Simple Pendulum

The objective here is to obtain an alternative relation of dimensionless quantities that represents the pendulum’s period T as a function of the the other quantities depicted in the figure below.

We start by importing the necessary classes from nodimo:

from nodimo import Quantity, Model

Next, we define the quantities that compose the problem. Note the arguments used:

• The arguments M, L and T represent the dimensions Mass, Length and Time, respectively

• The Period T is the dependent quantity, hence the argument dependent=True

• The quantities L and g are used as scaling parameters, therefore the argument scaling=True

T = Quantity('T', M=0, L=0, T=1, dependent=True)  # Period
L = Quantity('L', M=0, L=1, T=0, scaling=True)    # Length
m = Quantity('m', M=1, L=0, T=0)                  # Mass
g = Quantity('g', M=0, L=1, T=-2, scaling=True)   # Gravity
t0 = Quantity('theta_0')                          # Initial angle

Finally, the dimensionless model is defined. Note that, in the output of the cell, nodimo detects that the quantity m can not be part of the model, because the dimension M is not common among the group of quantities.

model = Model(T, L, m, g, t0)

NodimoWarning: Dimensionally irrelevant quantities (m)

The dimensional and dimensionless relations can be displayed by:

model.show()

$$\displaystyle T = f\left(L,\ g,\ \theta_{0}\right)$$

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$\displaystyle \mathtt{\text{Scaling group }}\left(L,\ g\right)$

$$\displaystyle \frac{T {g}^{\frac{1}{2}}}{{L}^{\frac{1}{2}}} = \Phi\left(\theta_{0}\right)$$

As you can see, the dimensional expression provides a relationship between 4 quantities, while the dimensionless one has 2 effective quantities. This is one the main advantages of building dimensionless relations.