Question

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Q : If ' T ' of a pendulum depends on ' m ', ' l ' and ' g '( usual parameters) , derive an expression for ' T ' .

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Answer 1

Here's the answer:-

We have,
Time period, T depends on - m, l and g

So, T is directly proportional to m^a × l^b ×g^c
T= km^a×l^b×g^c
where, k= dimensional constant
a, b, and c are exponent.

Putting dimensions on both sides:-
[T] = [M]^a [L]^b [LT^-2]^c
[M°L°T¹] = [M^a. L^b+c. T^-2c]

Comparing dimensions on both sides:-
a=0
b+c= 0
=>b= -c ......(i)
-2c=1
=>c=-1/2
b=1/2 {From (i)}

From equation:-
T =km° .l^1/2 .g^-1/2
T = k (l/g)^1/2
T = k√l/g




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