To determine the energy, velocity, and rotational velocity of the ring at different points along the path:
1. **Energy at point 2 in terms of h2 and v2**:
- At point 1, the total initial energy of the ring is potential energy due to its height: \( PE = mgh_1 \).
- At point 2, the total energy of the ring is the sum of potential energy (due to height \( h_2 \)) and kinetic energy: \( PE + KE = mgh_2 + \frac{1}{2}mv_2^2 \).
2. **Calculating the velocity at point 2**:
- Using the principle of conservation of energy, the initial potential energy at point 1 is converted into the sum of potential and kinetic energy at point 2: \( mgh_1 = mgh_2 + \frac{1}{2}mv_2^2 \).
- Solve for velocity \( v_2 \) to find the velocity at point 2.
3. **Rotational velocity at point 2**:
- For a rolling object, the velocity is related to the angular velocity by \( v = Rω \), where \( R \) is the radius of the ring.
- The angular velocity at point 2 can be found using the relationship between linear velocity and angular velocity.
4. **Linear velocity at point 3**:
- As the hill is frictionless and the ring's rotational velocity remains constant, the linear velocity at point 3 is equal to the product of the radius and the angular velocity (\( v = Rω \)).
By following these steps and applying the relevant formulas, you can calculate the energy, velocity, and rotational velocity of the ring at points 2 and 3.