Final answer:
The minimum speed at the top of a vertical circle is calculated using the formula v = √(g × r), where g is the gravitational acceleration and r is the radius of the circle. At the bottom, minimum speed depends on the energy gained from falling and other forces, if any.
Step-by-step explanation:
The minimum speed at the top of a vertical circle is determined by setting the gravitational force equal to the required centripetal force to keep the object in circular motion. This ensures that the object does not fall out of the circle when it is at the top. To calculate this, we use the equation:
v = √(g × r)
where v is the minimum speed, g is the acceleration due to gravity (9.8 m/s² on Earth), and r is the radius of the circle.
At the bottom of the circle, the minimum speed is determined by other factors such as energy conservation or additional forces. However, in the simplest case where only gravity acts, it can be calculated from the energy gained when descending from the top of the circle minus any energy lost due to factors like friction.
The equation W = 1/m(v² − v²) indicated in the question, represents the work done on the object, where m is the mass, v is the final speed, and v² is the initial speed.