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This problem involves energy conservation and circular motion dynamics. Let's solve it step-by-step.

### Given:
- A small block of mass \( m \) slides on a frictionless track.
- The block starts at height \( h = 3R \) (where \( R \) is the radius of the loop).
- We are asked to find the normal force at the top of the loop, point \( B \), where the block's height is \( 2R \) (since it is at the top of the loop).

### Step 1: Conservation of Mechanical Energy
Since the track is frictionless, mechanical energy is conserved. The block starts from rest, so the total mechanical energy is purely gravitational potential energy at point \( A \) (height \( h = 3R \)).

- **At point A (starting point):**
  \[
  E_{\text{total}} = \text{Potential Energy at } A = mgh = mg(3R)
  \]

- **At point B (top of the loop):**
  The block will have both kinetic energy and potential energy:
  \[
  E_{\text{total}} = \text{Kinetic Energy at } B + \text{Potential Energy at } B
  \]
  \[
  E_{\text{total}} = \frac{1}{2}mv_B^2 + mg(2R)
  \]
  where \( v_B \) is the velocity of the block at point \( B \).

### Step 2: Apply Conservation of Energy
Setting the total energy at \( A \) equal to the total energy at \( B \):
\[
mg(3R) = \frac{1}{2}mv_B^2 + mg(2R)
\]
Solve for \( v_B^2 \):
\[
mg(3R) - mg(2R) = \frac{1}{2}mv_B^2
\]
\[
mgR = \frac{1}{2}mv_B^2
\]
\[
v_B^2 = 2gR
\]

### Step 3: Forces at the Top of the Loop (Point B)
At the top of the loop, the block experiences two forces:
1. Gravitational force \( mg \) (downward).
2. Normal force \( N \) from the track (also downward at point \( B \)).

The net force provides the centripetal force required to keep the block moving in a circle:
\[
F_{\text{net}} = N + mg = \frac{mv_B^2}{R}
\]
Substitute \( v_B^2 = 2gR \) from the energy conservation step:
\[
N + mg = \frac{m(2gR)}{R}
\]
\[
N + mg = 2mg
\]
Solve for \( N \):
\[
N = 2mg - mg = mg
\]

### Conclusion:
The normal force at point \( B \) is \( mg \) and it acts **downward**. Thus, the correct answer is:
**(d) \( mg \) downward.**

Would you like any additional details or explanations?

### Related Questions:
1. How does the normal force change at different points in the loop?
2. What would happen if friction were introduced on the track?
3. How would the normal force change if the block started from a lower height, say \( h = 2R \)?
4. Can the block complete the loop if its starting height is less than \( 2R \)?
5. How does the velocity of the block change as it moves along the track?

### Tip:
When solving problems involving circular motion, always account for the centripetal force, which is the net inward force keeping an object moving in a circle.

A small block with mass m slides on the frictionless track shown in the sketch, starting from rest at point A. The circular loop has radius R and the block starts at an initial height equal to three times the radius, h = 3R. When the block passes through the top part of the loop (point B), what is the normal force (magnitude and direction) from the track acting on the block?

Learn how to calculate the normal force acting on a block at the top of a circular loop using the principles of energy conservation and circular motion. This problem involves key physics concepts such as mechanical energy conservation and centripetal force. Suitable for high school students in grades 10-12.

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