Pi (π) is a mathematical constant defining the ratio between a circle's circumference and its diameter. In other words, if you divide a circle's circumference by its diameter, you always get the same number, which is approximately equal to 3.14159.
Pi is an irrational number, meaning it has an infinite sequence of non-repeating decimals and cannot be expressed exactly as a fraction.
Pi (π) is present in numerous physics formulas, due to its relationship with circles, waves, and oscillations. Here are some examples of Pi's use in physics formulas:
Circular kinematics: For uniform circular motion, the angular velocity (ω) is defined as ω = 2πf, where f is the frequency. The linear velocity (v) is given by v = ωR, where R is the circle's radius.
Area and volume: The formulas for calculating the area of a disk (A = πR²) and the volume of a cylinder (V = πR²h), a sphere (V = 4/3πR³), or a cone (V = 1/3πR²h) all involve Pi.
Biot-Savart law: This law describes the magnetic field created by an electric current. In the case of an infinitely long and straight wire, the magnetic field at a distance R from the wire is B = (μ₀I)/(2πR), where μ₀ is the magnetic permeability of the vacuum and I is the current intensity.
Orbital period: The period (T) of an object in circular orbit around a central mass (M) is given by T = 2π√(a³/G(M+m)), where a is the average distance between the two objects (the semi-major axis of the orbit), G is the universal gravitational constant, and m is the mass of the orbiting object. This formula is derived from Kepler's third law and is valid for circular and elliptical orbits.
Harmonic oscillator formulas: In the case of a simple harmonic oscillator, such as a spring or a pendulum, the period (T) is defined by T = 2π√(m/k) for a spring and T = 2π√(l/g) for a pendulum, where m is the mass, k is the spring constant, l is the pendulum length, and g is the acceleration due to gravity.
Coulomb's law: The electrostatic force (F) between two point charges (q₁ and q₂) is given by F = (kq₁q₂)/r², where r is the distance between the charges and k = 1/(4πε₀), with ε₀ being the electric permittivity of the vacuum.
Heisenberg formula: Heisenberg's uncertainty principle states that it is impossible to know both the position (Δx) and the momentum (Δp) of a particle with infinite precision simultaneously. The uncertainty relation is given by ΔxΔp ≥ ħ/2, where ħ = h/(2π) is the reduced Planck constant.