Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Classical mechanics → dynamics ↓


Motion in two dimensions


In physics, dynamics is the study of motion, without considering the forces that cause it. When we explore motion in two dimensions, we analyze objects moving in a plane. This subject is fundamental in understanding how things move and is widely applied in a variety of fields, including engineering, astronomy, and everyday life.

Basic concepts of two-dimensional motion

Before diving into two-dimensional motion, let's recall the basic idea of motion in one dimension. In one-dimensional motion, an object moves forward or backward in a straight line. The elementary quantities that describe this motion are as follows:

  • Displacement - Change in position of an object.
  • Velocity - Rate of change of displacement relative to time.
  • Acceleration - The rate of change of velocity relative to time.

When we extend these concepts to two dimensions, the object can move in a plane, which is described by coordinates (x, y) on the Cartesian plane. The motion can be more complex as it can follow different paths such as straight lines, curves, or circles.

Representation of two-dimensional motion

To represent motion in two dimensions, we use vectors for displacement, velocity, and acceleration. Vectors are quantities that have both magnitude and direction.

Displacement vector

Displacement vector → d represents the change in position of the object in a plane and is given by:

→ d = (x 2 - x 1) →i + (y 2 - y 1) →j

where →i and →j are unit vectors along the x and y axes, respectively.

For example, if an object moves from point (1, 2) to point (4, 6), the displacement vector is:

→ d = (4 - 1) →i + (6 - 2) →j = 3 →i + 4 →j
→d (1, 2) (4, 6)

Velocity vector

Velocity in two dimensions is also a vector quantity. It tells how the displacement vector changes with time.

→ v = v x →i + v y →j

where v x and v y are the velocity components corresponding to the x and y directions, respectively.

Acceleration vector

Acceleration is the rate of change of velocity. In two dimensions, it is expressed as:

→ a = a x →i + a y →j

where a x and a y are the components of acceleration corresponding to the x and y directions.

Equations of motion in two dimensions

The equations of motion we use in two-dimensional dynamics are extensions of the one-dimensional equations. They involve both x and y components, and if the acceleration is constant, each component can be considered independently.

Uniform acceleration equation in two dimensions

  • x = x 0 + v x0 t + ½a x t2
  • y = y 0 + v y0 t + ½a y t2
  • v x = v x0 + a x t
  • v y = v y0 + a y t

These equations assume that the acceleration is constant in every direction.

Projectile motion

Projectile motion is a common example of motion in two dimensions. It combines horizontal motion with constant velocity and vertical motion with constant acceleration due to gravity.

Consider an object projected from the ground with an initial velocity v 0 at an angle θ to the horizontal. The equations describing this motion are as follows:

v x0 = v 0 cos(θ) v y0 = v 0 sin(θ) x = x 0 + v x0 ty = y 0 + v y0 t - ½gt2

Here, g is the acceleration due to gravity. For example, given an initial velocity of 20 m/s at an angle of 45 degrees, find the position of the object after 2 seconds.

First, calculate the horizontal and vertical components of the initial velocity:

v x0 = 20 cos(45°) = 14.14 m/s v y0 = 20 sin(45°) = 14.14 m/s

Using the kinematic equations for x and y:

x = 14.14 * 2 = 28.28 m y = 14.14 * 2 - 0.5 * 9.8 * 22 = 28.28 - 19.6 = 8.68 m
(28.28, 8.68)

Uniform circular motion

Motion in two dimensions also includes uniform circular motion, where an object moves around a circle at a constant speed. While the speed is constant, the direction of the velocity constantly changes, resulting in an acceleration known as centripetal acceleration.

The magnitude of the centripetal acceleration is given by:

a c = v2 / r

where a c is the centripetal acceleration, v is the speed of the object, and r is the radius of the circle.

→V → A C

Example: Car on a circular track

Suppose a car is moving at a speed of 10 m/s on a circular track of radius 50 m. Calculate the centripetal acceleration.

a c = 102 / 50 = 2 m/s2

The car experiences a centripetal acceleration of 2 m/s2 toward the center of the circle.

Relative motion in two dimensions

In some cases, the motion of objects is described relative to different reference frames. Understanding how to analyze relative motion is important for solving real-world problems, such as determining the speed of one object relative to another object.

The concept of relative velocity in two dimensions can be described by the following equation:

→ v AB = → v AB - → v AC

where → v XY is the velocity of object X relative to object Y.

Example: Boats crossing the river

Imagine a boat trying to cross a river flowing in an easterly direction at a speed of 3 m/s. If the boat moves at a speed of 5 m/s relative to the water in a direction perpendicular to the flow of the river (towards the north), find the resultant velocity of the boat relative to the observer on the shore.

Using the Pythagorean theorem, the resultant velocity is:

v resultant = sqrt((52) + (32)) = sqrt(25 + 9) = sqrt(34) ≈ 5.83 m/s

The direction θ of the resultant velocity relative to north can be calculated as:

θ = arctan(→v east / →v north) = arctan(3/5) = 30.96°

The boat moves at a speed of about 5.83 m/s, due east-north, at an angle of about 30.96 degrees.

Conclusion

Motion in two dimensions involves more complex analysis than one-dimensional motion, but the principles remain logically consistent. By understanding the vector representation and applying the kinematic equations, we can solve a variety of problems involving projectile motion, circular motion, and relative motion. Mastering these concepts lays the foundation for exploring more complex motions in physics and other disciplines.


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Motion in two dimensions

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