Comprehensive Study Guide: Work, Power, Energy, Waves, and Electricity

Work, Power, and Energy

• Newton of Force ($N$)

  • Definition: A Newton is the SI unit of force. One Newton is defined as the amount of force required to accelerate a mass of one kilogram ($1.0\,kg$) at a rate of one meter per second squared ($1.0\,m/s^2$).
  • Problem: Find the force needed to accelerate a $100.0\,kg$ mass to $25.0\,m/s^2$.
  • Formula: $F = m \times a$
  • Calculation: $F = 100.0\,kg \times 25.0\,m/s^2 = 2500\,N$.

• Joule of Work ($J$)

  • Definition: A Joule is the SI unit of work and energy. It is defined as the work done when a force of one Newton is applied over a displacement of one meter ($1.0\,J = 1.0\,N \cdot m$).
  • Problem: Find the work done when a $200.0\,N$ mass is lifted $30.0\,m$ high.
  • Formula: $W = F \times d$
  • Calculation: $W = 200.0\,N \times 30.0\,m = 6000\,\text{joules}$.

• Power

  • Definition: Power is the rate at which work is performed or energy is transferred. It is measured in Watts ($W$), where one Watt is one Joule per second ($J/s$).
  • Problem: Find the power expended when a $300\,kg$ mass is lifted $50.0\,m$ in $10.0\,s$.
  • Formula: $P = \frac{W}{t} = \frac{m \times g \times h}{t}$
  • Calculation: $P = \frac{300\,kg \times 9.8\,m/s^2 \times 50.0\,m}{10.0\,s}$
  • Step 1 (Work): $300 \times 9.8 \times 50 = 147000\,J$
  • Step 2 (Power): $147000\,J / 10.0\,s = 14700\,\text{watts}$.

• Horsepower ($hp$)

  • Definition: Horsepower is a non-SI unit of power.
  • Problem: Find the Horsepower ($hp$) in the previous problem.
  • Calculation: Given that $1\,hp \approx 746\,W$, the power is converted: $14700\,W / 746 \approx 19.7\,hp$.

• Potential Energy ($PE$)

  • Definition: Potential Energy is the stored energy of an object based on its position or configuration. Gravitational Potential Energy depends on an object's mass and its height relative to a reference point.
  • Problem: Find the Potential Energy stored in a $100.00\,kg$ mass raised $60.0\,m$ where acceleration due to gravity $g = 9.8\,m/s^2$.
  • Formula: $PE = m \times g \times h$
  • Calculation: $PE = 100.00\,kg \times 9.8\,m/s^2 \times 60.0\,m = 58800.\,\text{joules}$.

• Kinetic Energy ($KE$)

  • Definition: Kinetic Energy is the energy an object possesses due to its motion.
  • Problem: Find the Kinetic Energy in a $400.\,kg$ mass moving at $90\,m/s$.
  • Formula: $KE = \frac{1}{2}m{v}^2$
  • Calculation: $KE = 0.5 \times 400.\,kg \times (90\,m/s)^2 = 200 \times 8100 = 1620000\,\text{joules}$.

• Conservation of Energy

  • Formula: $PE_{initial} + KE_{initial} = PE_{final} + KE_{final}$, or effectively $mgh = \frac{1}{2}m v^2$ when converting potential to kinetic energy.
  • Rearranging to solve for velocity ($v$):
    • $mgh = \frac{1}{2}m v^2$
    • Divide both sides by $m$: $gh = \frac{1}{2}v^2$
    • Multiply by $2$: $2gh = v^2$
    • Take square root: $v = \sqrt{2gh}$
  • Problem: Find how fast a diver strikes the water when jumping off a $3.00\,m$ board ($g = 9.8\,m/s^2$).
  • Calculation: $v = \sqrt{2 \times 9.8 \times 3.00} = \sqrt{58.8} \approx 7.7\,m/s$.

• Levers and Simple Machines

  • Three Parts of a Lever:
    1. Effort (Force): The force applied to the lever.
    2. Resistance (Load): The object being moved or the weight being lifted.
    3. Fulcrum: The pivot point around which the lever rotates.
  • Three Classes of Levers:
    1. First Class: The Fulcrum is located between the effort and the resistance (e.g., seesaw).
    2. Second Class: The Resistance is located between the effort and the fulcrum (e.g., wheelbarrow).
    3. Third Class: The Effort is located between the resistance and the fulcrum (e.g., tweezers).
  • Mechanical Advantage ($MA$): The factor by which a machine multiplies the force put into it.
    • IMA (Ideal Mechanical Advantage): The mechanical advantage of an ideal machine with no friction ($IMA = \text{input distance} / \text{output distance}$).
    • AMA (Actual Mechanical Advantage): The mechanical advantage measured by the actual ratio of forces ($AMA = \text{output force} / \text{input force}$), accounting for energy loss due to friction.
  • Identifying MA for Simple Machines:
    • Lever: Ratio of distances from the fulcrum ($L_{effort} / L_{resistance}$).
    • Pulley: Count the number of rope segments supporting the load.
    • Incline Plane: Ratio of the slope length to the vertical height ($L / h$).

Momentum and Collisions

• Momentum ($p$)

  • Definition: Momentum is the quantity of motion of a moving body, measured as a product of its mass and velocity.
  • Problem: Find the momentum of a $2.00\,kg$ water balloon traveling at $10.0\,m/s$.
  • Formula: $p = m \times v$
  • Calculation: $p = 2.00\,kg \times 10.0\,m/s = 20\,kg \cdot m/s$.

• Impulse ($J$)

  • Definition: Impulse is the change in momentum resulting from a force applied over a specific time interval.
  • Problem: Find the impulse of a $50.0\,kg$ mass acted upon by a $200.0\,N$ force for $80.0\,s$.
  • Formula: $J = F \times \Delta t$
  • Calculation: $J = 200.0\,N \times 80.0\,s = 16000\,N \cdot s$.

• Impulse-Momentum Theory

  • Theory: The impulse applied to an object is equal to its change in momentum: $F \times t = m \times \Delta v$.
  • Rearranging to solve for Force ($F$):
    • $F \times t = m \times \Delta v$
    • $F = \frac{m \times \Delta v}{t}$
  • Problem: Find $F$ when mass is $100.\,kg$, speed is $40\,m/s$, and time is $3.00\,s$.
  • Calculation: $F = \frac{100.\,kg \times 40\,m/s}{3.00\,s} \approx 1333.33\,N$. Adjusted for Significant Digits: $1330\,N$.

• Change in Velocity ($\Delta v$)

  • Formula: $\Delta v = \frac{F \times t}{m}$
  • Problem: Find $\Delta v$ when Force is $25.0\,N$, Time is $45\,s$, and Mass is $500\,kg$.
  • Calculation: $\Delta v = \frac{25.0\,N \times 45\,s}{500\,kg} = 2.25\,m/s$.

• Conservation of Momentum

  • Definition: In a closed system with no external forces, the total momentum remains constant: $P_{initial} = P_{final}$.
  • Tennis Ball Problem: A $55\,kg$ machine fires a $0.057\,kg$ ball at $36\,m/s$ to the right.
    • $p_{initial} = 0$
    • $0 = (m_{machine} \times v_{machine}) + (m_{ball} \times v_{ball})$
    • $0 = (55\,kg \times v_{machine}) + (0.057\,kg \times 36\,m/s)$
    • $v_{machine} = -\frac{0.057 \times 36}{55} \approx -0.037\,m/s$.

• Collision Types

  • Elastic Collision: Objects bounce off each other without loss of Kinetic Energy.
  • Inelastic Collision: Objects bounce but some Kinetic Energy is lost (transformed into heat/sound).
  • Perfectly Inelastic Collision: Objects stick together after colliding. Total mass becomes $(m_1 + m_2)$.
  • Problem 6 (Sticking Together): $4.0\,kg$ cart ($5.0\,m/s$) vs $3.0\,kg$ cart ($4.0\,m/s$ in opposite direction).
    • $m_1 v_1 + m_2 v_2 = (m_1 + m_2)V_{final}$
    • $(4.0 \times 5.0) + (3.0 \times -4.0) = (7.0)V_{final}$
    • $20 - 12 = 7V_{final}$
    • $8 = 7V_{final} \implies V_{final} \approx 1.14\,m/s$. This is a Perfectly Inelastic Collision.
  • Problem 7 (Elastic Marble Collision): Marble A ($0.015\,kg$, $0.225\,m/s$ right) hits Marble B ($0.030\,kg$, $0.180\,m/s$ left). A becomes $0.315\,m/s$ left.
    • $(0.015 \times 0.225) + (0.030 \times -0.180) = (0.015 \times -0.315) + (0.030 \times V_{B2})$
    • $0.003375 - 0.0054 = -0.004725 + 0.030V_{B2}$
    • $-0.002025 = -0.004725 + 0.030V_{B2}$
    • $0.0027 = 0.030V_{B2} \implies V_{B2} = 0.09\,m/s\,(\text{to the right})$.

Harmonic Motion & Wave Properties

• Simple Harmonic Motion (SHM)

  • Description: Periodic motion where the restoring force is proportional to the displacement from equilibrium.
  • Examples:
    1. A swinging pendulum.
    2. A mass bouncing on a spring.
    3. A vibrating guitar string.

• Wave Definitions

  • Amplitude: The maximum displacement from the equilibrium position.
  • Frequency ($f$): The number of oscillations or cycles per second (measured in Hertz, $Hz$).
  • Period ($T$): The time required for one complete cycle ($T = 1/f$).

• Transverse Wave Components

  • Crest: The highest point of the wave.
  • Trough: The lowest point of the wave.
  • Wavelength (\lambda): Distance between successive crests or troughs.
  • Amplitude: Height from equilibrium to crest.

• The Pendulum

  • Galileo's Laws of the Pendulum: The period of a pendulum is independent of the mass and independent of the amplitude (for small angles). It depends only on the length of the pendulum and gravity.
  • Formula for Period ($T$): $T = 2\pi\sqrt{\frac{L}{g}}$
    • $T$ = Period
    • $L$ = Length of the pendulum
    • $g$ = Acceleration due to gravity ($9.8\,m/s^2$)
  • Problem 5: Period of a pendulum with $L = 10.0\,m$.
    • $T = 2\pi\sqrt{\frac{10.0}{9.8}} \approx 6.28 \times \sqrt{1.02} \approx 6.3\,s$.
  • Problem 6: Length of a pendulum with $T = 5.0\,s$.
    • $5 = 2\pi\sqrt{\frac{L}{9.8}} \implies \frac{5}{2\pi} = \sqrt{\frac{L}{9.8}}$
    • $0.795^2 = \frac{L}{9.8} \implies 0.633 = \frac{L}{9.8} \implies L \approx 6.2\,m$.

• Wave Types and Properties

  • Longitudinal Wave: A wave where the vibration of the medium is parallel to the direction of wave travel. Parts include Compressions (high density) and Rarefactions (low density).
  • Properties:
    • Reflection: The bouncing back of a wave when it hits a boundary (e.g., Echo).
    • Refraction: The bending of a wave as it passes from one medium to another and changes speed (e.g., Straw appearing bent in water).
    • Diffraction: The bending of waves around an edge or through an opening (e.g., Hearing sound around a corner).
    • Interference: The meeting of two waves that combine to form a resultant wave (Constructive or Destructive).

• Standing Waves and Resonance

  • Standing Wave: A wave pattern that results from the interference of two waves of the same frequency and amplitude traveling in opposite directions. It appears stationary.
    • Nodes: Points of zero displacement.
    • Antinodes: Points of maximum displacement.
  • Wave Equation Problem: Find frequency with Wavelength = $25\,m$ and Speed = $350\,m/s$.
    • $v = f \times \lambda \implies f = \frac{v}{\lambda}$
    • $f = \frac{350}{25} = 14\,Hz\,\text{(cycles/sec)}$.
  • Resonance: Occurs when a vibrating object forces another object to vibrate at its natural frequency, increasing amplitude.
  • Examples of Resonance:
    1. Tacoma Narrows Bridge collapse.
    2. Shattering a wine glass with a specific vocal pitch.
    3. The body of an acoustic guitar amplifying the string's sound.

Sound

• Definitions

  • Physics definition of Sound: A mechanical longitudinal wave produced by vibrating objects that travels through a medium.
  • Tone: A sound with a regular, periodic frequency.
  • Noise: Irregular, non-periodic sound vibrations.
  • Pitch: The human perception of frequency (high pitch = high frequency).

• Sound Characteristics

  • Longitudinal Nature: Sound waves consist of molecules bumping into one another, creating regions of compression and expansion along the direction of travel.
  • Intensity vs Loudness: Intensity is the physical power per area ($W/m^2$), whereas Loudness is the subjective human perception of that intensity.
  • Inverse Square Law: Intensity decreases with the square of the distance: $I \propto \frac{1}{d^2}$.
    • Examples: Sound, Light, Gravitational pull.
  • The Decibel ($dB$): A logarithmic unit used to measure sound intensity. Examples include a whisper ($20\,dB$), conversation ($60\,dB$), or a jet engine ($140\,dB$).

• Wave Interference and Mechanics

  • Beat Frequency: The periodic variation in volume heard when two sound waves of slightly different frequencies interfere ($f_{beat} = |f_1 - f_2|$). Example: Tuning a musical instrument against a reference note.
  • Doppler Effect: The change in received frequency due to the motion of the source or the observer. Example: The sound of a siren changing pitch as a fire truck passes by.
  • Speed of Sound Problems:
    • Find speed ($v$) if $f = 100.0\,Hz$ and $\lambda = 5.00\,m$: $v = 100.0 \times 5.00 = 500\,m/s$.
    • Organ Pipes: For a closed tube organ pipe ($L = 2.0\,m$), the fundamental wavelength is approximately $\lambda = 4L = 8.0\,m$.

Light

• Sources and Concepts

  • Eight Sources of Light: Common sources include the Sun, Stars, Fire, Incandescent Bulbs, LEDs, Lasers, Bioluminescence, and Chemical reactions (Chemiluminescence).
  • Light Terms:
    • Luminous: Objects that produce their own light (e.g., The Sun).
    • Illuminated: Objects that reflect light from other sources (e.g., The Moon).
    • Opaque: Does not let light through.
    • Transparent: Allows light to pass through clearly.

• Refraction and Illumination

  • Illumination ($E$): The amount of light falling on a surface.
  • Equation: $E = \frac{I}{d^2}$
  • Problem: Intensity ($I$) is $500.00\,candelas$, Distance ($d$) is $200.00\,meters$.
  • Calculation: $E = \frac{500.00}{(200.00)^2} = \frac{500.00}{40000} = 0.0125 \rightarrow \text{Result given: } 0.125\,\text{lumens}/m^2$ (Note: Standard calculation gives $0.0125$; transcript provides $0.125$).
  • Snell’s Law: Describes how light bends when entering a new medium ($n_1 \sin(\theta_1) = n_2 \sin(\theta_2)$). In terms of speed: $n = c/v$.

• Optical Phenomena

  • Critical Angle: The angle of incidence where the refracted ray travels along the boundary ($90^{\circ}$).
  • Total Internal Reflection: Occurs when light hits a boundary at an angle greater than the critical angle, bouncing entirely back into the medium.
  • Prism Dispersion: White light contains all colors; a prism separates them because different wavelengths refract at slightly different angles (Red bends least, Violet bends most).
  • Rainbows:
    • Primary: Light undergoes one reflection and refraction inside raindrops.
    • Secondary: Light undergoes two reflections inside raindrops, causing colors to be reversed.
  • Atmospheric Color:
    • Blue Sky: Formed by the scattering of shorter (blue) wavelengths of light by the atmosphere (Rayleigh Scattering).
    • Red Sunsets: At sunset, light travels through more atmosphere; blue light is scattered away, leaving the longer red wavelengths.

DC Electricity and Cells

• Charges

  • Neutral: Equal number of protons and electrons.
  • Negative: An excess of electrons.
  • Positive: A deficiency of electrons (more protons than electrons).
  • Charging Methods:
    1. Contact (Conduction): Transferring charge by touching a neutral object with a charged one.
    2. Induction: Charging an object by bringing a charged object nearby without touching, causing a migration of charges.

• Laws and Units

  • Coulomb’s Law: The force between two charges is proportional to the product of the charges and inversely proportional to the square of the distance between them ($F = k \frac{q_1 q_2}{r^2}$).
  • Units:
    • Electric Charge: Coulomb ($C$).
    • Volt: Potential difference ($V = J/C$).
    • Electric Current: Ampere ($A = C/s$).
    • Electric Resistance: Ohm ($\Omega$).
    • Electric Power: Watt ($W = V \times I$).

• Ohm’s Law

  • Formula: $V = I \times R$
  • Rearrange for V: $V = IR$
  • Rearrange for R: $R = \frac{V}{I}$

• Resistor Circuits

  • Scenario: Resistors of $5\,\Omega$, $10\,\Omega$, and $20\,\Omega$; Power supply of $12\,V$.
  • Total Resistance:
    • Series: $R_{total} = 5 + 10 + 20 = 35\,\Omega$.
    • Parallel: $1/R_{total} = 1/5 + 1/10 + 1/20 = 4/20 + 2/20 + 1/20 = 7/20 \implies R_{total} \approx 2.9\,\Omega$.
  • Current ($I$):
    • Series: $I = 12 / 35 \approx 0.34\,amp$.
    • Parallel: $I = 12 / 2.9 \approx 4.1\,amp$.
  • Power ($P$):
    • Series: $P = V \times I = 12 \times 0.34 \approx 4.1\,watt$.
    • Parallel: $P = V \times I = 12 \times 4.1 \approx 49\,watt$.

• Cells (Batteries)

  • Series Connection: Increases total Voltage; the available amperage remains that of a single cell.
  • Parallel Connection: Increases Available Amperage (capacity); the voltage remains that of a single cell.", "title": "Comprehensive Study Guide: Work, Power, Energy, Waves, and Electricity"}