A solid sphere of mass 0.1 kg and radius 2.5 cm rolls without sliding with a uniform velocity of `0.1 ms^(-1)` along a straight line on a smooth horizontal table. Find its total energy.

Answer

To find the total energy of a solid sphere rolling without sliding, we need to consider both its translational kinetic energy and its rotational kinetic energy.

### Step-by-step Solution:

1. **Identify the mass and radius of the sphere**:
- Mass (m) = 0.1 kg
- Radius (r) = 2.5 cm = 0.025 m (convert cm to m)

2. **Determine the velocity of the sphere**:
- Velocity (v) = 0.1 m/s

3. **Calculate the translational kinetic energy (TKE)**:
- The formula for translational kinetic energy is:
\[ TKE = \frac{1}{2} mv^2 \]
- Substituting the values:
\[ TKE = \frac{1}{2} \times 0.1 \, \text{kg} \times (0.1 \, \text{m/s})^2 \]
\[ TKE = \frac{1}{2} \times 0.1 \times 0.01 = 0.0005 \, \text{J} \]

4. **Calculate the moment of inertia (I) of the solid sphere**:
- The moment of inertia for a solid sphere is given by:
\[ I = \frac{2}{5} m r^2 \]
- Substituting the values:
\[ I = \frac{2}{5} \times 0.1 \, \text{kg} \times (0.025 \, \text{m})^2 \]
\[ I = \frac{2}{5} \times 0.1 \times 0.000625 = 0.000025 \, \text{kg m}^2 \]

5. **Calculate the angular velocity (ω)**:
- Since the sphere rolls without slipping, the relationship between linear velocity (v) and angular velocity (ω) is:
\[ v = r \omega \implies \omega = \frac{v}{r} \]
- Substituting the values:
\[ \omega = \frac{0.1 \, \text{m/s}}{0.025 \, \text{m}} = 4 \, \text{rad/s} \]

6. **Calculate the rotational kinetic energy (RKE)**:
- The formula for rotational kinetic energy is:
\[ RKE = \frac{1}{2} I \omega^2 \]
- Substituting the values:
\[ RKE = \frac{1}{2} \times 0.000025 \, \text{kg m}^2 \times (4 \, \text{rad/s})^2 \]
\[ RKE = \frac{1}{2} \times 0.000025 \times 16 = 0.0002 \, \text{J} \]

7. **Calculate the total energy (TE)**:
- The total energy is the sum of translational and rotational kinetic energy:
\[ TE = TKE + RKE \]
- Substituting the values:
\[ TE = 0.0005 \, \text{J} + 0.0002 \, \text{J} = 0.0007 \, \text{J} \]

### Final Answer:
The total energy of the solid sphere is **0.0007 J**.