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 → Dynamics ↓


Non-uniform circular motion


Non-uniform circular motion is a fascinating topic in the study of dynamics and mechanics. While in uniform circular motion an object moves in a circular path with constant speed, non-uniform circular motion occurs when the speed of the object changes while traveling on a circular path. This change in speed indicates that there is a tangential acceleration component acting on the object in addition to the centripetal acceleration that keeps it moving in a circle.

Key concepts

Circular motion

Circular motion refers to the motion of an object along the circumference of a circle or rotation on a circular path. It is important to understand that even in uniform circular motion, while the speed may remain constant, the velocity is not constant as the direction of motion of the object constantly changes.

Velocity Acceleration

Uniform vs. non-uniform circular motion

In uniform circular motion, the object moves along a circular path with constant speed. An example is a car traveling at a constant speed around a circular track. However, in non-uniform circular motion, the speed of the object varies at different points along the circular path. Imagine a carousel that starts slowly, speeds up, and then slows down; this is an example of non-uniform circular motion.

Components of non-uniform circular motion

To fully understand non-uniform circular motion it is important to focus on two main components: radial (or centripetal) acceleration and tangential acceleration.

Centripetal acceleration

Centripetal acceleration is responsible for keeping the object moving on a circular path. This acceleration is always directed towards the center of the circle. The mathematical representation of centripetal acceleration (a_c) is given as:

a_c = v² / r

where v is the tangential speed of the object, and r is the radius of the circle.

Tangential acceleration

In non-uniform circular motion, the tangential speed is not constant. Tangential acceleration describes the rate of change of the object's speed along a circular path. It occurs in the direction of motion and can increase or decrease the object's speed. The formula for tangential acceleration (a_t) is:

a_t = Δv / Δt

Where Δv is the change in tangential velocity, and Δt is the change in time.

Tangential velocity Centripetal acceleration

Pure acceleration

The net acceleration of an object in non-uniform circular motion is the vector sum of the tangential acceleration and the centripetal acceleration. These two components are perpendicular to each other. The magnitude of the net acceleration can be calculated using the Pythagorean theorem:

a_net = √(a_c² + a_t²)

Newton's second law in non-uniform circular motion

Newton's second law, F = ma, plays a key role in circular motion. There must be a net force acting on an object to make it accelerate. In non-uniform circular motion, two forces come into play: the radial (centripetal) force and the tangential force.

Radial force (centripetal force) keeps the object in circular motion and is calculated as follows:

F_c = m * a_c = m * (v² / r)

where m is the mass of the object.

The tangential force changes the speed of the object on a circular path, which is calculated as:

F_t = m * a_t

Applications and examples

Motion of planets

Perhaps the most spectacular example of non-uniform circular motion is the motion of planets around the Sun. Planets move in elliptical orbits at different speeds. Due to gravitational forces, tangential acceleration changes their speed as they orbit.

Vehicles on winding roads

When a car accelerates or decelerates while turning, it undergoes uneven circular motion. Drivers must adjust both the steering (centripetal force) and the gas/brake (tangential force) appropriately to maintain control without skidding.

Practical observation

Amusement park rides

A common example of non-uniform circular motion experienced in daily life is riding a merry-go-round. As the ride starts, it speeds up, and as it stops, it slows down, causing a change in acceleration patterns that affects the way riders feel the forces acting upon them.

Rider's path

Weather patterns

Cyclones and anticyclones also follow circular paths, but their motion patterns differ, influenced by atmospheric conditions. The tangential acceleration involved is what drives the constantly changing weather systems on Earth.

Conceptual summary and further thoughts

By understanding non-uniform circular motion, one can gain a deeper understanding of the forces and accelerations that govern the behavior of objects in circular paths with varying speeds. This topic is linked to many phenomena observed in nature and engineered systems. Engaging with these concepts expands the horizon for practical applications in transportation, physics, and diverse scientific studies.

In fact, the complexities of nonuniform circular motion suggest many opportunities for deeper investigation. The next time you ride a bicycle around a bend, drive a car around a roundabout, or observe a spinning storm system, remember these basic principles that explain the fascinating underlying mechanics.


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Non-uniform circular motion

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