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Summer 2012 PHYS 172: Modern Mechanics Lecture 12 – Internal Energy 7.1 – 7.9

TODAY Potential Energy of a spring Energy of a Spring/Mass system Path-Independence of Potential Energy Temperature and Energy Heat Capacity Power Open and Closed Systems Volume and Temperature

Potential energy of a spring x Potential energy of a spring: Assume U s = 0 for relaxed spring y

Real Life Physics: Potential energy of a spring

Interatomic potential energy between two atoms Spring-mass model: Problem: Energy becomes infinite at large distances! The Morse potential energy function is a realistic approximation Spring approximation:

Energy of an oscillating spring-mass system Neglect friction: Maximum speed: Speed at any point: x y x max -x max

Energy of an oscillating spring-mass system y x x max -x max Same as

HOME STUDY: Example, “a rebounding block” k s = 2000 N/m System: block, spring, Earth Assume: no interaction with surroundings 1. What is the lowest point reached by the block? L0=L0= One equation, one unknown 2. What is the highest point reached by the block? One equation, one unknown See details in the book (6.3), p. 230

Path independence of potential energy Change in potential energy does not depend on the path taken A B In a round trip the potential energy does not change At which point is the gravitational potential energy the largest? A B C

Clicker question 12 An object of mass m was pushed over a hill of height h from point 1 to point 2 as shown. The trajectory path length from point 1 to 2 is s. How much work was done by the gravitational force? h A.0 J B.mgh C.2mgh D.mgh/2 E.mgs s

Temperature and energy: the connection Heat capacity: amount of energy required to heat up a unit of mass by a unit of temperature

Specific heat capacity Specific heat capacity C: amount of energy required to heat up a unit of mass by a unit of temperature Usually expressed per gram: J/(g. K)

14 Demo: fire syringe Convert mechanical (kinetic) energy into heat

Example: heat capacity Niagara Falls – what is the rise in water temperature due to gravity if there were no losses of energy? C water = J/(K. g) h = 50 m Solution:

Thermal energy 1° K ~ J/molecule Thermal energy transfer: Sum over every atom!

Thermal transfer of energy A process in which energy moves between a system and surroundings with which it is in contact, due to a temperature difference Heat transferred to the system (microscopic work) Work done on the system Reserve the word heat specifically to denote thermal energy transferred from one object to another Flow of electromagnetic radiation can also change the energy Sign of Q: “+” Transfer into system “-” Transfer out of system

Example: heat Water bucket T wi = 100 o C = 373K m =10 kg C w = 4.2 J/(K. g) A perfectly insulated house has a volume of 500 m 3 (~1500 ft 2 house) and air temperature 0 0 C. You bring in a closed bucket of water (10 L) of temperature C. What will the temperature be in the house after equilibration? Air T ai = 0 o C = 273K V =500 m 3 C a = 1 J/(K. g) ρ air = 1.23 kg/m 3 Solution: Ignored: walls, yourself, heat capacity dependence on T

Power (P) Energy per unit time Power associated with work: Unit: Watt, 1 W = 1 J/s Clicker: If a car engine runs at constant power, which would take longer time – to accelerate from 0 to 10 m/s, or from 20 to 30 m/s? (~ mph) (~ mph) A)Need more time to accelerate from 0 to 10 m/s B)Need more time to accelerate from 20 to 30 m/s C)The same

Open and closed system Closed system: there is no energy flow between the system and surroundings Open system: there is energy flow between the system and surroundings SYSTEM

Volume and temperature When energy rises the average distance between atoms in general must also increase leading to an increase in volume. The spring law does not explain: - volume changes with temperature! - breaking

Interatomic Morse potential energy between two atoms The Morse potential energy function is a realistic approximation Spring approximation: When energy rises the average distance between atoms in general increases leading to increase in volume.

Thermometers based on expansion mercuryBi-metallicGalilean (buoyancy)

The choice of system and energy system Ball falls distance h toward Earth Work! Analyze this statement: “The Earth exerts a force mg through a distance h and does work mgh, and there is also a decrease in the potential energy –mgh, so the kinetic energy increases by 2mgh”. Avoid double counting! WRONG! HOME READING: 7.9

WHAT WE DID TODAY Path-Independence of Potential Energy Temperature and Energy Heat Capacity Power Open and Closed Systems Volume and Temperature