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 → Gravitational force → Universal gravitation ↓


Gravitational potential energy and energy in orbits


Introduction

The concepts of gravitational potential energy and energy in orbits are fundamental topics in understanding the dynamics of objects under the influence of gravity. When we talk about gravitational forces, we often think of the force that keeps planets moving around stars, just as the Earth revolves around the Sun. However, this is not the only role of forces; there is much more going on about the energy associated with these movements.

Gravitational potential energy

Gravitational potential energy (GPE) is the energy that an object has because of its position in a gravitational field. The most common context for discussing GPE is near the Earth's surface, where we use the formula:

        GPE = m * g * h

where m is the mass of the object, g is the gravitational acceleration (about 9.8 m/s2 on Earth), and h is the height above the reference point.

For example, if you pick up a book and place it on a shelf, you are increasing its gravitational potential energy, because you are increasing its height above the ground.

Example explained

Consider a rock at the top of a cliff. When it is at the edge, its height h is maximum, so its gravitational potential energy is also maximum. As it falls, its height decreases until it reaches the ground. At that point, the height is zero, and so the gravitational potential energy relative to that point is also zero. Mathematically,

        GPE = m * g * h = mass_of_rock * 9.8 * height_of_cliff

Gravitational potential energy in universal gravitation

When discussing universal gravitation, especially at the cosmic level, the formula for gravitational potential energy changes. In this context, the formula is:

        GPE = - (G * m1 * m2) / r

Here, G is the gravitational constant, approximately 6.674 * 10^-11 N(m/kg)^2. The values m1 and m2 are the masses of the two interacting objects, and r is the distance between their centers. The negative sign indicates that work must be done against the gravitational force to increase the distance between the masses.

Visual example: two-body system

M1 M2 R

As two objects get closer, the gravitational potential energy becomes more negative, meaning more energy will be required to separate them.

Energy in orbits

When an object orbits another object under the influence of gravity, its energy can be classified into two main types: kinetic energy (KE) and gravitational potential energy (GPE). The total mechanical energy in orbit is constant, obtained by the sum of these two types of energy.

        KE = (1/2) * m * v^2

where v is the orbital velocity. For an object in a stable orbit,

        Total Energy E = KE + GPE

Energy determines the shape and nature of the orbit. For circular orbits, momentum and energy remain constant. For elliptical orbits, momentum and energy vary, these principles still apply.

Circular orbits

In a circular orbit, the gravitational force provides the centripetal force to keep the object in orbit. Therefore, the equation for centripetal force is:

        F_gravitational = F_centripetal

Substituting the force expressions, you get:

        (G * m1 * m2) / r^2 = (m * v^2) / r

Simplifying, this leads to an expression for the orbital speed v:

        v = sqrt((G * m1) / r)

Elliptical orbits

Most celestial bodies follow elliptical orbits, where both kinetic and potential energy vary at different points. When the object is closest to the massive body (at perigee), its speed and hence kinetic energy is maximum, while the potential energy is least negative. Conversely, at apogee (the farthest point), the speed and kinetic energy are minimum, and the potential energy is most negative.

Visual example: elliptical orbit

culmination perigee

Energy exchange between the kinetic and potential forms is an example of the communication of energy principle in gravitational orbits.

Practical observations and applications

Gravitational potential energy and orbital mechanics are observed in the real world in a variety of phenomena, from the deployment of satellites to our understanding of galaxies. Understanding these concepts has allowed humanity to use energy principles to place satellites in stable, efficient, and predictable orbits around Earth.

Examples: satellite

When a satellite is put into orbit, mission planners calculate the required velocity and trajectory to ensure it reaches the desired altitude and speed, and to keep its kinetic and gravitational potential energy balanced.

        Total Energy of Satellite = KE + GPE

By analyzing these energies, they can predict the orbital path, duration and efficiency of the satellite's mission.

Conclusion

Gravitational potential energy and energy in orbits not only highlight the impact of the enormous, binding nature of gravity but also demonstrate the harmony of celestial mechanics. These principles have profound implications, from advancing our understanding of cosmic scales to enabling us to use and manipulate orbits for technological advancement, particularly in satellite telecommunications and space exploration.


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Gravitational potential energy and energy in orbits

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