Vertical Circular Motion

In vertical circular motion, an object moves in a circular path with its center of motion located above the ground. The object experiences a change in velocity due to the change in direction of its motion. Let's analyze the given scenario.

Initial Conditions

- The stone is tied to a string of length l.
- The other end of the string is fixed at the center.
- The stone is at its lowest position and has a speed u.

At the Lowest Position

When the stone is at its lowest position, it experiences a tension force in the string, directed towards the center of the circular path. This tension force provides the necessary centripetal force for the stone to move in a circular path.

Ascent from the Lowest Position

As the stone moves upwards from the lowest position, its speed decreases. This is because the tension force in the string is less than the weight of the stone, resulting in a net force directed downwards. This net force causes a decrease in speed as the stone moves against the direction of its motion.

At the Horizontal Position

When the stone reaches the horizontal position, its speed is minimum. This is because the tension force in the string is equal to the weight of the stone, resulting in zero net force. At this point, the stone experiences only the force of gravity acting vertically downwards.

Change in Velocity

The magnitude of the change in velocity as the stone reaches the horizontal position can be determined using the principle of conservation of energy. At the lowest position, the total mechanical energy (kinetic + potential) of the stone is given by:

E = (1/2)mv² + mgh

where m is the mass of the stone, v is its speed, g is the acceleration due to gravity, and h is the height from the lowest position to the horizontal position.

At the horizontal position, the total mechanical energy is given by:

E' = (1/2)mv'² + mgh'

where v' is the speed at the horizontal position and h' is the height from the lowest position to the ground.

Since energy is conserved, we have:

E = E'

(1/2)mv² + mgh = (1/2)mv'² + mgh'

Simplifying the equation, we find:

v'² = v² + 2gh

The magnitude of the change in velocity is given by:

Δv = |v' - v|

Substituting the values, we have:

Δv = √(v² + 2gh) - v

Conclusion

In conclusion, the magnitude of the change in velocity as the stone reaches a position where the string is horizontal is given by Δv = √(v² + 2gh) - v. This change in velocity is determined by the initial speed of the stone, the acceleration due to gravity, and the height difference between the lowest position and the horizontal position.