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Chapter 6: Work, Energy and PowerTuesday February 10th Finish Newton’s laws and circular motion Energy Work (definition) Examples of work Work and Kinetic Energy Conservative and non-conservative forces Work and Potential Energy Conservation of Energy As usual – iclicker, examples and demonstrationsReading: up to page 88 in the text book (Ch. 6)
Newton's 2nd law and uniform circular motion Although the speed, v, does notchange, the direction of themotion does, i.e., the velocity,which is a vector, does change. Thus, there is an accelerationassociated with the motion. We call this a centripetalacceleration.Centripetal acceleration:v2ac r(uniform circular motion) A vector that is always directed towards the center of thecircular motion, i.e., it’s direction changes constantly.
Newton's 2nd law and uniform circular motion Although the speed, v, does notchange, the direction of themotion does, i.e., the velocity,which is a vector, does change. Thus, there is an accelerationassociated with the motion. We call this a centripetalacceleration.Centripetal force:v2Fc mac mr2π rPeriod: T (sec)v(uniform circular motion)11 vFrequency: f (sec 1 )T 2π r
Newton's 2nd law and uniform circular motion! ! !!The vectors a, F, v and r are constantly changing The magnitudes a, F, v and r are constants of the motion. The frame in which the mass is moving is not inertial, i.e., itis accelerating. Therefore, one cannot apply Newton's laws in the movingframe associated with the mass. However, we can apply Newton's laws from the stationarylab frame. Examples of centripetal forces: gravity on an orbitingbody; the tension in a string when you swirl a mass inaround in a circle; friction between a car's tires and theracetrack as a racing car makes a tight turn.
So why do you appear weightless in orbit?ma gF mg
So why do you appear weightless in orbit? voma gF mg
So why do you appear weightless in orbit?ma gF mg v (t )
So why do you appear weightless in orbit?m
So why do you appear weightless in orbit?mFc mgYou are in constant free-fall!
Looping the loop2va r N Fg v (t )
v (t )2va r Fg , N
Daytona 500: the racetrack is covered in ice (!), so the physicist cannotrely on friction to prevent him/her from sliding off. How is it that he/she can continue the race? 0Nv
Daytona 500: the racetrack is covered in ice (!), so the physicist cannotrely on friction to prevent him/her from sliding off. How is it that he/she can continue the race? 0Nacv
Energy Energy is a scalar* quantity (a number) that we associatewith a system of objects, e.g., planets orbiting a sun, massesattached to springs, electrons bound to nuclei, etc. Forms of energy: kinetic, chemical, nuclear, thermal,electrostatic, gravitational. It turns out that energy possesses a fundamentalcharacteristic which makes it very useful for solvingproblems in physics: **Energy is ALWAYS conserved**Kinetic energy K is energy associated with the stateof motion of an object. The faster an object moves,the greater its kinetic energy.Potential energy U represents stored energy, e.g., in aspring. It can be released later as kinetic energy.*This can make certain kinds of problem much easier to solve mathematically.
Work - DefinitionWork W is the energy transferred to or from anobject by means of a force acting on the object.Energy transferred to the object is positive work, andenergy transferred from the object is negative work. If you accelerate an object to a greater speed byapplying a force on the object, you increase its kineticenergy K; you performed work on the object. Similarly, if you decelerate an object, you decrease itskinetic energy; in this situation, the object actually didwork on you (equivalent to you doing negative work).
Work - DefinitionWork W is the energy transferred to or from anobject by means of a force acting on the object.Energy transferred to the object is positive work, andenergy transferred from the object is negative work. If an object moves in response to your application of aforce, you have performed work. The further it moves under the influence of your force,the more work you perform. There are only two relevant variables in one dimension:the force, Fx, and the displacement, Δx.
Work - DefinitionWork W is the energy transferred to or from anobject by means of a force acting on the object.Energy transferred to the object is positive work, andenergy transferred from the object is negative work. There are only two relevant variables in one dimension:the force, Fx, and the displacement, Δx.Definition:W Fx Δx[Units: N.m or Joule (J)]Fx is the component of the force in the direction of the object’smotion, and Δx is its displacement. Examples: Pushing furniture across a room; Carrying boxes up to your attic.
Work - ExamplesNFxMgΔxFrictionless surfaceW Fx Δx
Work - ExamplesThese two seemingly similar examples are, in fact, quite differentfkNFxMgΔxRough surfaceWPull Fx ΔxWfric. f k Δx
Work - Examplesv v 2a Δx22What happens next?fixFxΔxFrictionless surface12mv mv m 2ax Δx2f122i12ΔK K f K i max Δx Fx Δx W
Kinetic Energy - DefinitionK mv12Fx2ΔxFrictionless surface12mv mv m 2ax Δx2f122i12ΔK K f K i max Δx Fx Δx W
Work-Kinetic Energy TheoremΔK K f K i Wnet change in the kinetic net work done on energy of a particle the particle K f K i Wnet kinetic energy after kinetic energy the net the net work is done before the net work work done
More on WorkNFθMgΔxFrictionless surfaceW Fx Δx F cosθ Δx
More on WorkTo calculate the work done on an object by a forceduring a displacement, we use only the force componentalong the object's displacement. The force componentperpendicular to the displacement does zero workFx F cos φW Fd cosφ! !W F d Caution: for all the equations we have derived so far, theforce must be constant, and the object must be rigid. I will discuss variable forces later.
The scalar product,ordotproduct! !a b abcos φ(a)(bcos φ ) (acos φ )(b)cos φ cos( φ )! ! ! ! a b b a The scalar product represents the product of themagnitude of one vector and the component of thesecond vector along the direction of the first!!If φ 0o , then a b ab! !oIf φ 90 , then a b 0
The scalar product,ordotproduct! !a b abcos φ(a)(bcos φ ) (acos φ )(b)cos φ cos( φ )! ! ! ! a b b a The scalar product becomes relevant in Chapter 6(pages 88 and 97) when considering work and power. There is also a vector product, or cross product, whichbecomes relevant in Chapter 11 (pages 176-178). I savediscussion of this until later in the semester. See also Appendix A.
c m v2 r (uniform circular motion) . problems in physics: **Energy is ALWAYS conserved** Kinetic energy K is energy associated with the state of motion of an object. The faster an object moves, the greater its kinetic energy. Potential energy U represents stored energy, e.g., in a spring. It can be released later as kinetic energy.
