A simple pendulum of length l and mass m free to oscillate in vertical plane. A nail is located at a distance d=l-a vertically below the point of suspension of a simple pendulum. The pendulum bob is released from the position where the string makes an angle of 90^@ from vertical. Discuss the motion of the bob if (a) l=2a and (b) l=2.5a.

A simple pendulum of length l and mass m free to oscillate in vertical

1.	A simple pendulum of length l and mass m free to oscillate in vertical plane. A nail is located at a distance d=l-a vertically below the point of suspension of a simple pendulum. The pendulum bob is released from the position where the string makes an angle of 90^@ from vertical. Discuss the motion of the bob if (a) l=2a and (b) l=2.5a.
Answer» Solution :a. `l=2a`, velocity at lowest point from ENERGY conservation `0+mg2a=1/2mv^2` `v=sqrt(4ga)` Here the radius of CIRCLE is a about nail and velocity at lowest point is not sufficient to COMPLETE the loop. Therefore, MOTION of bob is the combination of circular and projectile motion. Because the velocity at lowest point LIES between `sqrt(3ga)` and `sqrt(5ga)`. b. `l=2.5a`, Velocity at lowest point from energy conservation `0+mg(2.5a)=1/2mv^2impliesv=sqrt(5ga)` here radius of circle is a about nail and velocity at lowest point is just sufficient to complete the loop so that here looping the loop about nail.