Chapter 6: Work & Energy Lecture Notes Phy 201: General Physics I Chapter 6: Work & Energy Lecture Notes

What is Energy? Energy is a scalar quantity associated with the state of an object (or system of objects). Energy is a calculated value that appears in nature and whose total quantity in a system always remains constant and accounted for. Energy is said to be “Conserved” Loosely speaking, energy represents the “fuel” necessary for changes to occur in the universe and is often referred to as the capacity to perform work. We can think of energy as the “currency” associated with the “transactions” (forces!) that occur in nature. In mechanical systems, energy is “spent” as force transactions are conducted. Alternatively, the exertion of force requires an “expenditure” of energy. The SI units for energy are called Joules (J) In honor of James Prescott Joule

Work Work describes the energy transferred to/from an object by the exertion of a force Work is essentially a measure of useful physical output Definition: standard form: or component form: Notes: SI Units for Work: N.m Unit comparison: f Work = Effort x Outcome

Examples of Work (constant vs. variable force) Two forces (a constant force and a non-constant force) move an object 1.5 m: The area under the left graph is: Fconstant.Ds = 5N x 1.5m = 7.5 N.m = W (work) The area under the right graph is: Faverage.Ds = 5N x 1.5m = 7.5 N.m = W (work) F s AREA 5 N Ds = 1.5 m Constant force F s AREA Ds = 1.5 m 5 N Non-Constant force Consider f to be 0o for this example. Note that even for a variable force, the average force exerted over Ds is used to determine work.

Kinetic Energy The energy associated with an object’s state of motion Kinetic energy is a scalar quantity that is never negative in value Definition: Key Notes: The kinetic energy for the x, y, and z components are additive Kinetic energy is relative to the motion of observer’s reference frame, since speed and velocity are as well. An object’s kinetic energy depends more on its speed than its mass Any change in an object’s speed will affect a change in its kinetic energy Unit comparison:

The Work-Energy Theorem! Work & Kinetic Energy The net work performed on an object is related to the net force: When , a change in state of motion & kinetic energy is implied: Derivation: The Work-Energy Theorem!

Work Performed by Gravitational Force For a falling body (no air drag): Gravitational force only performs work in the vertical direction: Wg is + when Dy is – Wg is - when Dy is + What about gravitational force on an incline?

Work Performed during Lifting & Lowering Consider Joey “blasting” his pecs with a bench press workout (assume vlift = constant). Given: Applying Newton’s 2nd Law: The Work performed:

Energy & the Body Our bodies utilize energy to: Sustain cellular processes & maintain body temperature Overcome joint friction Produce motion In nutrition, energy is typically described in terms of Calories (i.e. the “calories” labeled on a cereal box): These “nutritional” calories are actually kilocalories: 1 Calorie = 1 kcal = 1000 cal The scientific calorie is related to the SI unit of energy: 1 cal = 4.186 J Question(s): The “average” person requires ~2000 kcal per day. How many joules are in 2000 kcal? How high could you climb with this amount of energy? It turns out the body uses only ~10% of its consumed energy for physical activity. How high could you actually climb after consuming 2000 kcal?

Power Power is a measure of work effectiveness Power is the time rate of energy transfer (work) due to an exerted force: Average Power: Average Net Power: SI units: The Watt (1 W = 1 J/s) Note: Power is also related to Force & Velocity:

Power (cont.) The same work output can be performed at various power rates. Example 1: Consider 100 J of work output accomplished over 2 different time intervals: 100 J over 1 s: 100 J over 100 s: Example 2a: An 900 kg automobile accelerates from 0 to 30 m/s in 5.8 s. What is the average net power? Example 2b: At the 30 m/s, how much force does the road exert on this vehicle? Use the same power as 2a.

Conservative vs. Nonconservative Forces When the configuration of a system is altered, a force performs work (W1). Reversing the configuration of the system results in the force performing work (W2). The force is conservative if: W1= -W2 A force that performs work independent of the path taken. A force in which the net work it performs around a closed path is always zero or: Wnet = 0 J {for closed path} Examples: Gravitational force (Fg), Elastic force (Fspring), and Electric force (FE) Nonconservative Forces: The work performed by the force depends on the path taken When the configuration of a system is altered then reversed, the net work performed by the force is not zero: Wnet ≠ 0 J {for closed path} Work performed results in energy transformed to thermal energy Air Drag (FDrag) and Kinetic friction (fkinetic)

Sliding on an Incline Example: No friction (conservative force) A 1 kg object (vo=5 m/s) travels up a 30o incline and back down. 1. The Wnet performed by Fg (up): 2. The Wnet performed by Fg (down): 3. The total Wnet performed: Example: With Kinetic Friction (nonconservative force) A 1 kg object (vo=5 m/s) travels up a 30o incline and back down against a 1.7 N kinetic friction force. Note: Block will not travel up as far as previous example. The Wup performed by fk (up): The Wdown performed by fk (down): The total Wnet performed:

Defining or Identifying a System A system is a defined object (or group of objects) that are considered distinct from the rest of its environment For a defined system: all forces associated strictly with objects within the defined system are deemed internal forces Internal forces do not transfer energy into/out of the system when performing work within the system Example: The attractive forces that hold the atoms of a ball together. These forces are ignored when applying Newton’s 2nd Law to the ball. all forces exerted from outside the defined system are deemed external forces External forces transfer energy into/out of the system when performing work on a system Example: The gravitational force that performs work on a falling object (the system) increases the ball system’s (kinetic) energy. Note: When the ball and the earth are together defined as the system, the work performed by the gravitational force on the ball does NOT transfer energy into the system. The appropriate of a system determines when a force is considered internal or external & can go a long way toward simplifying the analysis of a physics problem The total energy associated with a defined system:

Work Done on a System by External Forces For a defined system, external forces are forces that are not defined within the system yet perform work upon the system External forces transfer energy into or out of a system: Conservative external forces alter the U and K (a.k.a. the mechanical energy) of a system: Nonconservative external forces may alter the mechanical energy (U, K) as well as the non-mechanical energy (Einternal and/or Ethermal ) of a system:

Conservation of Energy In general, the total energy associated with a system of objects represents the complete state of the system: Work represents the transfer of energy into/out of a system: For an isolated system, the total energy within a system remains a constant value: or, for any 2 moments: Considering only the mechanical energy of the system: Conservation of Mechanical Energy!

Deeper Thoughts on Cons. of Energy Physicists have identified by experiment 3 fundamental conservation laws associated with isolated systems: Conservation of Energy Conservation of Mass Conservation of Electric Charge Treated as accepted “facts”, these laws have allowed for experimental predictions that would not have been foreseen otherwise: Conservation of Energy led to the discovery of the neutrino during neutron decay within the atomic nucleus Conservation of Mass is fundamental in the prediction of new substance formed during chemical processes Conservation of Electric Charge predicts the formation of neutrons do to the collision of protons with electrons, a process called Electron Capture. Considered as accepted “facts”, these laws have allowed for experimental predictions that would not have been foreseen otherwise.

Feynman on Energy "There is a fact, or if you wish, a law, governing natural phenomena that are known to date. There is no known exception to this law - it is exact so far we know. The law is called conservation of energy [it states that there is a certain quantity, which we call energy that does not change in manifold changes which nature undergoes]. That is a most abstract idea, Richard Feynman (1918-1988) because it is a mathematical principle; it says that there is a numerical quantity, which does not change when something happens. It is not a description of a mechanism, or anything concrete; it is just a strange fact that we can calculate some number, and when we finish watching nature go through her tricks and calculate the number again, it is the same...”

James Prescott Joule (1818-1889) English inventor & scientist Interested in the efficiency of electric motors Described the heat dissipated across a resistor in electrical circuits (now known as Joules’ Law) Demonstrated that heat is produced by the motion of atoms and/or molecules Credited with establishing the mechanical energy equivalent of heat Participated in establishing the “Law of Energy Conservation” "It is evident that an acquaintance with natural laws means no less than an acquaintance with the mind of God therein expressed."