A sphere of mass 1 kg has a radius of 10 cm. Calculate the moment of inertia (i) about the diameter and (ii) about the tangent.

Answer

To calculate the moment of inertia of a sphere about its diameter and about a tangent, we can follow these steps:

### Step 1: Calculate the Moment of Inertia about the Diameter
The formula for the moment of inertia \( I \) of a solid sphere about an axis through its center (diameter) is given by:

\[ I = \frac{2}{5} m R^2 \]

Where:
- \( m \) is the mass of the sphere
- \( R \) is the radius of the sphere

Given:
- Mass \( m = 1 \, \text{kg} \)
- Radius \( R = 10 \, \text{cm} = 0.1 \, \text{m} \)

Substituting the values into the formula:

\[ I = \frac{2}{5} \times 1 \, \text{kg} \times (0.1 \, \text{m})^2 \]

Calculating this:

\[ I = \frac{2}{5} \times 1 \times 0.01 = \frac{2}{500} \, \text{kg m}^2 = 0.004 \, \text{kg m}^2 \]

### Step 2: Calculate the Moment of Inertia about the Tangent
To find the moment of inertia about a tangent, we can use the Parallel Axis Theorem, which states:

\[ I = I_{\text{cm}} + m d^2 \]

Where:
- \( I_{\text{cm}} \) is the moment of inertia about the center of mass (which we calculated in Step 1)
- \( d \) is the distance from the center of mass to the new axis (which is equal to the radius \( R \) of the sphere)

Substituting the values:

\[ I_{\text{tangent}} = I_{\text{cm}} + m R^2 \]
\[ I_{\text{tangent}} = \frac{2}{5} m R^2 + m R^2 \]

Factoring out \( m R^2 \):

\[ I_{\text{tangent}} = m R^2 \left( \frac{2}{5} + 1 \right) = m R^2 \left( \frac{2}{5} + \frac{5}{5} \right) = m R^2 \left( \frac{7}{5} \right) \]

Now substituting \( m = 1 \, \text{kg} \) and \( R = 0.1 \, \text{m} \):

\[ I_{\text{tangent}} = 1 \, \text{kg} \times (0.1 \, \text{m})^2 \times \frac{7}{5} \]
\[ I_{\text{tangent}} = 1 \times 0.01 \times \frac{7}{5} = \frac{7}{500} \, \text{kg m}^2 = 0.014 \, \text{kg m}^2 \]

### Final Answers:
(i) Moment of inertia about the diameter: \( 0.004 \, \text{kg m}^2 \)

(ii) Moment of inertia about the tangent: \( 0.014 \, \text{kg m}^2 \)

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