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In vertical circular motion, the block moves in a circular path due to centripetal force. This force is necessary to keep the object moving in a circle. It acts towards the center of the circle. The concept of centripetal force can be denoted by the equation:\[ f_c = \frac{mv^2}{R} \]where:

• \( m \) is the mass of the object,
• \( v \) is the velocity,
• \( R \) is the radius of the circular path.

Looking at point A, the block initially has a velocity of \( \sqrt{6gR} \). At this point, we need to calculate the centripetal force required to keep it moving in a circle. By substituting the values into the centripetal force formula, we identify that the required force is \( 6mg \). This force is balanced by the normal force minus the gravitational force acting on the block, leading to the conclusion that the net force provides the centripetal acceleration needed for circular motion.

The principle of conservation of energy is foundational in analyzing mechanical systems. In the case of the block sliding along a vertical circular track, it dictates that the block's mechanical energy is conserved. This means that the sum of kinetic and potential energy remains constant, assuming no external forces like friction are doing work.At point A, the energy is mostly kinetic since the block is just beginning its journey around the circle. The kinetic energy is expressed as:\[ \frac{1}{2}mv_A^2 \]As the block moves to point B at the top of the circle, it gains potential energy and loses some kinetic energy. The potential energy at this point is \( 2mgR \), which corresponds to the weight of the block multiplied by the height (twice the radius). This energy transformation allows us to express energy conservation as:\[ \frac{1}{2}m v_A^2 = \frac{1}{2}mv_B^2 + 2mgR \]Solving this equation, we find that the velocity \( v_B \) squared at point B is equal to \( 2gR \). Conservation of energy helps predict how energy exchange between kinetic and potential energy affects the speed of the block as it moves through different points on the circular path.

Normal force is the contact force exerted by a surface on an object resting on it. In vertical circular motion, the normal force changes depending on the block's position on the track because it serves two needs: supporting the weight of the block and providing centripetal force.At point A, the normal force \( N_A \) works against gravity to provide the required centripetal force to keep the block in motion. The equation illustrating this combination is:\[ N_A - mg = 6mg \]This ensures that the normal force at A is \( 7mg \), demonstrating how much force is necessary to both counteract gravity and keep the block in its circular path.While at point B, the conditions change; the block is at the top of the loop. The forces need to be balanced such that they still furnish the necessary centripetal force. Here, gravity helps in maintaining the circular motion:\[ mg + N_B = 2mg \]Solving this, the net normal force \( N_B \) is \( mg \). This simplification illustrates how the normal force decreases as the block reaches the highest point, where gravity aids in centripetal acceleration. Observing normal force variations is crucial for understanding an object's dynamics in circular motion, particularly when balancing gravitational effects and the demand for centripetal forces.