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Energy Principle for a Single Particle

• The concept of a particle in physics refers to a simple object treated as having no smaller constituents, like a ball, Earth, or a person.

Types of Energy in a Particle

• A particle possesses two types of energy:

  • Rest Energy (E_rest): Associated with the mass of the particle, given by the equation E_rest = mc².

  • Kinetic Energy (KE): Associated with the motion of the particle, calculated as KE = (γ - 1)mc² for relativistic speeds, or KE =
    2mv² for low speeds (typically under 10% the speed of light).

• Total Energy (E_total): The sum of the rest energy and kinetic energy, expressed as E_total = γmc².

Example Calculations

• Example: A 2 kg ball moving at -3.4 m/s has a kinetic energy calculated by:

  • KE =
    2mv² =
    2(2 kg)(-3.4 m/s)² = 25 Joules.

• Units of Energy: A Joule (J) is defined as one Newton meter (N·m).

• Rest energy for the same ball is computed as:

  • E_rest = 2 kg * (3 × 10⁸ m/s)² = 1.8 × 10¹⁷ Joules, demonstrating that rest energy vastly exceeds kinetic energy for macroscopic objects.

Particle Behavior at High Speeds

• For an electron moving at 0.9c:

  • Rest energy (E_rest) is given by m * c², yielding about 8.1 × 10⁻¹⁴ Joules.

  • Total energy can be calculated using E_total = γmc², where γ is the Lorentz factor calculated from γ = 1 / √(1 - v²/c²).

  • Kinetic energy is determined by: KE = E_total - E_rest.

• For small, fast-moving particles, kinetic energy may approach or exceed rest energy, particularly at speeds approaching light speed.

Applications of Energy in Physics and Medicine

• High speed particles are utilized in nuclear medicine for treatments such as proton therapy, where high-energy particles preferentially target cancer cells.

• Energy is often expressed in electronvolts (eV), relevant for subatomic particles where 1 eV = 1.6 × 10⁻¹⁹ Joules.

  • An electron with a rest energy of 8.1 × 10⁻¹⁴ Joules corresponds to about 5.1 × 10⁵ eV or 0.51 MeV, highlighting the practical unit choice in nuclear physics.

Energy Principle for Particles

• The Energy Principle states:

  • The change in a particle's energy over time is the sum of energy inputs and outputs.

• Change in energy (ΔE) is influenced by external forces over a displacement (∆x):

  • Exerting a force parallel to the displacement increases kinetic energy;

  • A force acting perpendicular does not change energy (e.g., moon in circular orbit with constant speed).

Work Done on a Particle

• Work (W) is defined as the dot product of force and displacement:

  • W = F · d = |F| |d| cos(θ) where θ is the angle between force and displacement vectors.

• Positive work: If force and displacement are in the same direction, the kinetic energy increases (the object speeds up).

• Negative work: Force and displacement in opposite directions means the object loses energy (the object slows down).

• If force is perpendicular to displacement, no work is done, and energy remains constant.

Summary and Practical Considerations

• The total work done on a system is the sum of work from all forces acting on it. If multiple forces are acting (e.g., gravitational and drag forces on a skydiver), their contributions need to be combined to find the net work and resultant energy change.

• Understanding how to apply forces and determine work is crucial in calculating energy changes for systems in motion.