Physics · Intermediate · 35 min · Chapter 21 of 28
Energy, heat, and waves
Work and mechanical energy, the ideal gas, heat capacity, and the description of a traveling wave.
Read first: Classical mechanics
The idea
Work is energy transferred by a force acting through a displacement. For a constant force, W = F d cos θ. Kinetic energy ½mv² changes by exactly the net work done, which is the work–energy theorem. Potential energy is bookkeeping for conservative forces: gravity near Earth contributes mgh, and an ideal spring contributes ½kx². If friction does work, mechanical energy decreases by that amount and shows up as thermal energy.
An ideal gas is the model in which molecules do not interact except in elastic collisions, and the particles themselves take up no volume. Its equation of state is PV = nRT. At fixed pressure, volume is proportional to temperature; at fixed volume, pressure is. The kelvin scale is the one that makes this proportionality true, with 0 K the extrapolated point of zero pressure. The average translational kinetic energy of a molecule is (3/2) kT, so temperature is a measure of that microscopic energy, not of “how hot it feels.”
A wave carries energy and a pattern, not the medium. For a periodic wave the speed, frequency, and wavelength are tied by v = fλ. On a string the speed is √(T/μ), tension over linear density, which is why tightening a string raises the pitch. Standing waves on a string fixed at both ends fit an integer number of half-wavelengths into the length. Doubling the frequency at fixed speed halves the wavelength.
Hold onto this
- Work by a constant force is force times displacement times the cosine of the angle between them.
- Mechanical energy is conserved when the only forces doing work are conservative, such as gravity and springs.
- For an ideal gas, PV = nRT, and temperature measures average kinetic energy per molecule.
- A traveling wave has speed equal to frequency times wavelength.
Formulas
Work by a constant force
Only the component of force along the displacement does work.
- θ
- Angle between force and displacement
- d
- Distance moved
Mechanical energy
Kinetic energy, gravitational potential near Earth, and spring potential.
- k
- Spring constant
- h
- Height, measured from a zero you choose
Ideal gas law
Equation of state of an ideal gas. Temperature must be in kelvin.
- n
- Amount of substance in moles
- R
- 8.314 J/(mol·K)
- P, V, T
- Pressure, volume, absolute temperature
Sensible heat
Heat required to change the temperature of a mass m by ΔT, with no phase change.
- c
- Specific heat capacity
Wave speed
Frequency times wavelength. Holds for any periodic wave.
- f
- Frequency in cycles per time
- λ
- Wavelength
Speed on a string
Wave speed set by tension and mass per length.
- T
- Tension
- μ
- Linear mass density
Worked examples
1. How high does it go?
A 0.20 kg ball is thrown upward at 8.0 m/s. How high does it rise, ignoring air resistance?
- Take potential zero at the release point. Initial energy is all kinetic: ½(0.20)(64) = 6.4 J.
- At the top, velocity is 0, so 6.4 = mgh = 0.20 × 9.81 × h.
- h = 6.4 / 1.962 ≈ 3.26 m.
About 3.3 m. Using v² = v₀² − 2gh gives the same number.
2. A gas compressed at constant temperature
Two moles of ideal gas at 300 K occupy 0.050 m³. What is the pressure?
- P = nRT / V = (2)(8.314)(300) / 0.050.
- Numerator: 4988.4. Divide by 0.050: 99,768 Pa.
About 1.00 × 10⁵ Pa, essentially one atmosphere. A useful sanity check.
Try these
A wave on a lake has wavelength 2.0 m and frequency 0.50 Hz. What is its speed?
How much heat raises 0.50 kg of water by 10 K? Take c = 4184 J/(kg·K).
Common mistakes
- The ideal gas law needs absolute temperature. Celsius will give nonsense, including negative pressures.
- Potential energy depends on the zero you chose. Differences of potential energy do not.
- Frequency and angular frequency differ by 2π. In v = fλ, f is in hertz, not radians per second.