Mathematics for Physics I - Physics Applications

Harmonic Motion

Simple Harmonic Motion (SHM) is a fundamental concept in physics describing oscillatory motion where the restoring force is proportional to the displacement.

Equation of Motion

The differential equation for SHM is:

\[ m\frac{d^2x}{dt^2} = -kx \]

Where m is the mass, k is the spring constant, and x is the displacement.

The solution is of the form:

\[ x(t) = A\cos(\omega t + \phi) \]

Where A is the amplitude, ω = √(k/m) is the angular frequency, and φ is the phase constant.

Energy in SHM

The total energy in SHM is conserved and is given by:

\[ E = \frac{1}{2}kx^2 + \frac{1}{2}mv^2 = \frac{1}{2}kA^2 \]

This shows the interplay between potential energy (½kx²) and kinetic energy (½mv²).

Interactive visualization of Simple Harmonic Motion

Orbital Motion

Orbital motion describes the motion of celestial bodies under the influence of gravity, often following Kepler's Laws.

Kepler's Laws

Law of Orbits: Planets move in elliptical orbits with the Sun at one focus.
Law of Areas: A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.
Law of Periods: The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit (T² ∝ a³).

Gravitational Force

Newton's Law of Universal Gravitation describes the force between two masses:

\[ \vec{F} = -G\frac{Mm}{r^2}\hat{r} \]

Where G is the gravitational constant, M and m are the masses, r is the distance between them, and \(\hat{r}\) is the unit vector along the line joining them.

This force leads to elliptical, parabolic, or hyperbolic orbits depending on the total energy of the system.

Interactive visualization of orbital motion

Coordinate Transformations

Coordinate transformations are essential for describing physical systems in different reference frames or coordinate systems.

Cartesian to Polar

\[ r = \sqrt{x^2 + y^2} \] \[ \theta = \arctan\left(\frac{y}{x}\right) \]

Polar to Cartesian

\[ x = r\cos\theta \] \[ y = r\sin\theta \]

Jacobian Matrix

The Jacobian matrix describes how a transformation changes local areas or volumes:

\[ J = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix} \]

For polar coordinates, the Jacobian determinant is r, which is why \(dA = r \, dr \, d\theta\).

Applications

Coordinate transformations are used in:

Simplifying problems with certain symmetries (e.g., using spherical coordinates for spherically symmetric potentials)
Relativity (Lorentz transformations)
Robotics and computer graphics

Forces in Vector Systems

Forces are vector quantities, and their analysis often involves vector addition, resolution, and dot/cross products.

Vector Addition

The net force on an object is the vector sum of all individual forces acting on it:

\[ \vec{F}_{net} = \sum_i \vec{F}_i \]

This is crucial for applying Newton's Second Law (\(\vec{F}_{net} = m\vec{a}\)).

Work and Dot Product

The work done by a constant force \(\vec{F}\) over a displacement \(\vec{d}\) is:

\[ W = \vec{F} \cdot \vec{d} = Fd\cos\theta \]

This shows how the component of force in the direction of displacement contributes to work.

Torque and Cross Product

The torque \(\vec{\tau}\) due to a force \(\vec{F}\) acting at a position \(\vec{r}\) relative to an axis is:

\[ \vec{\tau} = \vec{r} \times \vec{F} \]

This is fundamental in rotational dynamics.

Equilibrium

For an object to be in static equilibrium:

\[ \sum \vec{F}_i = \vec{0} \quad \text{(translational equilibrium)} \] \[ \sum \vec{\tau}_i = \vec{0} \quad \text{(rotational equilibrium)} \]

These conditions are used to analyze structures and forces in static systems.

Calculation of Work, Energy, and Momentum

Work-Energy Theorem

The net work done on an object equals its change in kinetic energy:

\[ W_{net} = \Delta K = K_f - K_i \]

Where \(K = \frac{1}{2}mv^2\) is the kinetic energy.

Conservation of Energy

For a system with only conservative forces, the total mechanical energy (E = K + U) is conserved:

\[ E_i = E_f \]

If non-conservative forces (like friction) are present, then:

\[ W_{nc} = \Delta E = E_f - E_i \]

Momentum and Impulse

Linear momentum \(\vec{p} = m\vec{v}\).

Newton's Second Law can be written as \(\vec{F}_{net} = \frac{d\vec{p}}{dt}\).

Impulse \(\vec{J}\) is the change in momentum:

\[ \vec{J} = \int_{t_1}^{t_2} \vec{F}_{net} \, dt = \Delta \vec{p} = \vec{p}_f - \vec{p}_i \]

Conservation of Momentum

If the net external force on a system is zero, the total momentum of the system is conserved:

\[ \vec{P}_{total, i} = \vec{P}_{total, f} \]

This is a fundamental principle in physics, especially for analyzing collisions and explosions.

Key Insight:

The mathematical concepts covered in this course are not abstract tools but are deeply intertwined with the description and understanding of physical reality. From the oscillations of a spring to the orbits of planets, and from the forces in a bridge to the conservation of energy and momentum, these mathematical principles provide the language and framework for analyzing and predicting the behavior of the physical world.

Applications in Physics Contexts