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Lecture 15

Lecture 15. Chapter 10 Employ conservation of energy principle Chapter 11 Employ the dot product Employ conservative and non-conservative forces Relate force to potential energy Use the concept of power (i.e., energy per time). Goals:. Assignment: HW7 due Tuesday, Nov. 2 nd

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Lecture 15

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  1. Lecture 15 • Chapter 10 • Employ conservation of energy principle • Chapter 11 • Employ the dot product • Employ conservative and non-conservative forces • Relate force to potential energy • Use the concept of power (i.e., energy per time) Goals: Assignment: • HW7 due Tuesday, Nov. 2nd • For Wednesday: Read Chapter 12, Sections 1-3, 5 & 6 do not concern yourself with the integration process in regards to “center of mass” or “moment of inertia”

  2. Energy scaling question • I have a block a distance y above the floor (Ug≡ 0 when the block is on the floor). There is a spring attached to the block and, at y, the potential energy of the spring is equal to the gravitational energy (i.e., Ug = Us = mgy). I now raise the block to a distance 2y above the floor. The total potential energy of the system (spring and block) increases by a factor of A 2 B 3 C 4 D 5 E Can’t be determined with the information given

  3. 2 1 Mechanical Energy • Potential Energy (U) • Kinetic Energy (K) • If “conservative” forces (e.g, gravity, spring) then Emech = constant = K + U During  Uspring+K1+K2 = constant = Emech • Mechanical Energy conserved Before During After

  4. 1 2 h 3 0 mass: m -x Energy (with spring & gravity) • Emech = constant (only conservative forces) • At 1:y1 = h ; v1y = 0At 2:y2 = 0 ; v2y= ?At 3:y3 = -x ; v3 = 0 • Em1 = Ug1 + Us1 + K1 = mgh + 0 + 0 • Em2 = Ug2 + Us2 + K2 = 0 + 0 + ½ mv2 • Em3 = Ug3 + Us3 + K3 = -mgx + ½ kx2 + 0 Given m, g, h & k, how much does the spring compress?

  5. 1 2 h 3 0 mass: m -x Energy (with spring & gravity) • Emech = constant (only conservative forces) • At 1:y1 = h ; v1y = 0At 2:y2 = 0 ; v2y= ?At 3:y3 = -x ; v3 = 0 • Em1 = Ug1 + Us1 + K1 = mgh + 0 + 0 • Em2 = Ug2 + Us2 + K2 = 0 + 0 + ½ mv2 • Em3 = Ug3 + Us3 + K3 = -mgx + ½ kx2 + 0 • Given m, g, h & k, how much does the spring compress? • Em1 = Em3 = mgh = -mgx + ½ kx2 Solve ½ kx2 – mgx +mgh = 0 Given m, g, h & k, how much does the spring compress?

  6. 1 2 h 3 0 mass: m -x Energy (with spring & gravity) • Emech = constant (only conservative forces) • At 1:y1 = h ; v1y = 0At 2:y2 = 0 ; v2y= ?At 3:y3 = -x ; v3 = 0 • Em1 = Ug1 + Us1 + K1 = mgh + 0 + 0 • Em2 = Ug2 + Us2 + K2 = 0 + 0 + ½ mv2 • Em3 = Ug3 + Us3 + K3 = -mgx + ½ kx2 + 0 • Given m, g, h & k, how much does the spring compress? • Em1 = Em3 = mgh = -mgx + ½ kx2 Solve ½ kx2 – mgx +mgh = 0 Given m, g, h & k, how much does the spring compress?

  7. Energy (with spring & gravity) 1 mass: m • When is the child’s speed greatest? (A) At y1 (top of jump) (B) Between y1 & y2 (C) At y2 (child first contacts spring) (D) Between y2 & y3 (E) At y3 (maximum spring compression) 2 h 3 0 -x

  8. Before During 2 1 After Inelastic Processes • If non-conservative” forces (e.g, deformation, friction) then Emech is NOT constant • After  K1 + K2 < Emech (before) • Accounting for this “loss” we introduce • Thermal Energy (Eth , new) where Esys = Emech + Eth = K + U + Eth

  9. Chapter 11: Energy & Work • Impulse (Force over a time) describes momentum transfer • Work (Force over a distance) describes energy transfer • Any single acting force which changes the potential or kinetic energy of a system is said to have done work W on that system DEsys = W W can be positive or negative depending on the direction of energy transfer • Net work reflects changes in the kinetic energy Wnet = DK This is called the “Net” Work-Kinetic Energy Theorem

  10. Work performs energy transfer • If only conservative force present then work either increases or decreases the mechanical energy. • It is important to define what constitutes the system

  11. v Circular Motion • I swing a sling shot over my head. The tension in the rope keeps the shot moving at constant speed in a circle. • How much work is done after the ball makes one full revolution? (A) W > 0 Fc (B) W = 0 (C) W < 0 (D) need more info

  12. Examples of “Net” Work (Wnet) DK = Wnet • Pushing a box on a smooth floor with a constant force; there is an increase in the kinetic energy Examples of No “Net” Work DK = Wnet • Pushing a box on a rough floor at constant speed • Driving at constant speed in a horizontal circle • Holding a book at constant height This last statement reflects what we call the “system” ( Dropping a book is more complicated because it involves changes in U and K, U is transferred to K. The answer depends on what we call the system. )

  13. Changes in K with a constant F along a linear path • If F is constant

  14. Finish Start q = 0° F Net Work: 1-D Example (constant force) • A force F= 10 Npushes a box across a frictionless floor for a distance x= 5 m. • Net Work is F x= 10 x 5 N m = 50 J • 1 Nm ≡ 1 Joule and this is a unit of energy • Work reflects energy transfer x

  15. Units: Force x Distance = Work Newton x [M][L] / [T]2 Meter = Joule [L][M][L]2 / [T]2 mks cgs Other BTU = 1054 J calorie = 4.184 J foot-lb = 1.356 J eV = 1.6x10-19 J Dyne-cm or erg = 10-7 J N-m or Joule

  16. Net Work: 1-D 2nd Example (constant force) • Net Work is F x= -10 x 5 N m = -50 J • Work again reflects energy transfer • A forceF= 10 Nis opposite the motion of a box across a frictionless floor for a distance x = 5 m. Finish Start q = 180° F x

  17. Work in 3D…. • x, y and z with constant F:

  18. Work: “2-D” Example (constant force) • An angled force, F= 10 N,pushes a box across a frictionless floor for a distance x= 5 m and y= 0 m • (Net) Work is Fxx= F cos(-45°) x = 50 x 0.71 Nm = 35 J • Work reflects energy transfer Finish Start F q = -45° Fx x

  19. A q Ay Ax î Scalar Product (or Dot Product) • Useful for finding parallel components A î= Ax î  î = 1 î j = 0 • Calculation can be made in terms of components. A B = (Ax )(Bx) + (Ay )(By ) + (Az )(Bz ) Calculation also in terms of magnitudes and relative angles. A B≡ | A | | B | cosq You choose the way that works best for you!

  20. Scalar Product (or Dot Product) Compare: A B = (Ax )(Bx) + (Ay )(By ) + (Az )(Bz ) with A as force F, B as displacement Dr Notice if force is constant: F Dr = (Fx )(Dx) + (Fy )(Dz ) + (Fz )(Dz) FxDx +FyDy + FzDz = DK So here F Dr = DK = Wnet Again: A Parallel Force acting Over a Distance does Work

  21. “Scalar or Dot Product” Definition of Work, The basics Ingredients:Force (F ), displacement ( r) Work, W, of a constant force F acts through a displacement  r: W = F· r(Work is a scalar) F  r  displacement If we know the angle the force makes with the path, the dot product gives usF cos qandDr If the path is curved at each point and

  22. v ExerciseWork in the presence of friction and non-contact forces • A box is pulled up a rough (m > 0) incline by a rope-pulley-weight arrangement as shown below. • How many forces (including non-contact ones) are doing work on the box ? • Of these which are positive and which are negative? • State the system (here, just the box) • Use a Free Body Diagram • Compare force and path • 2 • 3 • 4 • 5

  23. Work and Varying Forces (1D) Area = Fx Dx F is increasing Here W = F·r becomes dW = Fdx • Consider a varying force F(x) Fx x Dx Finish Start F F q = 0° Dx Work has units of energy and is a scalar!

  24. to vo m spring at an equilibrium position x F V=0 t m spring compressed Example: Work Kinetic-Energy Theorem with variable force • How much will the spring compress (i.e. x) to bring the box to a stop (i.e., v = 0 ) if the object is moving initially at a constant velocity (vo) on frictionless surface as shown below ? Notice that the spring force is opposite the displacement For the mass m, work is negative For the spring, work is positive

  25. Uf rf rf ò U = Uf - Ui = - W = -F•dr ri Ui ri Conservative Forces & Potential Energy • For any conservative force F we can define a potential energy function U in the following way: The work done by a conservative force is equal and opposite to the change in the potential energy function. • This can be written as: ò W=F·dr= - U

  26. Exercise: Work & Friction • Two blocks having mass m1 and m2 where m1 > m2. They are sliding on a frictionless floor and have the same kinetic energy when they encounter a long rough stretch (i.e. m > 0) which slows them down to a stop. • Which one will go farther before stopping? • Hint: How much work does friction do on each block ? (A)m1(B)m2(C)They will go the same distance m1 v1 v2 m2

  27. Conservative Forces and Potential Energy • So we can also describe work and changes in potential energy (for conservative forces) DU = - W • Recalling (if 1D) W = Fx Dx • Combining these two, DU = - Fx Dx • Letting small quantities go to infinitesimals, dU = - Fx dx • Or, Fx = -dU / dx

  28. Recap Next time: Ch. 11 Power Start Chapter 12 Assignment: • HW7 due Wednesday, March 10 • For Tuesday: Read Chapter 12, Sections 1-3, 5 & 6 do not concern yourself with the integration process

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