In the given system, when the ball of mass m is released, it will swing down the dotted circular are. A nail is located at a distance d below the point of suspension as shown in figure. (a) How fast will it reach the lowest point in its swing? (b) Show that d must be at least 0.6 l, if the ball is to swing completely around a circle centered along the nail. (c) If d = 0.6 l, find the change in tension in the string just after it touches the nail.

In the given system, when the ball of mass m is released, it will swin

1.	In the given system, when the ball of mass m is released, it will swing down the dotted circular are. A nail is located at a distance d below the point of suspension as shown in figure. (a) How fast will it reach the lowest point in its swing? (b) Show that d must be at least 0.6 l, if the ball is to swing completely around a circle centered along the nail. (c) If d = 0.6 l, find the change in tension in the string just after it touches the nail.
Answer» Solution :From A to B : loss in GPE = Gain in KE mgl = 1/2 mv^(2) `RIGHTARROW v = sqrt(2gl)` (b) Radius of the circle centered at nail = L - d. To complete the circle centered at nail , the speed at the bottom MUST be at LEAST = `sqrt(5g(l -d)` `therefore, sqrt(2gl) = sqrt(5g(l -d) Rightarrow 5l - 5d = 2l` `Rightarrow d = 0.6 l` (c) Just before touching the nail, the ball is moving in a circle of radius l. `Rightarrow Tension - mg + (mv)6(2) /l = mg + 2mg = 3mg` Just after touching the nail, the ball is moving in a circle of radius (l - d). `Rightarrow Tension = mg + (mv)^(2) /(l-d) = mg + (m+(2gl)/(0.4 l) Rightarrow Tension = 6 mg` Hence the tension in the string changes from 3mg to 6 mg instantly as it touches the nail.