of inertia
The moment of inertia (I) about an axis through the center of mass for two point masses is given by:
\[ I = m_1 r^2 + m_2 r^2 \]
Since both masses are equal (both are hydrogen atoms), we can simplify this to:
\[ I = 2 \times m_H \times r^2 \]

### Step 5: Substitute the values into the formula
Substituting the values we have:
\[ I = 2 \times (1.7 \times 10^{-27} \text{ kg}) \times (2 \times 10^{-10} \text{ m})^2 \]

Calculating \( (2 \times 10^{-10})^2 \):
\[ (2 \times 10^{-10})^2 = 4 \times 10^{-20} \text{ m}^2 \]

Now substituting this back into the moment of inertia equation:
\[ I = 2 \times (1.7 \times 10^{-27}) \times (4 \times 10^{-20}) \]

### Step 6: Calculate the final moment of inertia
Calculating the product:
\[ I = 2 \times 1.7 \times 4 \times 10^{-27} \times 10^{-20} \]
\[ I = 13.6 \times 10^{-47} \text{ kg m}^2 \]

### Final Answer
Thus, the moment of inertia of the hydrogen molecule about the specified axis is:
\[ I = 1.36 \times 10^{-46} \text{ kg m}^2 \]