In the previous question the smallest kinetic energy at the bottom of the incline will be achieved by

Options

A. the solid sphere
B. the hollow sphere
C. the disc
D. all will achieve same kinetic energy

Answer

To solve the problem of determining which object achieves the smallest kinetic energy at the bottom of the incline, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Objects and Their Moments of Inertia**:
- We have three objects: a hollow sphere, a solid sphere, and a disk.
- The moments of inertia (I) for each object are:
- Hollow Sphere: \( I = \frac{2}{3} MR^2 \)
- Solid Sphere: \( I = \frac{2}{5} MR^2 \)
- Disk: \( I = \frac{1}{2} MR^2 \)

2. **Understanding the Kinetic Energy**:
- The total kinetic energy (KE) at the bottom of the incline consists of translational kinetic energy and rotational kinetic energy:
\[ KE = KE_{translational} + KE_{rotational} = \frac{1}{2} mv^2 + \frac{1}{2} I \omega^2 \]
- Here, \( v \) is the linear velocity and \( \omega \) is the angular velocity.

3. **Relate Angular Velocity to Linear Velocity**:
- For rolling objects without slipping, we have the relationship:
\[ v = R \omega \quad \Rightarrow \quad \omega = \frac{v}{R} \]

4. **Substituting Angular Velocity into Kinetic Energy**:
- Substitute \( \omega \) into the rotational kinetic energy term:
\[ KE_{rotational} = \frac{1}{2} I \left(\frac{v}{R}\right)^2 = \frac{1}{2} I \frac{v^2}{R^2} \]
- Therefore, the total kinetic energy becomes:
\[ KE = \frac{1}{2} mv^2 + \frac{1}{2} I \frac{v^2}{R^2} = \frac{1}{2} v^2 \left(m + \frac{I}{R^2}\right) \]

5. **Comparing the Moments of Inertia**:
- Since all objects have the same mass (m) and radius (R), we need to find which object has the largest moment of inertia (I) because the total kinetic energy is inversely proportional to \( I \).
- The moments of inertia are:
- Hollow Sphere: \( \frac{2}{3} MR^2 \)
- Solid Sphere: \( \frac{2}{5} MR^2 \)
- Disk: \( \frac{1}{2} MR^2 \)

6. **Determine the Largest Moment of Inertia**:
- Comparing the values:
- \( \frac{2}{3} \approx 0.67 \)
- \( \frac{2}{5} = 0.4 \)
- \( \frac{1}{2} = 0.5 \)
- The hollow sphere has the largest moment of inertia.

7. **Conclusion**:
- Since the hollow sphere has the largest moment of inertia, it will have the smallest kinetic energy at the bottom of the incline.

### Final Answer:
The object that achieves the smallest kinetic energy at the bottom of the incline is the **hollow sphere**.