Notes, summaries, assignments, exams, and problems for Physics

Kinematics: Understanding Motion

Scalars and Vectors in Motion

• Scalars: Quantities possessing magnitude only (e.g., speed, distance, time, mass).

• Vectors: Quantities possessing both magnitude and direction (e.g., velocity, displacement, acceleration, force). These are represented by arrows.

  • Position: Displacement (Vector)

  • Change: Distance (Scalar), Displacement (Vector)

  • Rate: Speed (Scalar), Velocity (Vector)

  • Change in Rate: Acceleration (Vector)

Constant Acceleration Equations

• Variables Used: Final velocity (v), Initial velocity (u), Acceleration (a), Displacement (s), Time (t).

• Key Equations:

  • v = u + at

  • v² = u² + 2as

  • s = &frac12(u+v)t

  • s = ut + &frac12at²

  • s = vt - &frac12at²

Graphing Motion Characteristics

• Displacement-Time (s-t) Graph:

  • A flat

Stokes' Theorem: Definition and Importance

Stokes' Theorem is a fundamental statement in multivariable calculus that relates the surface integral of the curl of a vector field over a surface to the line integral of the vector field around the boundary of the surface. This is a powerful tool that bridges the gap between line integrals and surface integrals. Stokes' Theorem is a higher-dimensional version of the two-dimensional Green's Theorem, and it is important in many fields of physics and engineering, including fluid dynamics, electromagnetism, and differential geometry. It is an effective tool for evaluating line integrals and investigating the behavior of vector fields in three dimensions.

The Stokes' Theorem Formula

The general formula for... Continue reading "Fundamental Theorems of Vector Calculus and Applied Mathematics" »

Fleming's Left-Hand Rule

This rule determines the direction of the force on a conductor when placed in a magnetic field. It is often used in the context of electric motors.

• If the index finger points in the direction of the magnetic field,
• And the middle finger points in the direction of the current,
• Then the direction of the thumb indicates the direction of the force on the conductor.

Fleming's Right-Hand Rule

This rule is used to identify the direction of induced current when a conductor moves within a magnetic field. It is fundamental to understanding electric generators.

• The thumb indicates the direction of the motion of the conductor.
• The index finger indicates the direction of the magnetic field.
• The middle finger indicates the direction of the

The Governor is the nominal head (constitutional head) of a state in India, appointed by the President of India under Article 155 of the Constitution. Though modeled on the President at the Union level, the Governor functions mainly on the aid and advice of the State Council of Ministers, headed by the Chief Minister.

Constitutional Powers of the State Governor

The powers of the Governor can be classified into the following categories:

1. Executive Functions and Appointments

• Appoints the Chief Minister and other ministers (Article 164).
• Appoints the Advocate General, State Election Commissioner, and the Chairman and Members of the State Public Service Commission.
• Administers the oaths of office to various officials.
• Acts as the Chancellor of state

Uniformly Distributed Load (UDL) and Varying Load (UVL)

Converting Distributed Loads to Equivalent Point Loads

1. Uniformly Distributed Load (UDL)

Definition: A load that is evenly spread across a specific length ($L$) of a beam or structure.

Magnitude of Equivalent Point Load ($W$):

$$W = w \times L$$

• $w$: Intensity of UDL (N/m or kN/m)
• $L$: Length over which the UDL acts

The equivalent point load $W$ acts at the geometric center of the distributed load (at $L/2$ from either end).

UDL Diagram and Equivalent Point Load:

|‾‾‾‾‾‾‾‾‾‾|
| w N/m |
|___________|

Equivalent:

| | ↓ W = w × L
| | (at L/2)
|_______________

2. Uniformly Varying Load (UVL)

Definition: A load whose intensity varies linearly across the beam length, typically... Continue reading "Structural Loads and Support Reactions in Engineering Mechanics" »

Science & the Universe                                                                                                                                       

Astronomy = study of celestial objects and their interactions.                                    

Scientific method: relies on observation, testing, and revision.

Distances measured in light-years; light travels at ~300,000 km/s

Scientific notation helps handle large/small numbers.  γ

Observing the Sky

Constellations = regions in the sky (88 official).

Sky appears to move due to Earth’s rotation (24h) and orbit (365 days).        

Zenith = overhead; meridian = N to S through zenith.

Ecliptic = Sun'... Continue reading "Fundamental Concepts in Astronomy and Astrophysics" »

1-D Motion

Can be described with zero displacement

Cannot be described with zero distance

Distance [x] = how far you move

Displacement [Δx] = distance from start to end

Speed = how fast [v = d/t]

Velocity = speed and direction [v = Δx/Δt]

Position/time: where we are at any given time

velocity = slope

v decreasing: A -> E

stationary: D

v increasing: A/none

greatest speed: A

Velocity/time: how fast we're going at any given time

acceleration/speeding up = slope

Stationary: A, L

Constant: H, E, D

Slowing down: K, J, I

Speeding up: B, C, F, G

Acceleration

Kinematic Equations:

1. V [end velocity] = V₀ [initial velocity] + at

ex. How fast do we hit the ground?

t = 20s

a = g = ~9.8 m/s²

x = 0m (x-axis intercept)

V₀... Continue reading "Essential Concepts in Classical Mechanics Physics" »

Introduction to Computational Fluid Dynamics (CFD)

Definition of CFD: CFD is the process of mathematically predicting physical fluid flow by solving the governing equations using computational power. Every CFD analysis uses a mathematical model and numerical method based on the Navier-Stokes (N-S) equations. Physical properties are calculated based on defined operating conditions.

Main objectives:

• Minimize the cost of the system
• Understanding and comprehension of the problem
• Improve behavior
• Reduce the time and cost of the design stage

3 Fundamental Principles:

1. Mass is conserved
2. F=m*a (Newton's 2nd Law)
3. Energy is conserved

Mass Conservation Principle: The rate of increase of mass in a fluid element equals the net rate of flow of mass into the fluid element.... Continue reading "CFD: Understanding Fluid Flow Through Computational Analysis" »

DC Motor Speed-Torque Characteristics

• Graph Interpretation

  • Y-axis: Speed (N).
  • X-axis: Torque (T).
  • Shape: Linear downward slope (speed decreases as torque increases).

• Speed-Torque Formula

  N=V−IaRaϕN=ϕV−Ia​Ra​​,
  where I_a = armature current, R_a = armature resistance, ϕ = flux.

• Key Performance Points

  • At No Load: High speed, low torque.
  • At Full Load: Low speed, high torque (due to armature reaction).

• Applications

  Used in electric vehicles and cranes for variable speed control.

Working Principle of 3-Phase Induction Motor

• Stator Function

  • A 3-phase AC supply produces a Rotating Magnetic Field (RMF).
  • RMF Speed (Synchronous Speed, N_s): Ns=120fPNs​=P120f​.

• Rotor Operation

  • Conductors (aluminum bars) are cut by the RMF, inducing current (Faraday’s

Capillary Action and Surface Tension

Capillary action describes the phenomenon where the level of a liquid inside a narrow tube (relative to its container) is either raised or lowered. This height difference is maintained by surface tension forces. The direction and magnitude of this change depend on the liquid's surface tension and its interaction with the tube material (wettability).

The vertical component of the surface tension force acting on the tube walls must balance the weight of the liquid column of height h. Horizontal forces typically cancel out.

The capillary height h can be determined by balancing these forces:

• Surface tension force (vertical component): Fv = γ · 2πR · cosθ
• Weight of liquid column: P = ρ · g · πR2h

Equating... Continue reading "Fundamental Fluid Properties and Transport Phenomena" »