A small ball of mass  is attached with upper end of a uniform straight rod of equal mass  and length  The rod is held vertical over a smooth horizontal surface as shown in fig. When the system is released, the lower end slips freely and the systems falls down. Assuming the initial position of the lower end to be origin and initially rod to be along -axis as shown in fig., the equation of trajectory of point  of the rod (  is at distance  from the lower end)is

a)

b)

c)

d)

(b)

Since the floor is smooth, therefore no horizontal force is acting on the system. Only three forces are acting on it, weight of the ball, weight  of the rod and vertically upward reaction of the floor

Since the rod is released from rest and no horizontal force is acting on it, therefore centre of mass of the system will not displace horizontally; it means it will fall vertically downward

Since masses of the rod and ball are equal, therefore centre of mass ‘ ’ is at mid-point of centre of the rod and the ball; it means at distance  from the ball or 3/4 from the lower end

If at an instant the rod makes angle  with the vertical, then point  and the system will be as shown in the figure

Co-ordinates of the point  are

Using  we have