Grade 11
  1. Mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two and three dimensions  1.1.3. Displacement and Distance  1.1.4. Speed and Velocity  1.1.5. Acceleration and deceleration  1.1.6. Graphical analysis of motion  1.1.7. Equations of motion (advanced derivations)  1.1.8. Free fall and terminal velocity  1.1.9. Projectile motion  1.1.10. Relative velocity in one and two dimensions  1.1.11. Motion of a car on an inclined plane  1.2. Dynamics  1.2.1. Newton's laws of motion and their applications  1.2.2. Inertia and Newton's first law  1.2.3. Force and types of force  1.2.4. Static and kinetic friction  1.2.5. Circular motion and centripetal acceleration  1.2.6. Non-uniform circular motion  1.2.7. Banking of roads and curves  1.2.8. Momentum and Impulse  1.2.9. Conservation of momentum in one and two dimensions  1.2.10. Applications of Newton's Third Law  1.3. Work, Energy and Power  1.3.1. Work done by a constant and variable force  1.3.2. Work–energy theorem  1.3.3. Kinetic and potential energy  1.3.4. Gravitational and elastic potential energy  1.3.5. Conservation of mechanical energy  1.3.6. Power and instantaneous power  1.3.7. Efficiency of machines  1.3.8. Work done by conservative and non-conservative forces  1.4. Rotational motion  1.4.1. Torque and angular acceleration  1.4.2. Rotational equilibrium  1.4.3. Moment of inertia and its applications  1.4.4. Angular momentum and its conservation  1.4.5. Kinetic energy of rolling motion and rotation  1.4.6. Parallel and Perpendicular Axis Theorem  1.4.7. Dynamics of rotational motion
  2. Gravitational force  2.1. Universal gravitation  2.1.1. Newton's law of universal gravitation  2.1.2. Gravitational field and potential  2.1.3. Change in acceleration due to gravity  2.1.4. Kepler's laws of planetary motion  2.1.5. Orbital mechanics and satellites  2.1.6. Escape velocity and orbital velocity  2.1.7. Weightlessness and free fall  2.1.8. Gravitational potential energy and energy in orbits  2.1.9. Black holes and gravitational lensing
  3. Properties of matter  3.1. Fluid mechanics  3.1.1. Density and pressure in liquids  3.1.2. Pascal's theory and applications  3.1.3. Archimedes' Principle and Buoyancy  3.1.4. Bernoulli's equation and applications  3.1.5. Continuity equation and the Venturi effect  3.1.6. Surface tension and capillarity  3.1.7. Viscosity and Poiseuille's Law  3.1.8. Reynolds number and turbulence  3.2. Elasticity and deformation  3.2.1. Hooke's law and the stress-strain relation  3.2.2. Elastic modulus and applications  3.2.3. Deformation and yield strength of solids
  4. Thermal physics  4.1. Heat and temperature  4.1.1. Thermometric properties and temperature scale  4.1.2. Thermal expansion of solids, liquids and gases  4.1.3. Heat transfer system  4.1.4. Specific heat capacity and calorimetry  4.1.5. Latent heat and phase change  4.2. Kinetic theory of gases  4.2.1. Molecular model of gases  4.2.2. Kinetic explanation of temperature  4.2.3. Ideal Gas Equation and Deviations  4.2.4. Mean free path and Maxwell–Boltzmann distribution  4.3. Laws of Thermodynamics  4.3.1. Zeroth Law and Thermal Equilibrium  4.3.2. First Law and Internal Energy  4.3.3. The Second Law and Entropy  4.3.4. Carnot cycle and heat engines  4.3.5. Refrigerators and heat pumps
  5. Waves and oscillations  5.1. Simple Harmonic Motion  5.1.1. Equations of SHM  5.1.2. Energy in SHM  5.1.3. Damped and forced oscillations  5.1.4. Resonance and applications  5.1.5. Pendulum and spring-mass system  5.2. Wave motion  5.2.1. Types of mechanical waves  5.2.2. Wave motion and energy transport  5.2.3. Superposition Principle and Standing Waves  5.2.4. Reflection, Refraction and Diffraction  5.2.5. Doppler effect in sound and light  5.2.6. Pulsations and interference
  6. Electricity and Magnetism  6.1. Electrostatics  6.1.1. Electric charge and conservation  6.1.2. Coulomb's law and its applications  6.1.3. Electric field and electric flux  6.1.4. Electric potential and potential energy  6.1.5. Capacitance and Energy Stored in a Capacitor  6.2. Current Electricity  6.2.1. Ohm's law and electrical resistance  6.2.2. Resistivity and temperature dependence  6.2.3. Kirchhoff's Laws and Circuit Analysis  6.2.4. Electrical power and efficiency  6.2.5. Combination of resistors  6.3. Magnetism and Electromagnetism  6.3.1. Magnetic fields and their sources  6.3.2. Magnetic force on moving charges  6.3.3. Electromagnetic induction  6.3.4. Faraday's and Lenz's laws  6.3.5. Inductance and Transformer  6.3.6. Alternating current and its applications
  7. Optics  7.1. Reflection and Refraction  7.1.1. laws of reflection and refraction  7.1.2. Mirrors and lenses  7.1.3. Total internal reflection and optical fiber  7.2. Wave optics  7.2.1. Interference and Diffraction  7.2.2. Young's double-slit experiment  7.2.3. Polarization of light
  8. Modern Physics  8.1.1. Photoelectric effect and Einstein's theory  8.1.2. Wave–particle duality  8.1.3. Heisenberg's uncertainty principle  8.1.4. Compton Effect  8.2. Atomic and Nuclear Physics  8.2.1. Atomic model and energy levels  8.2.2. Radioactive decay and half-life  8.2.3. Nuclear Fission and Fusion
  9. Electronics and Communication  9.1. Semiconductors  9.1.1. pn junction and diode  9.1.2. Transistors and Logic Gates  9.1.3. Integrated Circuits and Applications  9.2. Communication Systems  9.2.1. Basics of Analog and Digital Communication  9.2.2. Wireless and Optical Communication  9.2.3. Modulation and Demodulation

Grade 11 → Mechanics → Work, Energy and Power ↓


Work–energy theorem


The work-energy theorem is an essential concept in physics that connects the ideas of work and energy. By focusing on understanding how energy changes through the performance of work, this theorem gives us a useful tool for analyzing the motion of objects and the forces that act upon them. In this detailed explanation, we will explore the meaning of the work-energy theorem, provide examples and formulas, and clarify concepts through visual illustrations using code.

Basic definitions

Before delving deeper into the theorem, it is important to understand the fundamental definitions of work and energy.

Work

In physics, work is done when a force is applied to an object and makes it accelerate. The amount of work done is calculated by multiplying the applied force by the distance traveled by the object in the direction of the force. Mathematically, work can be expressed as:

Work (W) = Force (F) × Distance (d) × cos(θ)

Where:

  • W is the work done.
  • F is the applied force.
  • d is the distance travelled by the object.
  • θ is the angle between the force and the direction of motion.

Energy

Energy is the capacity to do work and it can exist in different forms such as kinetic energy, potential energy, thermal energy, etc. Energy is a conserved quantity, which means it cannot be created or destroyed but rather it can only change from one form to another.

Kinetic energy

Kinetic energy is the energy an object has due to its motion. It is given by the formula:

Kinetic Energy (KE) = 0.5 × mass (m) × velocity (v)²

Potential energy

Potential energy is energy that an object has because of its position relative to other objects. A common type of potential energy is gravitational potential energy, which is calculated as:

Potential Energy (PE) = mass (m) × gravitational acceleration (g) × height (h)

Work–energy theorem

The work-energy theorem states that the work done by all forces acting on an object is equal to the change in the kinetic energy of the object. Mathematically, it is expressed as:

Work (W) = ΔKE = KE_final - KE_initial

This theorem shows that the work done on an object changes the kinetic energy of the object. Let us understand this in more detail.

Explanation of the theorem

Let us imagine an object with an initial velocity of 1 moving under the influence of a resultant force. As the force acts on the object, it causes a change in the velocity of the object, which leads to a change in its kinetic energy. If the initial kinetic energy is 20 joules and the final kinetic energy is 50 joules, then according to the work-energy theorem, the work done on the object is equal to the change in kinetic energy, i.e. 30 joules.

Visual example 1: Force and displacement in the same direction

Imagine a cart being pushed on a frictionless surface.

Force Cart

In the above scenario, a force is applied to push the cart. Both the force and the displacement are in the same direction. The work-energy theorem tells us that the work done by this force is equal to the increase in the kinetic energy of the cart.

Visual example 2: Force and displacement at an angle

Consider a box being pulled up a ramp.

Force Box

Here, the force is applied at an angle to the displacement. The effective work contributed to the kinetic energy of the box is only the component of the force acting in the direction of motion of the box.

Real-life applications

Driving a car

When you accelerate the car, the engine does work on the car, increasing its speed. According to the work-energy theorem, the work done by the engine results in an increase in the kinetic energy of the car.

Lifting objects

When you lift a box from the ground, you do work against gravity, transferring energy to the box. If the box is stationary at some height, this work done is stored as gravitational potential energy.

Ball rolling down the hill

When a ball rolls down a hill, its potential energy is converted into kinetic energy. The work done by the gravitational force results in an increase in the speed of the ball, which demonstrates the work-energy theorem.

Conservative and non-conservative forces

For a broader understanding, let's differentiate between two types of forces:

Conservative forces

These forces, such as gravity, conserve mechanical energy. The work done by a conservative force is independent of the path taken. Energy can be fully recovered when an object returns to its starting point.

Non-conservative forces

These forces, such as friction, cause energy to be lost in other forms, such as heat. The work done by non-conservative forces depends on the path taken, and the energy cannot be fully recovered when the object returns to its starting point.

Example of conservative force

When you throw the ball straight up, the only force (assuming air resistance is negligible) is gravity, which is conservative. The work done by gravity turns kinetic energy into potential energy as it goes up and vice versa when it comes down.

Example of non-conservative force

If the ball rolls on a surface, friction (a non-conservative force) acts. Due to friction, some part of the kinetic energy is converted into thermal energy, which cannot be recovered.

Mathematical derivation of the work–energy theorem

Consider an object of mass m moving in a straight line with an initial velocity vi under the action of a constant force F. According to Newton's second law, the acceleration a is given by:

F = m * a

The work W done by the force over the displacement d is:

W = F * d

Substituting F from Newton's second law:

W = m * a * d

Using the equation of motion:

vf2 = vi2 + 2 * a * d

We rearrange:

d = (vf2 - vi2) / (2 * a)

Substitute d into the work equation:

W = m * a * [(vf2 - vi2) / (2 * a)]

On simplifying, we get:

W = 0.5 * m * vf2 - 0.5 * m * vi2

It is:

W = KEf - KEi

Implications and significance of the theorem

The work-energy theorem is very helpful in understanding various physical phenomena and mechanical systems. It allows us to solve complex problems by focusing on energy changes rather than forces and motions.

Conservation of mechanical energy

In the absence of non-conservative forces, the work–energy theorem is consistent with the principle of conservation of mechanical energy, which means that the total mechanical energy (the sum of kinetic and potential energies) of an isolated system remains constant.

Conclusion

In conclusion, the work-energy theorem is a powerful principle that connects the work done by forces to the change in kinetic energy. By understanding this relationship, we can predict how objects will move and interact with forces in different situations. It reveals the beautiful consistency in nature, where forces and motion follow predictable patterns governed by the laws of physics.


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Work–energy theorem

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