University Physics (AP Physics C Mechanics): Kinetic .
University Physics (AP Physics C Mechanics): Kinetic Energy andWorkWritten and Compiled by the ProcrastiNotetakers TeamLast Updated: December 28, 2020Contents1Kinetic Energy and Work in 1D31.1Energy and Energy Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31.2Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41.3Examples of Specific Forces and Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51.3.1Gravitational Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51.3.2Spring Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5Work done by a general force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51.42 Kinetic Energy and Work in 2 and 3D62.1Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62.2Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63 The Work-Kinetic Energy Theorem63.1Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63.2Significance and Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74 Power75 Conservative and Non-Conservative Forces*85.1Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .85.2Connections to Multivariable Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8A Credits9
AP Physics CA.1Kinetic Energy and WorkContributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9A.2 External Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9A.3 Image Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9B Extra Resources/Further Readings92
AP Physics C11.1Kinetic Energy and WorkKinetic Energy and Work in 1DEnergy and Energy TypesWhat is energy? It’s a super broad term that we use all the time, so it’s hard to put a definition on it, but we cansettle for a looser definition for now. Energy can be viewed as a special number assigned to a system that can beused to predict outcomes of experiments and construct machines. For example, if I push on a system, I exert aforce on the system and causes this scalar energy quantity to change.Common Types of EnergyCommon types of energy include:1. Kinetic Energy – The energy of motion2. Chemical Energy – Energy stored in chemical bonds3. Electrical Energy – Energy from electricity4. Mechanical Energy – Energy harnessed to do work For example, we would burn gasoline in our cars that releases chemical energy, which turns into mechanical then kinetic energy to move the car. (If you have an electric car, it would be electrical energy instead ofchemical energy)There’s a special property of energy called the principle of conservation of energy which states that energy is always conserved. This can be stated formally like the following:Conservation of EnergyWhen a system and surroundings undergo a transition from any initial state to any final state, the changein energy is zero: E Esystem Esurr 0(1) When the surroundings don’t undergo an energy change, we call the system a closed system.Kinetic Energy is going to be very important for our study in mechanics, so we’ll take a closer look at that.Kinetic EnergyThe kinetic energy KE of a non-rotating body of mass m moving with speed v is defined as the scalarquantity:KE 111mv 2 m( v · v ) m((vx )2 (vy )2 (vz )2 )222 The SI units for kinetic energy is the Joule: J kg·m2s23(2)
AP Physics CKinetic Energy and Work1.2 WorkLet’s define what we mean by the "work" done by a force. Hint: it’s not the common definition of the word.WorkThe work is energy transferred to or from an object by means of a force acting on the object. By convention, work to the object is denoted as positive work, while work from the object is denoted as negativework. This would mean that the work is essentially transferred energy.To intuitively grasp how to calculate work, we can ask two main question to help guide us to what work mightcontain:1. How strong is the force applied? This matters because a stronger force leads to a higher energy output.2. How "long" or how far is the force applied? This matters because the longer or farther a force is applied, thelarger the energy output. To visualize this, let’s look at an example. Suppose I want to exert a force and do work on a weight sled in aworkout. Well, the harder I push on the sled (the force is the push), the more energy I expend and thus themore work I do. Similarly, the longer I push the sled, the more energy I expend.Calculation of NotesThe work can be calculated as the force component multiplied by the displacement, or:W F · d F d cos(φ)Where φ is the angle between the force vector and the displacement vector. If we have multiple forces, we have two main methods of calculating the net work done on an object:1. Find the individual work for each force and sum them.2. Find the net force and substitute the net force into the equation for work.4(3)
AP Physics CKinetic Energy and Work1.3Examples of Specific Forces and Work1.3.1Gravitational ForceGravitational ForceThe work Wg done by the gravitational force is:Wg mgd cos(φ)where we substituted the force to be mass times the gravitational constant (we’ve seen this before) intothe standard expression for a work done by a constant force.1.3.2 Spring ForceThe spring force can be described by Hooke’s Law, which states that:Hooke’s LawF kxî1.4(4)Work done by a general forceAll of our analysis of work have been assuming the force F is a constant force, meaning that it does not changedue to time or position. However, it is much more useful to consider variable forces, as they are much more applicable in the real world. Consider Figure 1:Figure 1: Graph of a variable forceWe can view the work done in two ways:1. We want to find the force times the distance, which is area under the curve. Hmm.sounds like a job for theintegral!2. We want to find the net work by summing all the individual work segments done at each displacement x.In essence, we want to find the limit of the sum as we take shorter and shorter pieces. Hmm.limit? Sum ofareas? Smaller and smaller widths? Sounds like a job for the integral!5
AP Physics CKinetic Energy and WorkWork done by a variable forceThe work done by a variable force can be calculated as:ˆ xfF (x)dxW (5)xi2 Kinetic Energy and Work in 2 and 3D2.1Kinetic EnergyDid any of our work in 1D depend on the dimensions at all? Not really! The formula for kinetic energy still holdsin 2 and 3 dimensions. Kinetic energy is a scalar as well, so there is no need to separate it into components, because it has none!2.2 WorkTo calculate work in three dimensions, we would need the help of some multivariable calculus to generalize anexpression for work. In AP Physics C (which doesn’t use multivariable calculus), we can express a very limiteddefinition:Work in Three DimensionsSuppose we have a force F Fx î Fy ĵ Fz k̂ in which Fx only depends on x and not y or z, Fy dependsonly on y, and Fz only depends on z. The reason we do this is to ensure that x, y, z are independent ofeach other. The work done can then be calculated by:ˆ rfˆ xfˆW dW fx dx ri33.1xiˆyfyizfFy dy Fz dz(6)ziThe Work-Kinetic Energy TheoremStatementWork-Kinetic EnergyThe work-kinetic energy theorem says that the work done is equal to the change in kinetic energy:W KE(7) Wow! This theorem ties together everything we’ve discussed so far, but why must this formula be true?What does it tell us? These are the questions we will be investigating in the next section.6
AP Physics CKinetic Energy and Work3.2 Significance and ProofLet’s first look at why this theorem must be true. Consider the following: We know that work is the definite integral of the force, so:ˆ xfW F (x)dx and applying Newton’s 2nd Law givesxiˆˆxfW xfmadx ximxidvdxdt After using Newton’s second law to rewrite this expression, we have a slight issue: we have two terms dxand dt that need to be the same variable in order for us to integrate. Fortunately for us, we can use thechain rule to expressdvdt dvdx dxdt dvdx v Substituting this new expression back into our integrals yields:ˆ vfˆ vf11W mvdv mvdv and integrating gives usmvf 2 mvi 2 (KE)f (KE)i K22viviOkay, so now we’re convinced that this theorem is true, but why is this theorem important anyway? It ties kinetic energy with this new concept of work. Perhaps the most significant consequence of this result is that know we can visualize work in relation to kinetic energy. If the change in kinetic energy is positive, indicating that the final kinetic energy is higher than the initial kinetic energy, the amount of workdone must be positive.3.3ExamplesWorked Examples: View here!4PowerPowerThe rate at which work is done by a force is what we call the power due to the force. If we let W denotethe work and t denote the time, then:The Average Power:Pavg W t(8)and the Instantaneous Power:P dWdt(9) The unit for power is the watt (W), which is defined to be 1J/s. (Yes, I know we have so many things as ofright now denoted with W such as weight, work, and now the watt. It will usually require context to determine which concept W denotes)7
AP Physics CKinetic Energy and WorkPower as a Scalar ProductSimilar to how we can express work as a scalar product, we can also express power as a scalar product:P 55.1dWF cos(φ)dx F cos(φ)v F · vdtdt(10)Conservative and Non-Conservative Forces*StatementsBefore we investigate conservative and non-conservative forces, let’s review what the work is for a non-constantforce and an arbitrary path.Work as a Line IntegralˆF · d rW path(11)We can call a force conservative if the work does not depend on the path. In other words, if I take a random curvypath or go straight from point A to point B, the work stays the same.For a deeper explanation (optional), view this link.5.2 Connections to Multivariable CalculusConservative ForcesThe expression for work when the force is conservative is expressed as a closed line integral, or: F c · d r 0closed path(12)This has numerous connections in multivariable calculus, where an entire chapter of a course is usually designated to evaluate line integrals in a plane, and another chapter along with some of the fundamental theorems inmultivariable calculus, namely Stokes’ Theorem and Green’s Theorem, is dedicated to line integrals. To view amultivariable calculus course, click this link.8
AP Physics CKinetic Energy and WorkAppendicesACreditsA.1Contributions Drafted by Austin WangA.2 External SourcesMaterial from these notes are remixed and adapted from the following:1. Halliday and Resnick – Fundamentals of Physics Link2. MIT OCW Scholar - 8.01 Classical MechanicsMIT is not affiliated with us, and we are not affiliated with MIT. MIT OCW materials fall under a Creative Commons License; these notes remix and adapt the material, falling under fair usage. To see original terms and conditions, view this website.A.3Image Credits Clock : Veronica Cruz Figures 1: MIT OCW Scholar– To find original images, visit here.BExtra Resources/Further Readings All resources mentioned in the appendix retain the same licensing as the original creators intended. TheProcrastiNote Team has checked every single one of these to make sure that our usage remains within theirrights of usage. If you are the owner of these materials and you believe there is a mistake in our citations,please contact us at official.procrastinote@gmail.com. Thanks!– OpenStax Physics– Halliday and Resnick – Fundamentals of Physics– AP Physics C: Mech - College Board Playlist– 8.01 Classical Mechanics – MIT OCWScholar9
Dec 28, 2020 · Kinetic Energy is going to be very important for our study in mechanics, so we’ll take a closer look at that. Kinetic Energy The kinetic energy KE of a non-rotating body of mass mmoving with speed vis defined as the scalar quantity: KE 1 2 mv2 1 2 m( v· v) 1 2 m((v x)2 (v y)2 (v z)2) (2) Th
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