======== Examples ======== In the following sections we give few examples of what can be done with this module. Dimensional analysis ==================== We will start from Newton's second law .. math:: m a = F where :math:m, a and :math:F are the mass, the acceleration and the force respectively. Knowing the dimensions of :math:m (:math:M) and :math:a (:math:L T^{-2}), we will determine the dimension of :math:F; obviously we will find that it is a force: :math:M L T^{-2}. From there we will use the expression of the gravitational force between the particle of mass :math:m and the body of mass :math:M, at a distance :math:r .. math:: F = \frac{G m M}{r^2} to determine the dimension of the Newton's constant :math:G. The result should be :math:L^3 M^{-1} T^{-2}. >>> from sympy import symbols >>> from sympy.physics.units.systems import SI >>> from sympy.physics.units import length, mass, acceleration, force >>> from sympy.physics.units import gravitational_constant as G >>> from sympy.physics.units.systems.si import dimsys_SI >>> F = mass*acceleration >>> F Dimension(acceleration*mass) >>> dimsys_SI.get_dimensional_dependencies(F) {'length': 1, 'mass': 1, 'time': -2} >>> dimsys_SI.get_dimensional_dependencies(force) {'length': 1, 'mass': 1, 'time': -2} Dimensions cannot compared directly, even if in the SI convention they are the same: >>> F == force False Dimension system objects provide a way to test the equivalence of dimensions: >>> dimsys_SI.equivalent_dims(F, force) True >>> m1, m2, r = symbols("m1 m2 r") >>> grav_eq = G * m1 * m2 / r**2 >>> F2 = grav_eq.subs({m1: mass, m2: mass, r: length, G: G.dimension}) >>> F2 #doctest: +SKIP Dimension(mass*length*time**-2) >>> F2.get_dimensional_dependencies() #doctest: +SKIP {'length': 1, 'mass': 1, 'time': -2} Note that one should first solve the equation, and then substitute with the dimensions. Equation with quantities ======================== Using Kepler's third law .. math:: \frac{T^2}{a^3} = \frac{4 \pi^2}{GM} we can find the Venus orbital period using the known values for the other variables (taken from Wikipedia). The result should be 224.701 days. >>> from sympy import solve, symbols, pi, Eq >>> from sympy.physics.units import Quantity, length, mass >>> from sympy.physics.units import day, gravitational_constant as G >>> from sympy.physics.units import meter, kilogram >>> T = symbols("T") >>> a = Quantity("venus_a") Specify the dimension and scale in SI units: >>> SI.set_quantity_dimension(a, length) >>> SI.set_quantity_scale_factor(a, 108208000e3*meter) Add the solar mass as quantity: >>> M = Quantity("solar_mass") >>> SI.set_quantity_dimension(M, mass) >>> SI.set_quantity_scale_factor(M, 1.9891e30*kilogram) Now Kepler's law: >>> eq = Eq(T**2 / a**3, 4*pi**2 / G / M) >>> eq Eq(T**2/venus_a**3, 4*pi**2/(gravitational_constant*solar_mass)) >>> q = solve(eq, T)[1] >>> q 2*pi*venus_a**(3/2)/(sqrt(gravitational_constant)*sqrt(solar_mass)) To convert to days, use the convert_to function (and possibly approximate the outcoming result): >>> from sympy.physics.units import convert_to >>> convert_to(q, day) 71.5112118495813*pi*day >>> convert_to(q, day).n() 224.659097795948*day We could also have the solar mass and the day as units coming from the astrophysical system, but we wanted to show how to create a unit that one needs. We can see in this example that intermediate dimensions can be ill-defined, such as sqrt(G), but one should check that the final result - when all dimensions are combined - is well defined.