Conservative Forces, Potential Energy And Conservation Of Energy
CONSERVATIVE FORCES, POTENTIAL ENERGYAND CONSERVATION OF ENERGYToday’s Objectives:Students will be able to:1. Use the concept ofconservative forces anddetermine the potentialenergy of such forces.2. Apply the principle ofconservation of energy.Dynamics, Fourteenth EditionR.C. HibbelerCopyright 2016 by Pearson Education, Inc.All rights reserved.
READING QUIZ1. The potential energy of a spring isA) always negative.B) always positive.C) positive or negative.D) equal to ks.2. When the potential energy of a conservative systemincreases, the kinetic energyA) always decreases.B) always increases.C) could decrease orincrease.D) does not change.Dynamics, Fourteenth EditionR.C. HibbelerCopyright 2016 by Pearson Education, Inc.All rights reserved.
APPLICATIONSThe weight of the sacks resting onthis platform causes potential energyto be stored in the supporting springs.As each sack is removed, the platformwill rise slightly since some of thepotential energy within the springswill be transformed into an increasein gravitational potential energy of theremaining sacks.If the sacks weigh 100 lb and the equivalent spring constantis k 500 lb/ft, what is the energy stored in the springs?Dynamics, Fourteenth EditionR.C. HibbelerCopyright 2016 by Pearson Education, Inc.All rights reserved.
APPLICATIONS (continued)The young woman pulls thewater balloon launcher back,stretching each of the fourelastic cords.If we know the unstretched lengthand stiffness of each cord, can weestimate the maximum height andthe maximum range of the waterballoon when it is released from thecurrent position? Would we needto know any other information?Dynamics, Fourteenth EditionR.C. HibbelerCopyright 2016 by Pearson Education, Inc.All rights reserved.
APPLICATIONS (continued)The roller coaster is released from rest at the top of the hill A.As the coaster moves down the hill, potential energy istransformed into kinetic energy.What is the velocity of the coaster when it is at B and C?Also, how can we determine the minimum height of hill A sothat the car travels around both inside loops without leavingthe track?Dynamics, Fourteenth EditionR.C. HibbelerCopyright 2016 by Pearson Education, Inc.All rights reserved.
CONSERVATIVE FORCE (Section 14.5)A force F is said to be conservative if the work done isindependent of the path followed by the force acting on a particleas it moves from A to B. This also means that the work done bythe force F in a closed path (i.e., from A to B and then back to A)is zero.z F Fdr0·B Thus, we say the work is conserved.The work done by a conservativeforce depends only on the positionsof the particle, and is independent ofits velocity or acceleration.Dynamics, Fourteenth EditionR.C. HibbelerAxCopyright 2016 by Pearson Education, Inc.All rights reserved.y
CONSERVATIVE FORCE (continued)A more rigorous definition of a conservative force makesuse of a potential function (V) and partial differentialcalculus, as explained in the text. However, even withoutthe use of the these more complex mathematicalrelationships, much can be understood and accomplished.The “conservative” potential energy of a particle/system istypically written using the potential function V. There are twomajor components to V commonly encountered in mechanicalsystems, the potential energy from gravity and the potentialenergy from springs or other elastic elements.Vtotal Vgravity VspringsDynamics, Fourteenth EditionR.C. HibbelerCopyright 2016 by Pearson Education, Inc.All rights reserved.
POTENTIAL ENERGYPotential energy is a measure of the amount of work aconservative force will do when a body changes position.In general, for any conservative force system, we can definethe potential function (V) as a function of position. The workdone by conservative forces as the particle moves equals thechange in the value of the potential function (e.g., the sum ofVgravity and Vsprings).It is important to become familiar with the two types ofpotential energy and how to calculate their magnitudes.Dynamics, Fourteenth EditionR.C. HibbelerCopyright 2016 by Pearson Education, Inc.All rights reserved.
POTENTIAL ENERGY DUE TO GRAVITYThe potential function (formula) for a gravitational force, e.g.,weight (W mg), is the force multiplied by its elevation from adatum. The datum can be defined at any convenient location.Vg W yVg is positive if y is above thedatum and negative if y isbelow the datum. Remember,YOU get to set the datum.Dynamics, Fourteenth EditionR.C. HibbelerCopyright 2016 by Pearson Education, Inc.All rights reserved.
ELASTIC POTENTIAL ENERGYRecall that the force of an elastic spring is F ks. It isimportant to realize that the potential energy of a spring, whileit looks similar, is a different formula.Ve (where ‘e’ denotes anelastic spring) has the distance“s” raised to a power (theresult of an integration) or1 2 Veks2Notice that the potentialfunction Ve always yieldspositive energy.Dynamics, Fourteenth EditionR.C. HibbelerCopyright 2016 by Pearson Education, Inc.All rights reserved.
CONSERVATION OF ENERGY (Section 14.6)When a particle is acted upon by a system of conservativeforces, the work done by these forces is conserved and thesum of kinetic energy and potential energy remainsconstant. In other words, as the particle moves, kineticenergy is converted to potential energy and vice versa.This principle is called the principle of conservation ofenergy and is expressed asT1 V1 T2 V2 ConstantT1 stands for the kinetic energy at state 1 and V1 is thepotential energy function for state 1. T2 and V2represent these energy states at state 2. Recall, thekinetic energy is defined as T ½ mv2.Dynamics, Fourteenth EditionR.C. HibbelerCopyright 2016 by Pearson Education, Inc.All rights reserved.
EXAMPLEGiven: The 4 kg collar, C, has avelocity of 2 m/s at A.The spring constant is 400N/m. The unstretched lengthof the spring is 0.2 m.Find:The velocity of the collar atB.Plan: Apply the conservation of energy equation between A andB. Set the gravitational potential energy datum at point Aor point B (in this example, choose point A—why?).Dynamics, Fourteenth EditionR.C. HibbelerCopyright 2016 by Pearson Education, Inc.All rights reserved.
EXAMPLE (continued)Solution:.0.3 mDatum0.5 m.Note that the potential energy at B has twoparts.VB (VB)e (VB)gVB 0.5 (400) (0.5 – 0.2)2 – 4 (9.81) 0.4The kinetic energy at B isTB 0.5 (4) vB2Similarly, the potential and kinetic energies at A will beVA 0.5 (400) (0.1 – 0.2)2, TA 0.5 (4) 22The energy conservation equation becomes TA VA TB VB.[ 0.5(400) (0.5 – 0.2)2 – 4(9.81)0.4 ] 0.5 (4) vB2 [0.5 (400) (0.1 – 0.2)2 ] 0.5 (4) 22 vB 1.96 m/sDynamics, Fourteenth EditionR.C. HibbelerCopyright 2016 by Pearson Education, Inc.All rights reserved.
CONCEPT QUIZ1. If the work done by a conservative force on a particle as itmoves between two positions is –10 ft·lb, the change in itspotential energy isA) 0 ft·lb.B) -10 ft·lb.C) 10 ft·lb.D) None of the above.2. Recall that the work of a spring is U1-2 -½ k(s22 – s12) andcan be either positive or negative. The potential energy of aspring is V ½ ks2. Its value isA) always negative.C) always positive.Dynamics, Fourteenth EditionR.C. HibbelerB) either positive or negative.D) an imaginary number!Copyright 2016 by Pearson Education, Inc.All rights reserved.
GROUP PROBLEM SOLVING IGiven: The 800 kg rollercoaster car isreleased from restat A.Find: The minimum height, h, of Point A so that the car travelsaround inside loop at B without leaving the track. Also find thevelocity of the car at C for this height, h, of A.Plan: Note that only kinetic energy and potential energy dueto gravity are involved. Determine the velocity at B using theequation of motion and then apply the conservation of energyequation to find minimum height h .Dynamics, Fourteenth EditionR.C. HibbelerCopyright 2016 by Pearson Education, Inc.All rights reserved.
GROUP PROBLEM SOLVING I (continued)DatumSolution:1) Placing the datum at A:TA VA TB VB 0.5 (800) 02 0 0.5 (800) (vB)2 800(9.81) (h 20)(1)2) Find the required velocity of the coaster at B so it doesn’tleave the track.Equation of motion applied at B:NB 02v Fn man m r(vB)2 800 (9.81) 8007.5manmg vB 8.578 m/sDynamics, Fourteenth EditionR.C. HibbelerCopyright 2016 by Pearson Education, Inc.All rights reserved.
GROUP PROBLEM SOLVING I (continued)DatumNow using the energyconservation, eq. (1), theminimum h can be determined.0.5 (800) 02 0 0.5 (800) (8.578)2 800(9.81) (h 20) h 23.75 m3) Find the velocity at C applying the energy conservation.TA VA TC VC 0.5 (800) 02 0 0.5 (800) (vC)2 800(9.81) (23.75) VC 21.6 m/sDynamics, Fourteenth EditionR.C. HibbelerCopyright 2016 by Pearson Education, Inc.All rights reserved.
GROUP PROBLEM SOLVING IIGiven: The arm is pulled back such thats 100 mm and released.When s 0, the spring isunstretched.Assume all surfaces of contact tobe smooth. Neglect the mass ofthe spring and the size of the ball.Find: The speed of the 0.3-kg ball and the normal reaction of thecircular track on the ball when 60 .Plan: Determine the velocity at 60 using the conservationof energy equation and then apply the equation of motionto find the normal reaction on the ball.Dynamics, Fourteenth EditionR.C. HibbelerCopyright 2016 by Pearson Education, Inc.All rights reserved.
GROUP PROBLEM SOLVING II (continued)Solution:1) Placing the datum at A:TA VA TB VBwhereTA 0.5 (0.3) 02VA 0 0.5 (1500) 0.12TB 0.5 (0.3) 02VB 0.3 (9.81) 1.5 (1 cos 60 )60Datum A BThe conservation of energy equation is0 0.5 (1500) 0.12 0.5 (0.3) (vB)2 0.3 (9.81) 1.5 (1 cos 60 )vB 5.94 m/sDynamics, Fourteenth EditionR.C. HibbelerCopyright 2016 by Pearson Education, Inc.All rights reserved.
GROUP PROBLEM SOLVING II (continued)2) Find the normal reaction on the ball when 60 .Free-body diagramKinetic diagramnnWmat60 60 manttN Equation of motion applied at 60 :2v Fn man m Br5.942N 0.3 (9.81) cos 60 0.31.5N 8.53 NDynamics, Fourteenth EditionR.C. HibbelerCopyright 2016 by Pearson Education, Inc.All rights reserved.
ATTENTION QUIZ1. The principle of conservation of energy is usually toapply than the principle of work & energy.A) harderB) easierC) the same amount of workD) It is a mystery!2. If the pendulum is released from thehorizontal position, the velocity of itsbob in the vertical position isA) 3.8 m/s.B) 6.9 m/s.C) 14.7 m/s.D) 21 m/s.Dynamics, Fourteenth EditionR.C. HibbelerCopyright 2016 by Pearson Education, Inc.All rights reserved.
Dynamics, Fourteenth EditionR.C. HibbelerCopyright 2016 by Pearson Education, Inc.All rights reserved.
This principle is called the principle of conservation of energy and is expressed as T 1 V 1 T 2 V 2 Constant T 1 stands for the kinetic energy at state 1 and V 1 is the potential energy function for state 1. T 2 and V 2 represent these energy states at state 2. Recall, the kinetic energy is defined as T ½ mv2. CONSERVATION OF ENERGY .
Kinetic Energy (K) Potential Energy U: conservative forces (work independent of path) If only conservative forces present in system, conservation of mechanical energy: z ) constant Examples of conservative forces: Springs: elastic potential energy
Work-Kinetic energy theorem potential energy. Overview conservative and nonconservative forces conservative forces and potential energy . this section, we sharpen this distinction and explore some examples of conserva-tive and nonconservative forces. Perhaps the simplest c
KINETIC & POTENTIAL ENERGY 5.1 Conservative forces Examples 5.2 Potential with turning points 5.2.1 Oscillation about stable equilibrium 5.2.2 Bounded and unbounded potentials 2. 5.1 Conservative forces Wab Z b a F:dr U(a) U(b) For a conservative field of force, the work done depends only
Potential energy can be defined for conservative forces only Examples: gravitational potential energy spring elastic potential energy 6 Connection between energy and force - hint Left side – the kinetic energy has been
Chapter 6: Work, Energy and Power Tuesday February 10th Reading: up to page 88 in the text book (Ch. 6) 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
reduces Kinetic Energy and increase Potential Energy A: The energy is stored as potential energy. PE is like your saving account. Potential energy gain (mg h) during the rising part. We can get that energy back as kinetic E if the ball falls back off. During falling, Kinetic Energy will increase mg h. Potential energy will reduce mg h.
the potential energy of an object subject to this conservative force which depends only on the object's position. Potential energy is sometimes called the "energy of configuration" since it only depends on the position of objects in a system. Thus, for the conservative electric force, we can find a change in electrical potential energy
with an illustration of a sword and the main road located to the west of Sutton 6 7 Part of Thomas Jeffrey’s 1771 map of Yorkshire 6 8 Locations of the geophysical survey grid and the excavation trench 7 9 Results of the electrical earth resistance survey of the area across Old London Road, Towton 8 10 Results of geophysical survey shown superimposed over an aerial photograph 9 11 Electrical .
