Question

The period of oscillation of a compound pendulum is given by t=√k²+l²/lg,where l is the distance of the point of suspensions from its CG and k is the radium of gyration of pendulum about a parallel axis through CG. Test the validity of the above equation by the method of dimensions

Answer 1

Given ,
t = √{(k² + l²)/lg}

where , K is radius of gyration
dimension of radius of gyration = [L]
dimension of l = [ L ]
dimension of g = [L T-²]
dimension of t = [ T ]

now,
LHS = t = dimension [ T ]

RHS = √{(k² + l²)/lg}

= √{( dimension of k² or l²)/dimension of l × dimension of g }

= √{ [L²]/[L][LT-²]}

= √{ [L²-²]/[T-² ]

= √{1/[T-²] }

= [ T ]

here we see ,
dimension of LHS = dimension of RHS
hence, this equation is dimensionally correct .