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.