Work and Energy Math Based

Fundamentals of Work and Energy

• Energy is the capacity that makes things move.
• Work ($W$) is defined as the movement or transfer of energy from one system to another.
• The relationship between work and the change in energy is given by:   $W = \Delta E$
• Work is also defined mechanically as force multiplied by distance:   $W = F d$

Components of Work: Force and Distance

• Force ($F$):
  • A force is a push or a pull exerted on an object.
  • Contact forces occur through direct physical contact (e.g., touch or pulling a rope).
  • Action-at-a-distance forces act across space without direct contact (e.g., gravity).
  • Force is measured in Newtons ($\text{N}$).
  • One Newton is defined in base SI units as:     $1\,\text{N} = 1\,\text{kg}\cdot\text{m/s}^2$
• Distance ($d$):
  • Distance is measured in meters ($\text{m}$).
  • Distance is a scalar quantity, possessing magnitude without direction.
• Derived Units of Work:
  • Multiplying force by distance gives Newton-meters ($\text{N}\cdot\text{m}$):     $W = F d = \text{N}\cdot\text{m}$
  • One Newton-meter is equal to one Joule ($\text{J}$), the standard SI unit of work and energy:     $1\,\text{N}\cdot\text{m} = 1\,\text{J}$

Energy Units and Caloric Conversion

• Nutritional or food calories (Calories, capitalized) differ in scale from standard mechanical Joules.
• The conversion factor between Calories and Joules is defined as:   $1\,\text{Calorie} = 4180\,\text{Joules}$
• Dimensional analysis uses this equivalence as a conversion factor:   $\frac{4180\,\text{J}}{1\,\text{Cal}}$

Worked Calculation Examples

• Example 1: Energy Content of a Krispy Kreme Doughnut

  • Objective: Determine the total number of Joules of energy contained in $1$ Krispy Kreme doughnut rated at $190\,\text{Cal}$.
  • Conversion setup:     $190\,\text{Cal} \times \frac{4180\,\text{J}}{1\,\text{Cal}} = 794200\,\text{J}$
  • Result: One doughnut contains $794200\,\text{J}$ of energy.

• Example 2: Stair Climbing Height to Expend Doughnut Energy

  • Objective: Determine how high Mr. Last ($96\,\text{kg}$) must climb on stairs to completely expend the $794200\,\text{J}$ of energy from $1$ Krispy Kreme doughnut, assuming $100\%$ efficiency.
  • Given parameters:
  • Mass: $m = 96\,\text{kg}$
  • Acceleration due to gravity: $g = 9.81\,\text{m/s}^2$
  • Total energy to expend: $W = 794200\,\text{J}$
  • Efficiency: $100\%$
  • Step 1: Calculate the downward force of gravity (weight) that must be overcome:     $F = m g$$F = 96\,\text{kg} \times 9.81\,\text{m/s}^2 = 941.76\,\text{N}$$F \approx 942\,\text{N}$
  • Step 2: Set up the work formula to solve for vertical displacement ($d$):     $W = F d$$794200\,\text{J} = 942\,\text{N} \times d$
  • Step 3: Divide both sides by the force to isolate $d$:     $\frac{794200\,\text{J}}{942\,\text{N}} = \frac{942\,\text{N} \cdot d}{942\,\text{N}}$$d = 843\,\text{m}$
  • Result: Mr. Last would need to climb a vertical height of $843\,\text{m}$ to burn off the energy of the doughnut.

• Example 3: Dragging an Object Horizontally

  • Objective: Calculate the total work done when dragging a $10\,\text{kg}$ object for $100\,\text{m}$ using an average force of $10\,\text{N}$ pulling to the right.
  • Problem Diagram:     
  • Given parameters:
  • Mass of object: $m = 10\,\text{kg}$
  • Average pulling force: $F = 10\,\text{N}$
  • Distance: $d = 100\,\text{m}$
  • Note on mass: The mass value ($10\,\text{kg}$) is extraneous for this calculation because the horizontal pulling force is already explicitly given.
  • Step 1: Apply the formula for work:     $W = F d$
  • Step 2: Substitute the known values:     $W = 10\,\text{N} \times 100\,\text{m}$$W = 1000\,\text{J}$
  • Result: The total work performed on the object is $1000\,\text{J}$.