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 → Work and Energy ↓


Work–energy theorem


The work-energy theorem is a fundamental concept in classical mechanics that connects the work done by all forces acting on an object to the change in its kinetic energy. This theorem is a bridge that connects the concept of work and the energy principles. It is incredibly useful because it allows us to solve problems that involve changing velocities and the forces that produce them.

To understand this theorem better, let's take a deeper look at the terms involved:

Understanding the work

In physics, work is defined as the process of energy transfer through a force that accelerates an object. The mathematical expression of work when a constant force is applied is given as:

W = F · d · cos(θ)

Where:

  • W is the work done by the force (Joules).
  • F is the magnitude of the force (Newtons).
  • d is the displacement of the object (meters).
  • θ is the angle between the force vector and the displacement vector (degrees).

Understanding energy

Energy is the ability of an object to do work. There are different forms of energy, but in the context of the work-energy theorem, we are primarily interested in kinetic energy, which is the energy of motion. The kinetic energy (KE) of an object with mass m and velocity v is calculated using the formula:

KE = 0.5 · m · v 2

Work–energy theorem

According to the work-energy theorem, the work done by the net force on an object is equal to the change in its kinetic energy. In mathematical terms, it can be expressed as follows:

W_net = ΔKE = KE_final - KE_initial

This equation means that the total work done on an object is equal to the difference between its final and initial kinetic energy. It shows how forces acting over a certain distance can increase or decrease the speed of an object.

Visual example

F D Moving Object

Consider the example above, in which there is a blue box on a surface. A force F is applied to the box, causing it to move through a displacement d. According to the work-energy theorem, the work done by the force F (assuming it is the only force) leads to a change in the kinetic energy of the box.

Text example

Let us consider a practical example:

Suppose a car with a mass of 1000 kg moves at a speed of 15 m/s. Later, the car is accelerated to a speed of 25 m/s by applying the engine force. To find out how much work the engine does during this acceleration, we calculate the change in kinetic energy:

Initial kinetic energy:

KE_initial = 0.5 * 1000 * (15 2 ) = 112500 J

Final kinetic energy:

KE_final = 0.5 * 1000 * (25 2 ) = 312500 J

The work done by the engine is the change in kinetic energy:

W_net = KE_final - KE_initial = 312500 J - 112500 J = 200000 J

Therefore, the engine does 200,000 Joules of work to increase the speed of the car from 15 m/s to 25 m/s.

In-depth study on forces

When dealing with multiple forces, the work-energy principle is valid for the net -- or resultant -- force acting on the object. This means that if multiple forces act on the object, you determine the net force before applying the theorem. The net force is calculated as follows:

F_net = ΣF

where ΣF is the sum of all the individual forces acting on the object. Once determined, you can apply the work-energy theorem to find the change in kinetic energy.

Conservative and non-conservative forces

In physics, forces are often classified into two categories: conservative forces and non-conservative forces.

A conservative force (like gravity) is a force whose work does not depend on the path taken but only on the initial and final conditions. In the presence of conservative forces, energy is conserved within the system, converting between potential and kinetic forms.

On the other hand, a non-conservative force (such as friction) dissipates energy from the system, usually as heat. The work done by non-conservative forces changes the total mechanical energy (the sum of potential and kinetic energy) of the system.

Applications of the work-energy theorem

The work-energy theorem is used in various fields of physics and engineering. Below are some scenarios where it is particularly useful:

  • Vehicle dynamics: Understanding the power output required from the engine to achieve a given speed and acceleration.
  • Design of machines: Estimating the energy requirements and efficiency of machines such as cranes and lifts.
  • Sports science: Analyzing the motions and forces in athletic performances to enhance training regimes.
  • Astronomy: Calculating energy changes during the motion of celestial bodies.

Conclusion

The work-energy theorem is a powerful and versatile tool in the physicist's toolkit. By understanding the relationship between work done and changes in kinetic energy, scientists and engineers can solve complex problems involving the motion of objects subjected to various forces. Through its applications, the theorem enhances our ability to predict and manipulate the physical world, leading to practical results in technology, sports, space exploration, and beyond.


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Work–energy theorem

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