A U - tube of uniform cross section contains mercury upto a height h in either limb. The mercury in one limbe is depressed a little and then relased. Obtain an expression for time period (T) of oscillation, assuming that T depends on h,`rho` and g, where `rho` is density of mercury.

A U - tube of uniform cross section contains mercury upto a height h i

1.	A U - tube of uniform cross section contains mercury upto a height h in either limb. The mercury in one limbe is depressed a little and then relased. Obtain an expression for time period (T) of oscillation, assuming that T depends on h,`rho` and g, where `rho` is density of mercury.
Answer» Correct Answer - `T =Ksqrt((h)/(g))` Let `T = Kh^a rho^b g^c` where a,b,c are the dimensions and k is dimensionless constant. Writing the dimensions in (i) we get `[M^0L^0T^1] = L^a(ML^(-3))^b (LT^(-2))^c` `=M^b L^(a -3b +c T^(-2c)` Applying the principle of homogeneity of dimensions, we get `b= 0, a-3b +c =0, -2c =1, c = (1)/(2).` From `a-3b + c =0, a-3xx0 - (1)/(2) =0, a =(1)/(2)`1 Putting these value in (i) we get `T = K h^(1//2) rho^0 g^(-1//2)` `T = K(sqrth/sqrtg)`