Engineering Mechanics: Derivation of Kinetic Energy, Potential Forces, and Electrical Units

Leibniz Notation for Derivatives: The notation used for the derivative of a variable with respect to time represents the rate of change. * If displacement is defined as $x$ , then the derivative of displacement with respect to time ( $t$ ) yields velocity ( $v$ ): * $v = \frac{dx}{dt}$
Separation of Variables: This mathematical technique involves treating the derivative components as separable algebraic terms. * From the equation $v = \frac{dx}{dt}$ , one can multiply both sides by $dt$ to isolate variables. * Application: $v \times dt = dx$ .

Newton's Second Law Application: In the context of a small distance ( $ds$ ), the resultant force ( $f$ ) is substituted with the consequence of Newton's second law ( $\text{Resultant Force} = m \times a$ ). * The resultant force is expressed as: $m \times \frac{dv}{dt}$ .
Conceptual Treatment of Infinitesimals: The small time increment ( $dt$ ) is treated as a separate numerical entity (variable separation) rather than just a symbolic part of the fraction $\frac{dv}{dt}$ .
Process of Cancellation: * When the resultant force is multiplied by the displacement ( $ds$ ), the variable $dt$ appears in both the numerator and denominator of the combined expression. * By canceling out the $dt$ terms ( $\frac{dv}{dt} \times v \times dt$ ), a new relationship is formed.
Definition of Work and Kinetic Energy: * Small work done over the distance $ds$ by the resultant force is equivalent to the difference between two terms. * Kinetic Energy: Each of these terms in the difference is formally defined as "kinetic energy." * The Work-Energy Principle: Multiplied resultant force times displacement equals the work done, and this work is explicitly equal to the change ( $\text{change}$ ) in kinetic energy.

Rotational Description: Kinetic energy in rotation involves the sum of the velocity of a specific point ( $c$ ) and additional variables described as $bx$ and $dy$ .
Position of the Mass Center: Within the mathematical integral for rotational motion, the position of the mass center is differentiated with respect to time.
Reference Point: In this specific derivation, the position of the mass center ( $r \text{ vector}$ ) is measured with respect to point $c$ .

Condition for a Potential Force: For a force to be classified as potential, it must depend strictly and exclusively on the location of the body.
Gravity in a Uniform Field: Gravity ( $m \times g$ ) is considered a potential force because it behaves as a constant function of location.
Work Calculation in Gravity: * When moving from point $a$ to point $b$ , and then from point $b$ to point $b$ (completing a vertical or varied path), work is only calculated for the vertical displacement. * The formula for vertical work done is $m \times g \times h$ .
Horizontal Displacement and Zero Work: * When moving horizontally, the angle between the displacement and the weight (force of gravity) is exactly $90^{\text{o}}$ . * Since $\text{cos}(90^{\text{o}}) = 0$ , the gravity force does not perform any work during horizontal movement.

Energy Loss and Stopping: Calculations are possible when the amount of energy lost (the work done by forces that waste energy) is known.
The Volt (V): Defined as the amount of energy lost or gained by each unit of charge in a device. * Example (Battery): A $1.5 \text{ volt}$ battery indicates that each Coulomb ( $C$ ) of charge receives $1.5 \text{ joules}$ ( $J$ ) of energy. * Example (Light): Measuring $1.5 \text{ volts}$ across a light indicates it extracts $1.5 \text{ joules}$ from each Coulomb of charge flowing through it.
Unit Derivations: * Ampere ( $A$ ): The unit of charge over time. * Charge Conversion: Multiplying Amperes by time yields Charge. * Ampere-hours ( $Ah$ ): This is a unit of charge equal to Amperes multiplied by hours. * Mathematical Conversion (16 Ah): * $\text{Charge} = 16 \text{ hours} \times 3600 \text{ seconds}$ . * Since $\text{Amperes} \times \text{Seconds} = \text{Coulombs}$ , the final unit of the calculation is in Coulombs ( $C$ ).

The Skater Dialogue: There was a discussion regarding a "skater" (likely referencing a specific person or image) and various "deal breakers" discussed in a social context. * Social/Dating Deal Breakers Listed: * "Weird fingers." * Putting a phone face down on a table. * Eating before getting one's own plate. * Not offering a jacket on a date. * Talking too much about personal income/money. * Not leaving a tip. * Interaction: An individual was told a specific image "looks a bit like you," to which they responded, "That’s not a compliment."