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 → Rotational motion ↓


Parallel and Perpendicular Axis Theorem


In the world of rotational motion, we often deal with objects rotating around a particular axis. To properly understand these motions, we need to know about two important theorems: the parallel axis theorem and the perpendicular axis theorem. These theorems help us calculate moments of inertia, which play a vital role in analyzing and predicting the behavior of rotating objects.

Moment of inertia

Before diving into the two theorems, let's first understand what moment of inertia is. Moment of inertia is a measure of an object's resistance to a change in its rotation. It's similar to mass in linear motion - just as more mass makes it harder to start or stop moving, more moment of inertia makes it harder to start or stop rotating.

The moment of inertia depends on how the mass of the object is distributed with respect to the axis of rotation. In mathematical terms, the moment of inertia I is calculated as:

I=Σmr²

where m is the mass of a small element of the object and r is the distance of this mass from the axis of rotation.

Parallel axis theorem

When the axis of rotation is shifted from the center of mass of a body to another parallel axis, the calculation of the moment of inertia becomes a little more complicated. The parallel axis theorem provides a solution to this problem. It states that:

I = Icm + Md²

In this equation:

  • I is the moment of inertia about the new axis.
  • Icm is the moment of inertia about the fundamental axis through the centre of mass.
  • M is the total mass of the object.
  • d is the perpendicular distance between the original axis and the new axis through the center of mass.

Example: Rod rotation

Consider a uniform rod of mass M and length L rotating about an axis perpendicular to its length and pivoted at one end. The moment of inertia of the rod about an axis passing through its centre is:

Icm = (1/12) ml²

To find the moment of inertia about the end of the rod (parallel axis), we use the parallel axis theorem:

I = Icm + Md²

Here, d = L/2 (since the center of mass is at the middle of the rod). Substituting the values, we get:

I = (1/12)ML² + M(L/2)² = (1/12)ML² + (1/4)ML² = (1/3)ML²

Visual example

New pivot Center of mass axis D = L/2 L

Perpendicular axis theorem

The perpendicular axis theorem applies to plane objects, commonly called plane objects. It states that if two perpendicular axes, x and y, exist in the plane of an object, and z is the axis perpendicular to the plane at the point where the two axes intersect, then the sum of the moments of inertia about the two plane axes is equal to the moment of inertia about the perpendicular axis. Mathematically:

Iz = Ix + Iy

Example: Disk rotation

Let us consider a disc of mass M and radius R. We want to find the moment of inertia about an axis z passing through the centre of the disc using the perpendicular axis theorem.

For a symmetric disc, the moments of inertia about x and y axes (both in the plane of the disc) are equal.

Ix = Iy = (1/4)MR²

Using the perpendicular axis theorem:

Iz = Ix + Iy = (1/4)MR² + (1/4)MR² = (1/2)MR²

Visual example

X-axis Shaft Z-axis

Combining theorems: Analysis

In practical scenarios, these two theorems can often be combined to simplify complex problems involving rotation. For example, consider a plane object such as a disk or rectangle whose moment of inertia must be calculated about an axis away from the center. By strategically employing both the parallel axis and perpendicular axis theorems, one can efficiently determine the required moment of inertia without having to solve the entire integration procedure from scratch.

Example scenario

Suppose we have a thin rectangular plate with width w and height h, and we need to find its moment of inertia about an axis parallel to its width but 1 m above the center of mass x = 0. To solve this, we can do this:

  1. Calculate the moment of inertia passing through the center of mass using the perpendicular axis theorem.
  2. Use the parallel axis theorem for the shift.

Let's explore these steps for clarity:

Step 1: Moment of inertia through the center of mass

If you look at the object from its plane view, along the axis of interest, the perpendicular axis theorem holds:

Iz = Ix + Iy

For a thin rectangle:

Ix = (1/12)wh³
Iy = (1/12)hw³

Step 2: Applying the parallel axis theorem

Now use the following to move the axis 1 meter vertically:

I = Icm + M(d)²

Substitute d = 1 for the axis displacement.

Understanding the parallel and perpendicular theorems in rotational dynamics can greatly reduce some of the complexities that arise from rotational motion situations. Understanding these basic principles allows for much simpler problem-solving approaches within physics.


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Parallel and Perpendicular Axis Theorem

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