https://file.aiquickdraw.com/mathbot/file/1727983279985gra8si98.jpg

Let's break down the problem from the image:

The scenario involves a block of mass \( m \) moving from rest at a height \( h \) at point \( A \) and going through a vertical loop of radius \( R \). There are some key questions related to forces and the energy conservation involved.

### Part a: Vector Force Diagram at Points B and C

For **Point B** (at the lowest point of the loop):
- **Gravitational force** (\( mg \)): This always points downward.
- **Normal force** (\( F_n \)): This will point upward because it's perpendicular to the surface of the loop, supporting the block against gravity.

For **Point C** (at the top of the loop):
- **Gravitational force** (\( mg \)): This again points downward.
- **Normal force** (\( F_n \)): This also points downward, as it acts towards the center of the circular loop to keep the block in contact with the track.

### Part b: Minimum Height \( h_{\text{min}} \)

To find the minimum height \( h_{\text{min}} \) necessary for the block to maintain contact at point \( C \) (the top of the loop), we can use energy conservation and centripetal force concepts.

1. **Energy conservation** from point \( A \) to point \( C \):
   - At point \( A \), the block starts with only potential energy \( mgh \).
   - At point \( C \), it has potential energy \( mg(2R) \) (since it is \( 2R \) above the ground) and kinetic energy \( \frac{1}{2} mv_C^2 \).

   Using energy conservation:
   \[
   mgh = mg(2R) + \frac{1}{2} mv_C^2
   \]

   Simplifying, we get:
   \[
   gh = 2gR + \frac{1}{2} v_C^2
   \]

2. **Centripetal force** at the top of the loop:
   - For the block to stay in contact at point \( C \), the centripetal force requirement must be met. At point \( C \), the gravitational force and the normal force together provide the centripetal force:
   \[
   mg + F_n = \frac{mv_C^2}{R}
   \]
   For minimum height, the normal force \( F_n \) should be zero (since we want the block to just stay in contact), so:
   \[
   mg = \frac{mv_C^2}{R}
   \]
   This simplifies to:
   \[
   v_C^2 = gR
   \]

3. Substituting \( v_C^2 = gR \) into the energy conservation equation:
   \[
   gh = 2gR + \frac{1}{2} gR
   \]
   \[
   gh = \frac{5}{2} gR
   \]

   Therefore, the minimum height \( h_{\text{min}} \) is:
   \[
   h_{\text{min}} = \frac{5R}{2}
   \]

This concludes the solution to the problem.

### Relative Questions:
1. How does the normal force at point B compare to the gravitational force?
2. What happens if the height \( h \) is less than \( \frac{5R}{2} \)?
3. Why is the normal force zero at point C for the minimum height?
4. How would friction affect the minimum height calculation?
5. Could you calculate the velocity at point B, given \( h = \frac{5R}{2} \)?

### Tip:
When dealing with energy conservation, always check if there are forces doing work that could add or remove mechanical energy (e.g., friction).

A block of mass m starts from rest at point A and travels with negligible friction through a loop onto a horizontal surface, where it makes contact with a spring. The problem asks for force diagrams at specific points and to derive the minimum height needed for the block to maintain contact at the top of the loop.

This physics problem explores the forces acting on a block moving through a vertical loop and determines the minimum height needed for the block to maintain contact at the loop's top using energy conservation and centripetal force equations. Suitable for advanced high school physics students.

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