Comprehensive Lecture Notes on General Physics 1: Mechanics, Work-Energy, Gravitation, and Elasticity

General Course Information and Mechanics Curriculum

• Course Title: General Physics 1 (3 Units)
• Course Code: PHS 101 (PART A) / Physics
• Instructor: Professor Akinlami Johnson O.
• Platform: SONRAL LEARNING MANAGEMENT SYSTEM (LMS)
• Core Discipline: Mechanics
• Curriculum Module Outline:
  • Units and Dimension
  • Rectilinear Motion
  • Newton's Laws
  • Friction
  • Work, Energy and Conservation of Energy
  • Gravitation
  • Elasticity, Young Modulus and Bulk Modulus

Principles of Work

• Fundamental Definition of Work:
  • Work is said to be done only when the point of application of the force moves through a distance.
  • Basic scalar formulation:     $\text{Work done} = \text{force} \times \text{distance}$
• Work with Force at an Angle:
  • For a body acted upon by a force inclined at an angle $\theta$ to the displacement, the work done is given by:     $W = F s \cos(\theta)$
• Vector Formulation of Work:
  • Force vector in three-dimensional Cartesian coordinates:     $\mathbf{F} = F_x \mathbf{i} + F_y \mathbf{j} + F_z \mathbf{k}$
  • Displacement vector in three-dimensional Cartesian coordinates:     $\mathbf{s} = \Delta x \mathbf{i} + \Delta y \mathbf{j} + \Delta z \mathbf{k}$
  • Total work done represented by the scalar (dot) product of force and displacement:     $W = \mathbf{F} \cdot \mathbf{s}$$W = F_x \Delta x + F_y \Delta y + F_z \Delta z$

Gravitational Work and Integral Formulations

• Work Done by Gravity:
  • Gravitational work near the surface of the Earth:     $W = mgh$
  • Elevation change between two vertical coordinates $h_1$ and $h_2$:     $W = mg(h_2 - h_1)$
  • Path Independence: The work done by the force of gravity depends only on the initial and final vertical coordinates, not on the path taken.
• Integral Form of Work Done:
  • When a force $F_x$ acts along a continuous path from an initial position $x_A$ (point $A$) to a final position $x_B$ (point $B$), the total work done is the line integral:     $W_{A \rightarrow B} = \int_{x_A}^{x_B} F_x\,dx$

Kinetic Energy and the Work-Energy Relation

• Derivation of the Kinetic Energy Expression:
  • Applying Newton's second law along the direction of displacement:     $W = F \Delta x$$W = m a \Delta x$
  • For a body undergoing uniform acceleration, the acceleration $a$ is constant, satisfying the kinematic equation:     $v_2^2 = v_1^2 + 2 a (\Delta x)$
  • Rearranging the kinematic equation to solve for the product $a \Delta x$:     $a \Delta x = \frac{1}{2} (v_2^2 - v_1^2)$
  • Substituting this relation directly into the work expression:     $W = \frac{1}{2} m (v_2^2 - v_1^2)$$W = \frac{1}{2} m v_2^2 - \frac{1}{2} m v_1^2$
• Definition of Kinetic Energy:
  • The physical quantity $\frac{1}{2} m v^2$ is called the Kinetic energy.

Potential Energy and Power

• Potential Energy:
  • Gravitational potential energy relationship:     $W = mgh = \mathbf{F} \cdot \mathbf{x}$
  • Concept: Potential energy is the energy associated with the relative positions of two or more interacting particles.
• Power Definitions:
  • Power is defined as the rate at which work is done.
  • Average Power ($P_{av}$):     $P_{av} = \frac{\Delta W}{\Delta t}$
  • Instantaneous Power ($P_{inst}$):     $P_{inst} = \frac{dW}{dt}$
  • Infinitesimal work element $dW$ expressed in terms of force and displacement:     $dW = \mathbf{F} \cdot d\mathbf{s}$
  • Instantaneous power expressed in terms of velocity:     $P = \mathbf{F} \cdot \frac{d\mathbf{s}}{dt}$
• Alternative Definitions of Power:
  • Power can also be defined as the rate of energy transfer from one body to another.
  • Power can be defined as the rate at which energy is transferred from one form to another:     $P = \frac{dE}{dt}$

Water Pump Power Calculation

• Example Problem Statement:
  • A pump raises water from a well of depth $20\,\text{m}$ at a rate of $10\,\text{kg}\,\text{s}^{-1}$ and discharges it at $6\,\text{m}\,\text{s}^{-1}$. What is the power of the motor?
• Formulation and Mathematical Solution:
  • Total work performed in raising and discharging the mass of water:     $\text{Total Work} = mgh + \frac{1}{2} m v^2$
  • Determining the motor power by differentiating total work with respect to time:     $P = \frac{d}{dt}\left(mgh + \frac{1}{2} m v^2\right)$$P = \left(\frac{dm}{dt}\right)(gh) + \frac{1}{2}\left(\frac{dm}{dt}\right) v^2$
  • Given parameters:
  • Mass flow rate: $\frac{dm}{dt} = 10\,\text{kg}\,\text{s}^{-1}$
  • Well depth: $h = 20\,\text{m}$
  • Acceleration due to gravity: $g = 10\,\text{m}\,\text{s}^{-2}$
  • Discharge speed: $v = 6\,\text{m}\,\text{s}^{-1}$
  • Intermediate calculations:     $gh = (10\,\text{m}\,\text{s}^{-2})(20\,\text{m}) = 200\,\text{m}^2\,\text{s}^{-2}$$\frac{1}{2} v^2 = \frac{1}{2}(6\,\text{m}\,\text{s}^{-1})^2 = \frac{1}{2}(36\,\text{m}^2\,\text{s}^{-2}) = 18\,\text{m}^2\,\text{s}^{-2}$
  • Numerical substitution and final value:     $P = 10(200 + 18)$$P = 10(218) = 2180\,\text{W}$

Gravitation and Gravitational Field Strength

• Newton's Law of Gravitation:
  • Statement: The force of attraction between two masses $m_1$ and $m_2$ separated by a distance $r$ is proportional to the product of their masses and inversely proportional to the square of the distance of separation between them.
  • Mathematical expression:     $F = -\frac{G m_1 m_2}{r^2}$
• Gravitational Field Strength or Intensity ($g$):
  • Definition: Gravitational field strength is the gravitational force per unit mass:     $F = mg$$g = \frac{F}{m}$
  • Substituting Newton's gravitational force into the field equation:     $g = -\frac{G M m}{m r^2}$$g = -\frac{G M}{r^2}$
  • Surface of the Earth: The magnitude of the field strength at the surface of the earth is:     $g(R_E) = \frac{G M_E}{R_E^2}$     where $M_E$ is the mass of the Earth and $R_E$ is the radius of the Earth.

Kepler's Laws of Planetary Motion

• First Law:
  • States that the planets move around the sun in elliptical orbits with the sun at one focus.
• Second Law:
  • States that the line joining the sun to a planet sweeps out equal areas in equal times.
• Third Law:
  • States that the square of the period of a planet is proportional to the cube of the semi-major axis of its orbit:     $T^2 \propto a^3$

Elasticity and Elastic Moduli

• Mechanics of Elastic Deformation:
  • A force applied to an object can change its shape.
  • In general, the response of a material to a given type of deforming force is characterized by an elastic modulus.
  • General definition of an elastic modulus:     $\text{Elastic modulus} = \frac{\text{stress}}{\text{strain}}$$\text{Elastic modulus} = \frac{\text{stress}_{\text{tensile}}}{\text{strain}_{\text{tensile}}}$
• Fundamental Quantities:
  • Tensile Stress:
  • Defined as a force per unit area.
  • Measured in units of $\text{N}\,\text{m}^{-2}$.
  • Tensile Strain:
  • Defined as a fractional change in a dimension or volume.
  • Being a ratio of identical physical quantities, strain is dimensionless.
• Three Primary Elastic Moduli:
  • Young's modulus (governs changes in length in solids).
  • Shear modulus (governs shear distortion and tangential force response in solids).
  • Bulk modulus (governs uniform volume changes in solids and fluids).

Young's Modulus and Longitudinal Strain

• Physical Meaning:
  • Young's modulus is the measure of the resistance of a solid to a change in its length when a force is applied perpendicular to a face.
• Theoretical Setup:
  • Consider a solid rod with an unstressed length $l_0$ and cross-sectional area $A$.
  • The rod changes in length by $\Delta l$ when subjected to equal and opposite forces $F$ directed along its axis and perpendicular to the end faces.
  • These axial tensile forces tend to stretch the rod.
• Mathematical Expressions:
  • Tensile stress on the rod:     $\text{Tensile stress} = \frac{F}{A}$
  • Tensile strain on the rod:     $\text{Tensile strain} = \frac{\Delta l}{l_0}$
  • Young's Modulus definition:     $Y = \frac{\text{Tensile stress}}{\text{Tensile strain}}$$Y = \frac{\frac{F}{A}}{\frac{\Delta l}{l_0}} = \frac{F l_0}{A \Delta l}$

Shear Modulus and Tangential Stress

• Physical Meaning:
  • The shear modulus of a solid indicates its resistance to a shearing force, which is a force applied tangentially to a surface.
• Shear Stress:
  • Defined as the ratio of tangential deforming force to the surface area over which it is applied:     $\text{Shear stress} = \frac{\text{Tangential Force } F}{\text{Area } A} = \frac{F}{A}$
• Shear Strain:
  • Defined as the lateral displacement divided by the transverse height:     $\text{Shear strain} = \frac{\Delta x}{h}$     where $\Delta x$ is the horizontal displacement of the sheared surface and $h$ is the vertical separation between the top and bottom surfaces.
• Shear Modulus ($S$):
  • Defined as the ratio of shear stress to shear strain:     $S = \frac{\text{shear stress}}{\text{shear strain}}$$S = \frac{\frac{F}{A}}{\frac{\Delta x}{h}} = \frac{F h}{A \Delta x}$

Bulk Modulus and Compressibility

• Physical Meaning:
  • The bulk modulus of a solid or a fluid indicates its resistance to a change in volume.
• Model System:
  • Consider a cube of a material (solid or fluid) whose faces each experience the same uniform force $F$ applied normally across each face of area $A$.
• Pressure:
  • The pressure on the cube is the normal force per unit area:     $P = \frac{F}{A}$
• Response to Pressure Changes:
  • When the pressure on a body is increased, its volume decreases.
  • Volume Stress: The change in pressure $\Delta P$ applied to the body is called the volume stress.
  • Volume Strain: The resulting fractional change in volume $\frac{\Delta V}{V}$ is called the volume strain.
• Bulk Modulus ($B$) Expression:
  • The bulk modulus of the material is defined as:     $\text{Bulk Modulus} = \frac{\text{Volume Stress}}{\text{Volume Strain}}$$B = -\frac{\Delta P}{\frac{\Delta V}{V}}$
• Compressibility ($K$):
  • Compressibility is the inverse of the bulk modulus:     $K = \frac{1}{B}$