When a pendulum bob is projected in such a
way that it moves in a circular motion on vertical plane, its speed as well as
tension in string continuously vary due to effect of gravity. Analysis of this
motion is very important as on many such situations varieties of problems are
framed.
When a pendulum bob is imparted an initial
velocity at its bottom most stable position, it starts following the vertical
circular motion and as it goes up due to negative work of gravity its kinetic
energy decreases and the decrease in kinetic energy decreases the centrifugal
force acting on it in rotating reference frame due to which string tension also
decreases.
Depending upon the initial velocity of
projection when it goes up in the upper part of circular trajectory, its tension
may become zero and it may no longer continue the circular motion.
See this video for analysis of this
vertical circular motion – https://youtu.be/9hivNCWgwt8
So for different initial velocities of
projection there can be different cases of motion of the bob which we can study
one by one. See this video for the different cases of projection of the pendulum bob in vertical
circular motion –https://youtu.be/htiFN9EHCgA
In different cases we have seen that if
initial velocity of projection is not sufficient, the string may slack in upper
half of circular trajectory and bob may leave the trajectory and follow the
parabolic trajectory of projectile motion.
If instead of a string we use a light rod
in making the pendulum, it will never be slack and the bob will always follow
the circular path but may return if its velocity become zero anywhere in the
motion.
See this video for this motion analysis of bob –