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Chapter 6 Work and Energy What is Energy? Transferred into/from different forms Heat Light Sound Electrical (batteries) Mechanical Chemical ○ Food – Calories! Physics definition: “ability to do work” ○ Doesn’t apply to all forms but will apply to mechanical Introduction Forms of energy: Mechanical ○ focus for now chemical electromagnetic nuclear Energy can be transformed from one form to another Essential to the study of physics, chemistry, biology, geology, astronomy Can be used in place of Newton’s laws to solve certain problems more simply (no vectors!) Work Provides a link between force and energy The work, W, done by a constant force on an object is defined as the product of the component of the force along the direction of displacement and the magnitude of the displacement W (F co s ) x (F cos θ) is the component of the force in the direction of the displacement Δx is the displacement Work Work in 1D (along the line of force) W = Fd Work Done = Force (on object) x Displacement Force acting on a particle moving along x does an amount of Work When multiple forces are present, sum the individual contributions Work is energy transferred to or from a system via a force Work If the force is not aligned with the x-axis, we take the x-component W = (Fcosθ)d ○ F is the magnitude of the force ○ Δ x is the magnitude of the object’s displacement Work This gives no information about the time it took for the displacement to occur the velocity or acceleration of the object Note: work is zero when ○ there is no displacement (holding a bucket) ○ force and displacement are perpendicular to each other, as cos 90° = 0 (if we are carrying the bucket horizontally, gravity does no work) W (F co s ) x (different from everyday “definition” of work) More About Work Scalar quantity Units of Work SI joule (J=N m) CGS erg (erg=dyne cm) US Customary foot-pound (foot-pound=ft lb) If there are multiple forces acting on an object, the total work done is the algebraic sum of the amount of work done by each force More About Work Work can be positive or negative Positive if the force and the displacement are in the same direction Negative if the force and the displacement are in the opposite direction Example 1: lifting a cement block… Work done by the person: is positive when lifting the box is negative when lowering the box Work done by gravity: is negative when lifting the box is positive when lowering the box Example 2: … then moving it horizontally is zero when moving it horizontally T o ta l w o rk :W W 1 W 2 W 3 m g h m g h 0 0 lifting lowering moving total Problem: cleaning the dorm room Eric decided to clean his dorm room with his vacuum cleaner. While doing so, he pulls the canister of the vacuum cleaner with a force of magnitude F=50.0 N at an angle 30.0°. He moves the vacuum cleaner a distance of 3.00 meters. Calculate the work done by all the forces acting on the canister. Problem: cleaning the dorm room Given: angle: force: a=30° F=55.0 N 1. Introduce coordinate frame: Oy: y is directed up Ox: x is directed right Find: Work WF=? Wn=? Wmg=? 2. Note: horizontal displacement only, Work: W=(F cos ) s WF ( F cos )x (50.0 N )(cos 30.0 )(3.00m) 130 N m 130 J Wn ( N cos )x (n)(cos 90.0 )(3.00m) 0 J Wmg (mg cos )x (n)(cos( 90.0 ))(3.00m) 0 J No work as force is perpendicular to the displacement Graphical Representation of Work Split total displacement (xf-xi) into many small displacements x For each small displacement: W i (F cos ) xi Thus, total work is: W to t W i i Fx xi i which is total area under the F(x) curve! Work Example 6.8 A 330-kg piano slides 3.6 m down a 28º incline and is kept from accelerating by a man who is pushing back on it parallel to the incline (Fig. 6–36). The effective coefficient of kinetic friction is 0.40. Calculate: (a) the force exerted by the man, (b) the work done by the man on the piano, (c) the work done by the friction force, (d) the work done by the force of gravity, and (e) the net work done on the piano. To Work or Not to Work Is it possible to do work on an 1) yes object that remains at rest? 2) no To Work or Not to Work Is it possible to do work on an 1) yes object that remains at rest? 2) no Work requires that a force acts over a distance. If an object does not move at all, there is no displacement, and therefore no work done. Kinetic Energy Energy associated with the motion of an object Scalar quantity with the same units as work Work is related to kinetic energy 1 2 KE mv 2 This quantity is called kinetic energy: Work-Kinetic Energy Theorem When work is done by a net force on an object and the only change in the object is its speed, the work done is equal to the change in the object’s kinetic energy W n et KE f KEi KE Speed will increase if work is positive Speed will decrease if work is negative KE and Work-Energy Theorem When work is done by a net force on an object and the only change in the object is its speed, the work done is equal to the change in the object’s kinetic energy W = KEf – KEi = ΔKE Speed will increase if work is positive Speed will decrease if work is negative Work and Kinetic Energy An object’s kinetic energy can also be thought of as the amount of work the moving object could do in coming to rest The moving hammer has kinetic energy and can do work on the nail Potential Energy Potential energy is associated with the position of the object within some system Potential energy is a property of the system, not the object A system is a collection of objects or particles interacting via forces or processes that are internal to the system Units of Potential Energy are the same as those of Work and Kinetic Energy Gravitational Potential Energy Gravitational Potential Energy is the energy associated with the relative position of an object in space near the Earth’s surface Objects interact with the earth through the gravitational force Actually the potential energy of the earthobject system Work and Gravitational Potential Energy Consider block of mass m at initial height yi Work done by the gravitational force W grav F cos y ( mg cos ) y , but : y yi y f , cos 1, Thus : W grav mg yi y f mgyi mgy f . This quantity is called potential energy: PE m gy Note: W g ra vity PEi PE f Important: work is related to the difference in PE’s! Reference Levels for Gravitational Potential Energy A location where the gravitational potential energy is zero must be chosen for each problem The choice is arbitrary since the change in the potential energy is the important quantity Choose a convenient location for the zero reference height ○ often the Earth’s surface ○ may be some other point suggested by the problem Reference Levels for Gravitational Potential Energy A location where the gravitational potential energy is zero must be chosen for each problem The choice is arbitrary since the change in the potential energy gives the work done Wgrav1 mgyi1 mgy f1 , Wgrav2 mgyi2 mgy f 2 , Wgrav3 mgyi3 mgy f 3 . Wgrav1 Wgrav2 Wgrav3 . Potential Energy Stored in a Spring Involves the spring constant (or force constant), k Hooke’s Law gives the force F=-kx ○ F is the restoring force ○ F is in the opposite direction of x ○ k depends on how the spring was formed, the material it is made from, thickness of the wire, etc. Spring Example Spring is slowly stretched from 0 to xmax Fapplied = -Frestoring = kx W = {area under the curve} = ½(kx) x = ½kx² Potential Energy Example 6.31 A 55-kg hiker starts at an elevation of 1600 m and climbs to the top of a 3300m peak. (a) What is the hiker’s change in potential energy? (b) What is the minimum work required of the hiker? (c) Can the actual work done be more than this? Explain why. KE and Work-Energy Theorem There are two general types of forces Conservative ○ Work and energy associated with the force can be recovered Nonconservative ○ The forces are generally dissipative and work done against it cannot easily be recovered Conservative Forces A force is conservative if the work it does on an object moving between two points is independent of the path the objects take between the points The work depends only upon the initial and final positions of the object Any conservative force can have a potential energy function associated with it Note: a force is conservative if the work it does on an object moving through any closed path is zero. Examples of Conservative Forces: Examples of conservative forces include: Gravity Spring force Electromagnetic forces Since work is independent of the path: W c PEi PE f : only initial and final points KE and Work-Energy Theorem Conservative Forces Work done is independent of path 1 Wf Wi 2 Examples of conservative forces include: Gravity Spring force Electromagnetic forces Potential energy is another way of looking at the work done by conservative forces Nonconservative Forces A force is nonconservative if the work it does on an object depends on the path taken by the object between its final and starting points. Examples of nonconservative forces kinetic friction, air drag, propulsive forces KE and Work-Energy Theorem Nonconservative Forces Work done is dependent of path 1 Wi Examples include: 2 of nonconservative forces Kinetic friction Air drag Energy Heat Wf is transformed Example: Friction as a Nonconservative Force The friction force transforms kinetic energy of the object into a type of energy associated with temperature the objects are warmer than they were before the movement Internal Energy is the term used for the energy associated with an object’s temperature Friction Depends on the Path The blue path is shorter than the orange path The work required is less on the blue path than on the red path Friction depends on the path and so is a nonconservative force KE and Work-Energy Theorem To account for conservative and nonconservative forces, the work-energy theorem can be rewritten as Wc + Wnc = ΔKE Conservation of Mechanical Energy Conservation in general To say a physical quantity is conserved is to say that the numerical value of the quantity remains constant In Conservation of Energy, the total mechanical energy remains constant In any isolated system of objects that interact only through conservative forces, the total mechanical energy of the system remains constant. Conservation of Energy Total mechanical energy is the sum of the kinetic and potential energies in the system Ei E f K Ei P Ei K E f P E f Other types of energy can be added to modify this equation Systems and Energy Conservation Recall Wnc + Wc = ΔKE From Work-Energy Theorem Wnc = ΔKE + ΔPE = (KEf – KEi) + (PEf – PEi) = (KEf + PEf) - (KEi + PEi) = Ef – Ei = ΔE Systems and Energy Conservation Principle of Conservation of Energy In an isolated system, energy may be transferred from one type to another, but the total Etot of the system always remains constant. ΔEtot = ΔKE + ΔPE + ΔEint (changes in other forms of energy) = 0 If the system is not isolated then ≠ 0 “Energy cannot be created or destroyed” Systems and Energy Conservation Conservation of Energy Consider a simple pendulum system ○ ΔEtot = ΔKE + ΔPE = 0 At the highest points, all PE ○ Velocity = 0 ○ Δy is maximum At the bottom, all KE ○ Velocity is maximum ○ Δy = 0 V =0 V =Vmax Energy Conversion/Conservation Example 10 m P.E. = 98 J K.E. = 0 J Drop 1 kg ball from 10 m starts out with mgh = (1 kg)(9.8 m/s2)(10 8m P.E. = 73.5 J K.E. = 24.5 J 6m P.E. = 49 J K.E. = 49 J 4m 2m 0m m) = 98 J of gravitational potential energy halfway down (5 m from floor), has given up half its potential energy (49 J) to kinetic energy ○ ½mv2 = 49 J v2 = 98 m2/s2 v 10 m/s at floor (0 m), all potential energy is given P.E. = 24.5 J K.E. = 73.5 J P.E. = 0 J K.E. = 98 J up to kinetic energy ○ ½mv2 = 98 J v2 = 196 m2/s2 v = 14 m/s ConcepTest 6.16 Down the Hill Three balls of equal mass start from rest and roll down different ramps. All ramps have the same height. Which ball has the greater speed at the bottom of its ramp? 4) same speed for all balls 1 2 3 ConcepTest 6.16 Down the Hill Three balls of equal mass start from rest and roll down different ramps. All ramps have the same height. Which ball has the greater speed at the bottom of its ramp? 4) same speed for all balls 1 2 3 All of the balls have the same initial gravitational PE, since they are all at the same height (PE = mgh). Thus, when they get to the bottom, they all have the same final KE, and hence the same speed (KE = 1/2 mv2). Follow-up: Which ball takes longer to get down the ramp? Systems and Energy Conservation The changes occur in all forms of energy ΔK + ΔP + Δ(all other forms of energy) = 0 Newton’s Laws fails in submicroscopic world of atoms and nuclei Law of Conservation of Energy still applies! ○ One of the great unifying principles of science Spring Potential Energy Conservation of Energy with Spring The PE of the spring is added to both sides of the conservation of energy equation, Wnc = 0 (KE + PEg+PEs)i = (KE + PEg + PEs)f The same problem-solving strategies apply Systems and Energy Conservation Nonconservation of Energy Force is not conservative ○ Thermal energy from friction Internal Energy Positive work ○ Energy transferred from environment to system Negative work ○ Energy transferred from system to environment Nonconservative Forces with Energy Considerations When nonconservative forces are present, the total mechanical energy of the system is not constant The work done by all nonconservative forces acting on parts of a system equals the change in the mechanical energy of the system W n o n c o n s e r va t i v e E n erg y Nonconservative Forces and Energy In equation form: W n c K E f K E i ( P E i P E f ) or W nc ( K E f P E f ) ( K Ei P Ei ) The energy can either cross a boundary or the energy is transformed into a form not yet accounted for Friction is an example of a nonconservative force Transferring Energy By Work By applying a force Produces a displacement of the system Transferring Energy Heat The process of transferring heat by collisions between molecules Transferring Energy Mechanical Waves a disturbance propagates through a medium Examples include sound, water, seismic Transferring Energy Electrical transmission transfer by means of electrical current Transferring Energy Electromagnetic radiation any form of electromagnetic waves ○ Light, microwaves, radio waves Systems and Energy Conservation Energy Units Electron volts - An electron volt is the amount of work done on an electron when it moves through a potential difference of one volt ○ Describes energies of atoms and molecules ○ 1 electron volt = 1.60217646 × 10-19 joules Examples Energy for the dissociation of an NaCl molecule into Na+ and Cl- ions: 4.2 eV Ionization energy of atomic hydrogen: 13.6 eV Calories - The energy needed to increase the temperature of 1 g of water by 1 °C ○ Labels on food refer to kilocalories ○ 1 calorie = 4.18400 joules Notes About Conservation of Energy We can neither create nor destroy energy Another way of saying energy is conserved If the total energy of the system does not remain constant, the energy must have crossed the boundary by some mechanism Applies to areas other than physics Work-Energy Theorem Example Waterslides are nearly frictionless, and one such slide named Der Stuka, is 72.0 feet high, found at Six Flags in Dallas, Texas and Wet’n Wild in Orlando, Florida. a) Determine the speed of a 60kg woman at the bottom of such a slide, assuming no friction. b) If the woman is clocked at 18.0 m/s at the bottom of the slide, how much mechanical energy was loss to friction? Systems and Energy Conservation Example: Problem A bead of mass m = 5.00kg is released from point A and slides on the frictionless track shown in the figure. Determine a) the bead‘s speed at B and C and b) the net work done by the force of gravity in moving the bead from A to C. Power Often also interested in the rate at which the energy transfer takes place Power is defined as this rate of energy transfer W P Fv t SI units are Watts (W) ○ J kg m 2 W s s2 Power US Customary units are generally hp Need a conversion factor ○ 1 hp = 550 (ft x lb)/s = 746 W Can define units of work or energy in terms of units of power: ○ 1kWh = (103J/s)(3600s) = 3.60 x 106 J kilowatt hours (kWh) are often used in electric bills This is a unit of energy, not power Power Example: A 1.00x 103 kg elevator carries a maximum load of 8.00 x 102kg. A frictional force of 4.00 x 103 N impedes the motion upward. What minimum power (in kW and hp), must the motor deliver to lift the full elevator at constant speed of 3.00 m/s?