Final answer:
The value of tension (T) at the top of a vertical circular path is negligible, and to find the minimum velocity for continued uniform circular motion, one considers the point where gravity alone provides the necessary centripetal force. At the bottom of the circle, the tension is the sum of forces needed for circular motion and the object's weight. The formula T = (mv²/r) + mg can be used to calculate this maximum tension.
Step-by-step explanation:
For an object being swung in a vertical circle on a string, to determine the minimum velocity for continued uniform circular motion, we need the value of tension in the string (T) when it is at the top of its path. At this point, if the object is moving at its slowest speed without falling, the tension is effectively zero because all of the force needed to keep the object in circular motion is provided by gravity. To calculate the tension at the bottom of the circle, we assume that there is no additional energy added to the ball during rotation and use the principles of conservation of energy and circular motion dynamics.
At the bottom of the vertical circle, the tension in the string is at its maximum because centripetal force required for circular motion is the sum of the object's weight and the force provided by the tension in the string. The tension can be calculated using the formula T = (mv²/r) + mg, where m is the mass of the object, v is the tangential velocity at the bottom, r is the radius of the circle, and g is the acceleration due to gravity.