Work, Kinetic Energy, and Thermodynamics: PHYS 111 Study Notes

Work and Kinetic Energy

Review of Vectors and Motion

Understanding the motion of particles requires knowledge of vectors such as acceleration, velocity, and displacement. Forces are also vectors, and Newton's Laws allow us to determine acceleration at any time by resolving force components.

• Acceleration, velocity, and displacement are all vector quantities.

• Forces are vectors and can be decomposed into components.

• Vector equations are sometimes converted to scalar equations for simplicity.

Work-Kinetic Energy Theorem

The work done by all forces on a mass is equal to the change in its kinetic energy. This theorem provides a scalar approach to analyzing energy changes in a system.

• Work ($W$): The energy transferred by a force acting over a displacement.

• Kinetic Energy ($KE$): The energy of motion, $KE = \frac{1}{2}mv^2$.

• Work-Kinetic Energy Theorem: $W_{net} = \Delta KE$

Definition of Work

Work is defined as the product of force and displacement when the force is parallel to the displacement.

• $W = F \cdot d$

• Unit of work: Newton-meter (N·m), also called a Joule (J).

Work is Energy

Work is a means of transferring energy between systems or changing the energy of a system. Energy can be transformed between different forms such as gravitational, spring, chemical, kinetic, and rotational.

• To do anything requires work.

• Forces are applied to change the energy of a system.

• Work enables energy transfer between forms.

Activity	Equivalent work (J)
Annual U.S. energy use	$10^{19}$
Mt. St. Helens eruption	$10^{17}$
Burning one gallon of gas	$10^8$
Human food intake/day	$10^7$
Melting an ice cube	$10^3$
Lighting a 100-W bulb for 1 minute	6000
Heartbeat	1
Turning page of a book	$10^{-2}$
Hop of a flea	$10^{-3}$
Breaking a bond in DNA	$10^{-19}$

Forces at an Angle

When a force is applied at an angle to the displacement, only the component of the force parallel to the displacement does work.

• $W = F d \cos \theta$

• If $\theta = 90^\circ$, $W = 0$ (no work is done).

• Normal and centripetal forces typically do no work.

The Angle Matters

The sign and magnitude of work depend on the angle between force and displacement vectors.

• If $\theta < 90^\circ$, $W > 0$ (work is positive).

• If $\theta = 90^\circ$, $W = 0$ (no work).

• If $\theta > 90^\circ$, $W < 0$ (work is negative).

The Sign of Work

The sign of work affects the speed of an object:

• If $W > 0$, the object's speed increases.

• If $W < 0$, the object's speed decreases.

Example: Work on an Incline

Consider a vehicle moving down an incline. The work done by each force (weight, normal, friction) is calculated and summed. The angle used for the weight is the angle between the force of gravity and the displacement along the incline.

• Work by weight: $W = Mg \sin \theta \cdot d$

• Change in height: $\Delta h = -d \sin \theta$

• Work by gravity: $W = -Mg \Delta h = -Mg(h_1 - h_2)$

Kinematic Equations and Work

The total work can be related to kinematic equations and kinetic energy:

• $v_f^2 = v_i^2 + 2a(x - x_0)$

• $\frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2 = ma(x - x_0)$

• $W_{net} = \Delta KE$

Conservative Forces and Energy Conservation

For conservative forces, the work done depends only on the initial and final positions, not the path taken. This allows the work-kinetic energy theorem to be rewritten as conservation of energy.

• Conservative forces: gravity, springs, electricity.

• Work done by conservative forces: $W_{conservative} = \Delta PE$

Thermodynamics

Work and Energy in Thermodynamics

Work and energy concepts apply to gases and fluids. The work-kinetic energy theorem becomes the Bernoulli equation in fluids, and the area under a pressure-volume graph represents work done by a gas.

• Bernoulli equation governs fluid flow and flight.

• Buoyancy is the weight of the displaced fluid.

Introduction to Heat and Temperature

Heat and temperature are fundamental concepts in thermodynamics. Heat is energy transfer due to temperature difference, while temperature measures the average kinetic energy of particles.

• Heat: Thermal energy and radiation.

• Temperature: Related to average kinetic energy of atoms/molecules.

Temperature Scales: Fahrenheit, Celsius, Kelvin

Different temperature scales are used in physics. Kelvin is preferred for scientific calculations, especially with gases.

• Water freezes at $0^\circ$C and boils at $100^\circ$C.

• Water freezes at $32^\circ$F and boils at $212^\circ$F.

• Conversion formulas:

  • $T_F = \left(\frac{9}{5}\right)T_C + 32$

  • $T_C = \left(\frac{5}{9}\right)(T_F - 32)$

  • $T = T_C + 273.15$ (Kelvin)

Thermal Expansion

Materials expand when heated. The coefficient of linear and volume expansion quantifies this effect.

• Linear Expansion: $\Delta L = \alpha L_0 \Delta T$

• Volume Expansion: $\Delta V = \beta V_0 \Delta T$

• SI unit for $\alpha$: $K^{-1}$; for $\beta$: $K^{-1}$

Substance	Coefficient of Linear Expansion ($\alpha$)
Ether	1.13 × 10-3
Carbon tetrachloride	1.18 × 10-3
Alcohol	1.12 × 10-3
Gasoline	0.96 × 10-3
Olive oil	0.68 × 10-3
Water	0.21 × 10-3
Mercury	0.18 × 10-3

Heat Transfer by Conduction

Heat flows from hot to cold regions by conduction. The rate of heat transfer depends on the material's thermal conductivity.

• Heat Flow by Conduction: $Q = kA \left(\frac{\Delta T}{L}\right)t$

• Materials with low thermal conductivity are good insulators.

Substance	Thermal Conductivity (W/m·K)
Silver	417
Copper	391
Gold	291
Aluminum	237
Steel, low carbon	54
Ice	2.2
Glass	0.8
Water	0.6
Air	0.023

Latent Heats and Specific Heat

Latent heat is the energy required for phase changes, while specific heat is the energy needed to change temperature.

• Latent Heat: $Q = mL$

• Specific Heat: $Q = mC\Delta T$

• Used to predict time to boil or freeze substances.

Stefan-Boltzmann Law

This law describes the power radiated by a blackbody as a function of its temperature.

• $P = \epsilon \sigma A T^4$

• Important in climate science and energy transfer.

Laws of Thermodynamics

The laws of thermodynamics govern energy and entropy in physical systems.

• 0th Law: If two systems are each in thermal equilibrium with a third, they are in equilibrium with each other.

• 1st Law: Conservation of energy applies.

• 2nd Law: Entropy of the universe always increases; heat flows from hot to cold.

• 3rd Law: Entropy approaches a constant as temperature approaches zero.

Thermodynamics of Gases

Gases are central to heat engines and refrigerators, which transfer heat and do work.

• Heat engines: Take in heat, do work, and emit heat.

• Refrigerators: Extract heat from one place and emit it elsewhere.

Boyle's Law

For ideal gases at constant temperature, pressure is inversely proportional to volume.

• $P \propto \frac{1}{V}$ (at constant $T$)

• Paths on a P-V diagram are called isotherms.

Charles' Law and Guy-Lussac's Law

These laws describe the relationship between temperature, volume, and pressure for gases.

• Charles' Law: At constant pressure, $V \propto T$

• Guy-Lussac's Law: At constant volume, $P \propto T$

Ideal Gas Law

The ideal gas law combines the relationships between pressure, volume, temperature, and number of particles.

• $PV = N k T$

• $PV = nRT$

Microscopic View of Gases

Pressure results from collisions of gas molecules with container walls. The Maxwell-Boltzmann distribution describes the spread of molecular speeds.

• $P = \frac{F}{A} = \frac{m(v_x^2/L)}{L^2} = \frac{mv_x^2}{L^3} = \frac{mv_x^2}{V}$

• $P = \frac{m(v_x^2)_{av}}{V}$

• $P = \frac{1}{3}\left(\frac{N}{V}\right)m(v^2)_{av}$

• Kinetic energy and temperature: $\left(\frac{1}{2}mv^2\right)_{av} = K_{av} = \frac{3}{2}kT$

Heat Engines and Internal Energy

Heat engines use the internal energy of gases to do work. The internal energy of a monatomic ideal gas is given by:

• $U = \frac{3}{2}N k T$

• SI unit: Joule (J)