Work and Energy Flashcards

Fundamental Concepts of Work and Energy

• Introduction to Utility: Work and energy are critical concepts for quantifying the ability of objects to perform useful tasks, such as moving a piston, turning a wheel, or spinning a turbine for electricity generation.

• Definitions: These concepts have intertwined, circular definitions:

  • Energy: The capability of an object to do useful work.

  • Work: The transfer of energy by applying a force to an object over a distance.

• Applications: Understanding these principles allows for the analysis of simple machines (pulleys, levers), projectiles, planetary motion, and roller coasters.

• Quantity Type: Both work and energy are scalar quantities, meaning they possess magnitude (strength) but no associated direction.

• Units of Measurement:

  • The primary unit for both is the Joule ($J$).

  • A Joule is defined as a Newton times a meter ($N \times m$).

  • Base unit breakdown: $1\,J = 1\,kg \cdot m^2 / s^2$.

  • Note: While a Joule is a Newton-meter, the unit "Newton-meter" is typically reserved for torque (discussed in Lecture 8) to avoid confusion.

Forms of Energy

• General Categories:

  • Potential Energy: Stored energy based on the arrangement or configuration of a system.

  • Kinetic Energy: Energy resulting from motion.

• Specific Potential Energy Types:

  • Electric Potential Energy: Derived from the distance between charged particles (detailed in Lecture 22).

  • Chemical Energy: Based on the configuration of atoms or molecules; examples include gasoline and the energy humans obtain from food via digestion.

  • Nuclear Energy: Energy stored in the nucleus of an atom, released through radioactive decay, fission, or fusion (Lectures 33 and 34).

  • Elastic Potential Energy: Stored in deformable objects, such as a spring.

  • Gravitational Potential Energy (GPE): Stored energy due to the gravitational attraction between Earth and other masses.

• Oscillation Example: A mass bouncing on a spring demonstrates the continuous conversion between elastic potential energy, kinetic energy, and gravitational potential energy.

Gravitational Potential Energy ($GPE$)

• Variables and Calculation: $GPE$ is directly proportional to mass, height, and gravitational acceleration.

• Standard Equation:   $GPE = m \times g \times h$

  • $m$: Mass of the object in kilograms ($kg$).

  • $g$: Gravitational acceleration in meters per second-squared ($m/s^2$).

  • $h$: Height above the reference level (e.g., ground or sea level) in meters ($m$).

• Observations:

  • Increasing mass or height increases $GPE$.

  • $GPE$ varies by location; an object has more $GPE$ on Jupiter than on Mars due to differences in gravitational acceleration.

Kinetic Energy ($KE$)

• Definition: Energy objects possess by virtue of their motion.

• Equation:   $KE = \frac{1}{2} \times m \times v^2$

  • $m$: Mass.

  • $v$: Velocity.

• Calculational Warnings: When squaring velocity, parentheses should always be used. Velocity squared is always positive; therefore, negative kinetic energy is impossible.

Thermal Energy and Energy Conversion

• Heat (Thermal Energy): Energy that moves between objects of different temperatures. While heat can do work, it is often considered "wasted energy" (e.g., heat generated by friction).

• Conservation of Energy: Energy cannot be created or destroyed. The total energy in the universe, or any closed system, remains constant.

• Conversion Examples:

  • Steel Ball Collision: Kinetic energy is converted into sound (clacking) and heat (burning a hole in paper).

  • Rocket Launch: Chemical energy converts to heat, sound, $KE$, and $GPE$.

  • Bed of Nails Demo: Dr. Fazzini (College of DuPage) demonstrated energy transfer by breaking a cinder block. $GPE$ of a sledgehammer becomes $KE$, which then rearranges molecules in the block and produces heat and sound. Pressure (weight distribution over many nails) prevents injury.

Mechanical Energy and Conservation

• Definition: Mechanical energy is the sum of $GPE$ and $KE$. In the absence of friction and air drag, mechanical energy is conserved.

• Track Demonstration: Balls on tracks with different paths but identical starting and ending heights will have the same final velocity and landing distance, though their transit times differ based on speed durations.

• Roller Coaster Example: A cart starts with maximum $GPE$ at the highest hill. This is converted to maximum $KE$ (speed) at the lowest point. Without external energy (drive chains), the cart cannot ascend a hill higher than its starting point.

• Ballistic Cart Experiment:

  • Launch mass: $9.7\,g$.

  • Initial velocity: $3.6\,m/s$.

  • Initial $KE$ (at height $0\,m$): $0.5 \times 9.7\,g \times (3.6\,m/s)^2 = 62.86\,g \cdot m^2/s^2$.

  • Measured height: $0.66\,m$.

  • Theoretical calculation: $h = \frac{62.86}{9.7 \times 9.8} = 0.66\,m$.

• Dissipative Forces: A pendulum experiment (starting at a person's nose) shows that air drag and friction reduce mechanical energy over time, preventing the pendulum from returning to its original height.

Physics Definition of Work

• Definition: Energy transferred to an object by exerting a force over a distance ($d$).

• Equation:   $W = F_{\parallel} \times d$

  • $F_{\parallel}$: The component of force that is parallel to the displacement.

• Parallel Force Component: In cases like a wheeled suitcase, force is applied to the handle at an angle. Only the horizontal component of that force does work. The perpendicular component supports weight/remains upright but does zero work.

• Positive vs. Negative Work:

  • Positive: Force and displacement are in the same direction.

  • Negative: Force and displacement are in opposite directions (e.g., friction).

  • Example: If a traveller pulls a suitcase with $100\,N$ of parallel force over $2\,m$ ($200\,J$ positive work) but friction exerts $60\,N$ over the same distance ($120\,J$ negative work), the net work is $80\,J$.

• Zero Work: Walking with a bowling ball results in zero work on the ball because the upward force is perpendicular to the horizontal displacement.

Work-Energy Theorem

• Theorem: Work is equal to the change in kinetic energy ($\Delta KE$).   $W = KE_{final} - KE_{initial}$

• Experimental Proof (Cart on Ramp):

  • Cart mass: $0.349\,kg$.

  • Initial velocity: $0.621\,m/s$ (at distance $0.295\,m$).

  • Final velocity: $1.002\,m/s$ (at distance $0.667\,m$).

  • Calculation: $\Delta KE = 0.108\,J$.

  • Calculated work by gravity ($F_{\parallel} = 0.298\,N$ over $d = 0.372\,m$): $0.111\,J$.

  • Minor discrepancies are due to dissipative forces (heat/sound).

Power

• Definition: The rate at which work is performed.

• Unit: Watts ($W$), where $1\,W = 1\,J/s$.

• Equation:   $P = \frac{W}{t}$

• Relative Comparison (Horsepower is a non-SI unit used to relate motors to horses):

  • Golf Cart: $500\,kg$ mass, $0$ to $9\,m/s$ in $60\,s$.

    • $W = KE = \frac{1}{2} \times 500 \times 9^2 = 20,250\,J$.

    • $P = 337.5\,W$.

  • Sports Car: $1600\,kg$ mass, $0$ to $27\,m/s$ in $3\,s$.

    • $W = KE = \frac{1}{2} \times 1600 \times 27^2 = 583,200\,J$.

    • $P = 194,400\,W$.

Simple Machines

• Function: Used to change the direction, magnitude, or both, of an applied force.

• Types: Levers, wheel and axles, pulleys, inclined planes, wedges, and screws.

• Mechanical Advantage ($MA$): The ratio of force multiplication.   $MA = \frac{\text{Output Force}}{\text{Input Force}}$

• Trade-off: Force and distance are inversely proportional (Conservation of Work). What is gained in force is lost in distance.

• Efficiency ($e$): Quantifies how well work input is converted to work output.   $e = \frac{\text{Work Output}}{\text{Work Input}} \times 100$

• Machine Specifics:

  • Levers: Lifting a $1\,kg$ mass ($9.8\,N$) with $2.6\,N$ input force resulted in $MA = 3.8$. Due to bending, efficiency was $83.8\%$.

  • Pulleys: $MA$ is related to the number of load strings. Single fixed pulleys only change direction, not force magnitude.

  • Inclined Planes: Reduce force required to move height; $MA$ relates to steepness.

  • Wedge: Transmits force from blunt to pointy edge (e.g., axes, scissors).

  • Screws: Convert rotational motion into linear motion.