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2000, W. E. Haisler Energy Principles for Kinetic Problems ENGR 211 Principles of Engineering I (Conservation Principles in Engineering Mechanics) Conservation of Energy Kinetic Energy Potential Energy Internal Energy Conservation Equation Energy Analysis 1 2000, W. E. Haisler 2 Energy Principles for Kinetic Problems The conservation of energy follows the same general format as for mass, linear momentum and angular momentum: Accumulation of Energy within system during time period Energy entering system during time period Energy leaving - system during time period where Accumulation of Energy Energy in system Energy in system within system at end of - at beginning of during dime period time period time period In the above, we have not considered any generation or consumption of energy, i.e., we have not considered certain chemical or nuclear reactions although these may be important in special cases. 2000, W. E. Haisler 3 Energy Principles for Kinetic Problems Energy may take many forms including Kinetic Energy (KE) Potential Energy (PE) Internal Energy (U) Heat Energy (Q) Work Energy (W) Kinetic Energy (KE) Kinetic energy is that energy that a body possesses as a result of mass and its motion, specifically, its velocity. The Kinetic Energy is defined by and KE 1 mv 2 1 m(vx2 v 2y vz2 ) 2 2 KE 1 (v v )dm 2 concentrated mass distributed mass 2000, W. E. Haisler 4 Energy Principles for Kinetic Problems For a rigid body, it is convenient to think of the kinetic energy in terms of its translational and rotational components. So we write Rotational Translational kinetic energy Kinetic Energy of kinetic energy with respect to rigid body of center of mass center of mass of rigid body of rigid body We need the position vectors as before: any particle r rG G r y v vG G v ua r G r O z system G rG x r = rG + Gr axis through the center of mass 2000, W. E. Haisler Energy Principles for Kinetic Problems The integrand of the KE becomes v v (vG G v ) (vG G v ) vG vG 2vG G v G v G v and the KE becomes KE 1 (v v )dm 2 1 (vG vG )dm (vG G v )dm 1 ( G v G v )dm 2 2 The velocity of the center of mass vG does not vary within the body (mass) and may be taken outside of the integral. Thus, KE 1 mvG2 vG 2 G v dm 12 ( G v G v )dm 5 2000, W. E. Haisler Energy Principles for Kinetic Problems 6 d Gr The integrand in the second term can be written G v and dt d Gr thus Gvdm dm d G rdm 0 . The integral rdm 0 dt dt when r is measured from the center of mass.. Thus the KE becomes KE 1 mvG2 1 ( G v G v )dm 2 2 For a rigid body we can write Gv G r This says that with respect to the center of mass, Gv has only a tangential component (no radial component). Then the integrand in the last term of KE can be written (using a vector identity): 2000, W. E. Haisler G Now v Gv ( G r )( G r ) ( )(G r G r ) ( G r )(G r ) 2 system ua 2 r r r G G G particle G rn y and ua so that ra G r G r x r = rG + Gr z ( G r )( G r ) ( G ra)(G ra) 2G ra2 And finally: G rG G r (ua) G r G ra Thus 7 Energy Principles for Kinetic Problems G v G v G r 2 2 r 2 G a 2(G r 2 G ra2) 2 G rn2 2 axis through center of mass 2000, W. E. Haisler Energy Principles for Kinetic Problems 8 Substituting this result back into the previous KE equation gives KE 1 mvG2 1 ( 2 G rn2 )dm 2 2 1 mvG2 1 2 G rn2 dm 2 2 1 mvG2 1 2 G I a 2 2 Thus, the KE for a rigid body translating with velocity vG and rotating about its a-axis with angular velocity is given by KE 1 mvG2 1 G I a 2 2 2 It is sometimes useful to define the specific kinetic energy, or the kinetic energy per unit mass as I KEˆ KE / m 1 vG2 1 ( G a ) 2 2 2 m 2000, W. E. Haisler 9 Energy Principles for Kinetic Problems Potential Energy (PE) A body possesses potential energy as a result of its position in a potential force field. For conservative forces, the change in _ potential energy is defined by the negative z s ds F of the work the force does to move from one position to another in the direction of path s: PE PE f c(s)ds 0 y PE = potential energy at the end of the x path followed PE0 = potential energy at the beginning of the path f c = force s = path force follows The dot product is required in order to obtain the magnitude of the force in the direction of the path followed. 2000, W. E. Haisler Energy Principles for Kinetic Problems 10 Gravity is an example of an important conservative force field where the force is given by f c mg . The the potential energy definition becomes PE PE mg ds . 0 Suppose that the gravity vector is directed in the -z coordinate direction with magnitude g. Then g gk . The path can be described in a Cartesian coordinate system as ds dxi dyj dzk . The potential energy equation becomes z PE PE m ( g )dz mg (z z0) 0 0 The above assumes that m and g do not change in moving from 0 to z. 2000, W. E. Haisler Energy Principles for Kinetic Problems 11 Another way to write this is to say PE PE mg(z z ) end beginning end beginning For simplicity, PEbeginning is usually taken to be zero at zbeg so that PE mgz mgh end end where h is distance in the +z direction. 2000, W. E. Haisler 12 Energy Principles for Kinetic Problems Notes on Sign Convention and Sign of PE The expression for potentional energy of a mass, PE mgh , has a sign convention associated with with it. When the direction of h is opposite the direction of gravity, g, PE is assumed positive: PE mgh1 Thus, when the direction of h is in the same direction of gravity, g, PE is negative: mg h1 h2 PE mgh2 mg 2000, W. E. Haisler Energy Principles for Kinetic Problems 13 Internal Energy (U) A deformable body also possesses internal potential energy as a result of atomic bonds being stretched. These bonds create forces between particles. When the particles are moved relative to one another, these forces do work and result in energy being stored in the body. We use the symbol U to define this internal potential energy. An example is a spring. When we apply an external force to the end of a spring, the spring is stretched (or compressed) and internal potential energy is stored in the spring. If the force is removed, the internal energy within the spring is converted to translational energy and the spring returns to its original shape. We can also define specific internal energy or internal energy per unit mass as Uˆ U . m 2000, W. E. Haisler Energy Principles for Kinetic Problems 14 k F Internal Energy, U, of a Linear, Elastic Spring F For a linear, elastic spring, we observe experimentally that the stretching of the spring, , is proportional to the amount of force, F, applied to the spring. The proportionality constant is called the spring constant, k, and we thus have F k . F The amount of internal energy U stored in k the spring is equal to the area under the F vs. curve. Hence, 2 2 1 1 U Fd k d 1 k 2 2 2 1 F k k F k U 12 k 2 U 12 k ( 22 12 ) 2000, W. E. Haisler Energy Principles for Kinetic Problems 15 Work (W) Work done on a body by an external force is defined to be the (product of the force component in the direction of the resultant motion of the body) times (the displacement it produces). The work that we will be interested in is that due to forces that act on the external boundary of the system. W f ext ds Work is considered positive when the external force is in the direction of the displacement. We can also define the work rate (or power) as W dW dt 2000, W. E. Haisler Energy Principles for Kinetic Problems 16 Work is defined to be positive when energy is added to the system and negative when energy leaves the system. Note: Work is also produced then a torque causes a rotation. Note: Gravity forces also do work on the volume of the body but they are included in the potential energy (PE) term. 2000, W. E. Haisler 17 Energy Principles for Kinetic Problems Examples of Work, W, Done on System Work is positive if force and motion are in same direction. S A C A 10 lb When block B moves through the distance S B , the force F will move through the same distance. Hence the work done by F is W FS B When block A moves through the distance S A (note, down the plane), the work done by the frictional force Fk is W Fk S A . The negative is because the frictional force and motion are always in opposite direction. SB 20o B 6 lb F 100 lbf 10 lb y x 20o Fk F N 2000, W. E. Haisler Energy Principles for Kinetic Problems 18 Heat Energy (Q) Heat energy (defined by the symbol Q) may enter or leave the system through the system boundary at a rate Q dQ . We may dt also consider internal heat sources or sinks within the body (to be discussed in ENGR 214). Heat energy has units of Joules and is defined to be positive when it enters the system. 2000, W. E. Haisler Energy Principles for Kinetic Problems Conservation of Energy Equation The total energy of the system is defined to be Esys and consists of the internal, kinetic and potential energy terms: Esys U KE PE For a complete system, we can write the conservation of energy equation in the same manner as was done for conservation of linear and angular momentum. We write dEsys (mUˆ mKEˆ mPEˆ ) Q WTOT in / out dt The first term on the right represents energy entering or leaving the system due to mass flow across the system boundary. The second term is the net rate of heat entering the system. The last term is the net rate of work applied to the system boundary. 19 2000, W. E. Haisler Energy Principles for Kinetic Problems 20 Integral Form of Conservation of Energy The conservation of energy equation may be integrated over a time period (tbeg to tend) to obtain: tend dEsys tend ( )dt ( (mUˆ mKEˆ mPEˆ ) )dt tbeg dt tbeg in / out tend tend tbeg ( Q)dt tbeg ( WTOT )dt If the energy flow terms are constant over the time period, then (Esys ) (Esys) (U KE PE) end beg in / out Q WTOT Note: ˆ dmUdt ˆ mUˆ U mUdt dt 2000, W. E. Haisler Energy Principles for Kinetic Problems 21 Closed System For a closed system with no mass crossing the system boundary, the only work is non-flow work. In other words, when m 0 , there is no work associated with the mass flow terms ( mUˆ ,mKEˆ ,mPEˆ ). Thus, dEsys Q Wnon flow dt W means the work done by forces, pressures, etc. that non flow act on the system boundary (not due to mass flow across a boundary). Steady State For steady state, the system properties do no change with time and the rate equation becomes 0 (U KE PE) Q Wnon - flow in / out 2000, W. E. Haisler Energy Principles for Kinetic Problems Energy Analysis of Dynamic Systems We now return to the consideration of a rigid body undergoing translational and rotational motion. For a closed system, we can write the conservation of energy equation as dEsys dt Q Wnon flow where Esys U KE PE The kinetic energy (KE) is given by KE 1 mvG2 1 G I a 2 2 2 The potential energy (PE) due to gravitational forces is given by PE mgh G where hG=position of center of mass relative to the position where PE=0. 22 2000, W. E. Haisler Energy Principles for Kinetic Problems The work energy due to boundary forces is given by W f ext ds For a rigid body, the internal energy (U) is zero. We take the special case when there is no heat flow through the boundary ( Q 0 ). The system energy becomes Esys 1 mvG2 1 G I a 2 mghG 2 2 The right side of COE becomes W d( f ds) ( f ds ) non flow dt surface surface dt ( f v ) surface surface 23 2000, W. E. Haisler 24 Energy Principles for Kinetic Problems The Conservation of Energy equation (for a rigid body and no heat flow) becomes d ( 1 mv 2 1 I 2 mgh ) ( f vsurface ) G G a G surface dt 2 2 RIGID BODY The term on the right side will be non-zero only if the surface boundary is moving where the surface forces are applied. For friction without slipping, the frictional forces does no work since it does not move relative to the point of contact. Conservation of Energy is more conveniently written as: ( 1 mvG2 1 G I a 2 mghG )end ( 1 mvG2 1 G I a 2 mghG )beg WTOT 2 2 2 2 OR FOR RIGID BODY ONLY ( KE PE )end ( KE PE )beg WTOT 2000, W. E. Haisler Energy Principles for Kinetic Problems Remember that the Conservation of Energy equation for a general system is given by d (KE PE U ) (mKEˆ mPEˆ mUˆ ) Q WTOT in / out dt where KE 1 (v v )dm 2 ; for rigid body: KE 1 mvG2 1 G I a 2 2 2 PE f c(s)ds ; for gravity, PE mghG U = internal energy (a further topic for ENGR 214) Q = heat flow rate into body WTOT ( f surface vsurface) 25 2000, W. E. Haisler Energy Principles for Kinetic Problems When the system has no mass flow in/out and no heat input, above reduces to: d (KE PE U ) W TOT dt Note that the term on the right is the work rate of the external forces, or power. Integration of the above (between the "end" and "beginning" states) gives: ( KE PE U )end ( KE PE U )beg W 26 2000, W. E. Haisler Energy Principles for Kinetic Problems 27 E For many problems, we k D B must simply "count up" all A the energy contributions of the system. For the problem shown, we have: KE of mass E KE of disk AB C KE of mass C PE of mass C F W done by force F W done be frictional force on E U for spring k There could also be: W due to frictional force on pin at disk support W due to a torque applied to disk If rope has mass, then KE and PE of rope, etc., etc., …