Example - the oscillating pendulum

Consider the case of a pendulum, oscillating within the gravity field, whose oscillation frequency we wish to determine. The a priori parameters of the problem are the angular frequency, a (s ) - the quantity to be determined - the pendulum s mass, M (kg), the length of the wire, / (m), the gravitational acceleration, g (m-s ), and the maximum angle of oscillation, (dimensionless). Based on five parameters and three dimensions, there should therefore be two dimensionless numbers. The first is obviously The mass A/, therefore, does not intervene in the physical problem, as no other parameter depends on the unit of mass. Consequent to this, the only option is to write  

For small angles, we obtain the well-known pendulum relation  

Constant A is undetermined as per dimensional analysis. However, based on dimensional analysis the fact that the oscillation fiequency is not dependent on the 

2 This difficulty is encountered when solving Exercise 3.1 at the end of this chapter. 

3 Is the notion that simple An angle is expressed by a unit (radians, degrees, etc.). For a dimensional analysis problem, we consider that an angle is a dimensionless quantity, because the length of a circular arc is the product of the angle by the radius. The angle, therefore, has no dimension as such. 

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