Question

Time period T of a simple pendulum may depend upon the mass of the bob m;length of the simple pendulum l and g,the acceleration due to gravity,i.e., T directly proportional to mlg(raised to powers a.b,c),the values ofa,b,c are
a)-1/2, 0,1/2
b)1/2,0, -1/2
c)0,-1/2,1/2
d)0,1/2 1/2
choose the correct answer with derivation also

Answer 1

T = k*m^a*l^b*g^c           (where k is dimensionless constant)
[M0L0T1] = [M]^a[L]^b[LT-2]^c
[M0L0T1] = [M^a*L^(b+c)*T^-2c]
compare the power of the both side
a = 0
b+c = 0
-2c = 1
c = -1/2
b = 1/2
 hence the value of a,b,c is 0,1/2,-1/2.

Answer 2

We use the dimension analysis to derive the formula.

$As\ given\ T \alpha \ m^x\ L^y\ g^z \\ \\ \ [ T^1 ] \alpha \ [ M^x ] [ L^y ] [ L^z T^{-2z} ] \\ \\ = [ M^{x} L^{y+z} T^{-2z} ] \\ \\ \ equating\ the\ powers\ of\ dimensions\ on\ both\ sides\ \\ \\ x = 0,\ \ y+z = 0, y = -z, \ \ -2z = 1\ \ \ \ z = -\frac{1}{2} \\ T \alpha \ \ m^0 L^\frac{1}{2} T^{-\frac{1}{2}}$