A block of mass 'm' is attached to one end of a light inextensible string passing over a smooth light Pulley B and under another smooth light Pulley A as shown is figure. The other end of a string is fixed to a ceilling . A and B are held by springs of spring constant K_1 and K_2 . Find the angular frequency of small oscillation of the system

A block of mass 'm' is attached to one end of a light inextensible str

1.	A block of mass 'm' is attached to one end of a light inextensible string passing over a smooth light Pulley B and under another smooth light Pulley A as shown is figure. The other end of a string is fixed to a ceilling . A and B are held by springs of spring constant K_1 and K_2 . Find the angular frequency of small oscillation of the system
Answer» Solution :Initially SYSTEM is in equilibrium Assume that BLOCK is PULLED down by a SMALL displacement and released Let T be the extra tension which provides the restoring force. T= ma If x is the small displacement of the block , `x_1 and x_2` are extension of the springs. Here `x = 2x_1 + 2x_2` `K_1 x_1 = 2T` `K_2 x_2 = 2T` `x = ((4T)/(K_1) + (4T)/(K_2))` from T = ma `4m (1/(K_1) + 1/(K_2)) = x `(or) `a=-{(1)/(4m((1)/(K_1) + 1/(K_2)))}x , omega = SQRT(a/x) = sqrt((K_1 K_2)/(4m(K_1 + k_2)))`