Question

The formula for the time period of the revolution of a pendulum is T= Root of length÷ root of gravitational acceleration. So the time period is said to be independent of the mass of the pendulum. But, I can derive another formula by 'Dimensional analysis' that comes out to be T=1/momentum × density × length⁴ × constant. Thus the time period will also be dependent on the mass of the pendulum. Isn't this contradictory ?

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Answer 1

Given:

Dimensional formula for time period of pendulum is T = √l/√g.

The time period is also represented as

T = (constant× l⁴ ×density)/(momentum)

To prove:

Time period is not dependent on mass of pendulum.

Solution:

In first case :

$T \propto \sqrt{\dfrac{l}{g} }$

In the second case :

$T = \dfrac{constant \: \times \: \rho \times {(l)}^{4} }{ momentum}$

Seeing this Equation, it apparently looks like that the time period of a pendulum depends upon the mass of the pendulum.

• But , in reality, it is not so because momentum and density are derived quantities (not basic quantities).

• The derived quantities (in this case) are themselves dependent upon mass.

• So, simplification of the derived quantities into basic physical quantities like mass , length and time will reveal the fact that time period is not dependent on mass .

• For example , in this case , if we simplify Momentum and density we will observe that the mass factor will get cancelled in the Equation.

Hope It Helps.