A ball is attached to a horizontal cord of length L whose other end is fixed. A

Question

A ball is attached to a horizontal cord of length L whose other end is fixed. A peg is located at a distance d directly below the fixed end of the cord. The ball is released from rest when the string is horizontal. ?? a. If d = 0.75L, find the speed of the ball when it reaches the top of the circular path about the peg. b. Will the ball be able to make a complete circle about the peg if d = 0.5L? (Hint: what speed does the ball need to have at the top of its arc if it is to continue to move in a circle?) What is the minimum distance d such that the ball will be able to make a complete circle around the peg after the string catches on the peg?

Explanation / Answer

Consider the particle when it is at the point P and the string makes an angle ? with vertical. Forces acting on the particle are: T = tension in the string along its length, and mg = weight of the particle vertically downward. forces-acting-on-the-particle Hence, net radial force on the particle is FR = T - mg cos ? => T - mg cos ? = mv2/R => T = mv2/R + mg cos ? Since speed of the particle decreases with height, hence tension is maximum at the bottom, where cos ? = 1 (as ? = 0). => Tmax = mv2/R + mg; Tmin = mv'2/R - mg (at the top) Here, v' = speed of the particle at the top. Critical Velocity It is the minimum velocity given to the particle at the lowest point to complete the circle. The tendency of the string to become slack is maximum when the particle is at the topmost point of the circle. At the top, tension is given by T = mvT2/R - mg, where vT = speed of the particle at the top. => mvT2/R = T + mg For vT to be minimum, T ? 0 => vT = ?gR put the correseponding value and get result

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