11. Suppose that period of oscillation of the simple pendulum depends on (i) mass of the bob (m) (ii) length L of the pendulum and (ii) acceleration due to gravity g at the plate. Derive expression for its time period, using method of dimensions.

Dear Student ,

Let  t  ma lb gc

Where a, b and c are the powers of m (mass), l (length) and g (acceleration due to gravity)

or, t= k ma lb gc   ..........(i)

Where k is dimensionless constant of proportionality.
Writing the dimensions in terms of M, L and T on each side of equation (i), we get

M0 L0 T1=Ma Lb L T-2c                    =Ma Lb+c  T-2c

Applying the principal of homogeneity of dimensions, we get

a=0                    .......(ii)b+c=0             ...........(iii)-2c=1             or, c=-12      ...........(iv)

Putting the value of c in equation (iii), we get
                                                                 b+-12=0or, b=12

Now, putting the values of a, b and c in equation (i), we get

t=k m0 l12 g-12or,  t=k lg

Using other methods, we calculate the value of k=2π

So, t=2πlg


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