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Engineering Physics : Lecture 10 (Chapter 7 Halliday )

Engineering Physics : Lecture 10 (Chapter 7 Halliday ). Work done by variable force Spring Problem involving spring & friction Work done by variable force in 3-D Newton’s gravitational force. Review: Constant Force. W = F  d No work done if  = 90 o . No work done by T .

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Engineering Physics : Lecture 10 (Chapter 7 Halliday )

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  1. Engineering Physics : Lecture 10(Chapter 7 Halliday) • Work done by variable force • Spring • Problem involving spring & friction • Work done by variable force in 3-D • Newton’s gravitational force

  2. Review: Constant Force... W = Fd • No work done if = 90o. • No work done by T. • No work done by N. T v v N

  3. v1 v2 F x Review: Work/Kinetic Energy Theorem: {NetWork done on object} = {change in kinetic energy of object} WF = K = 1/2mv22 - 1/2mv12 WF = Fx m

  4. F(x) x1 x2 dx Work done by Variable Force: (1D) • When the force was constant, we wrote W = F x • area under F vs. x plot: • For variable force, we find the areaby integrating: • dW = F(x) dx. F Wg x x

  5. dv Work/Kinetic Energy Theorem for a Variable Force dv F dx F dx dv dv dv v (chain rule) dx = = dx dt dx dt dv dx v dx v dv v22 v12 v22 v12

  6. 1-D Variable Force Example: Spring • For a spring we know that Fx = -kx. F(x) x1 x2 x relaxed position -kx F= - k x1 F= - k x2

  7. Spring... • The work done by the spring Wsduring a displacement from x1to x2 is the area under the F(x) vs x plot between x1and x2. F(x) x1 x2 x Ws relaxed position -kx

  8. Spring... • The work done by the spring Wsduring a displacement from x1to x2 is the area under the F(x) vs x plot between x1and x2. F(x) x1 x2 x Ws -kx

  9. Work & Energy • A box sliding on a horizontal frictionless surface runs into a fixed spring, compressing it a distance x1 from its relaxed position while momentarily coming to rest. • If the initial speed of the box were doubled and its mass were halved, how far x2 would the spring compress ? (a)(b) (c) x

  10. In the case of x1 Solution • Again, use the fact that WNET = DK. In this case, WNET =WSPRING = -1/2 kx2 and K = -1/2 mv2 so kx2 = mv2 x1 v1 m1 m1

  11. Solution So if v2 = 2v1 and m2 = m1/2 x2 v2 m2 m2

  12. Problem: Spring pulls on mass. • A spring (constant k) is stretched a distance d, and a mass m is hooked to its end. The mass is released (from rest). What is the speed of the mass when it returns to the relaxed position if it slides without friction? m relaxed position stretched position (at rest) m d after release m v back at relaxed position m vr

  13. Problem: Spring pulls on mass. • First find the net work done on the mass during the motion from x = d to x = 0 (only due to the spring): stretched position (at rest) m d relaxed position m i vr

  14. Problem: Spring pulls on mass. • Now find the change in kinetic energy of the mass: stretched position (at rest) m d relaxed position m i vr

  15. Problem: Spring pulls on mass. • Now use work kinetic-energy theorem: Wnet = WS = K. stretched position (at rest) m d relaxed position m i vr

  16. Problem: Spring pulls on mass. • Now suppose there is a coefficient of friction  between the block and the floor • The total work done on the block is now the sum of the work done by the spring WS (same as before) and the work done by friction Wf.Wf = f.Δr = -mgd r stretched position (at rest) m d relaxed position f= mg m i vr

  17. Problem: Spring pulls on mass. • Again use Wnet = WS + Wf = KWf = -mgd r stretched position (at rest) m d relaxed position f= mg m i vr

  18. F dr Work by variable force in 3-D: • Work dWF of a force F acting • through an infinitesimal • displacement dris: • dW = F.dr • The work of a big displacement through a variable force will be the integral of a set of infinitesimal displacements: • WTOT = F.dr ò

  19. m Work by variable force in 3-D:Newton’s Gravitational Force • Work dWg done on an object by gravity in a displacement drisgiven by: • dWg = Fg.dr= (-GMm / R2r).(dR r + Rd)dWg = (-GMm / R2)dR (sincer.= 0, r.r = 1) ^ ^ ^ ^ ^ ^ ^ ^ dR  ^ r dr Rd Fg d R M

  20. R2 R2 R1 R1 m Work by variable force in 3-D:Newton’s Gravitational Force • Integrate dWg to find the total work done by gravity in a “big”displacement: • Wg = dWg= (-GMm / R2)dR = GMm (1/R2 - 1/R1) Fg(R2) R2 Fg(R1) R1 M

  21. Work by variable force in 3-D:Newton’s Gravitational Force • Work done depends only on R1 and R2, not on the path taken. m R2 R1 M

  22. Newton’s Gravitational ForceNear the Earth’s Surface: • Suppose R1 = REand R2 = RE + ybut we have learned thatSo: Wg = -mgy m RE+ y RE M

  23. Recap of today’s lecture • Review • Work done by gravity near the Earth’s surface • Examples: • pendulum, inclined plane, free fall • Work done by variable force • Spring • Problem involving spring & friction • Work done by variable force in 3-D • Newton’s gravitational force • Look at textbook problems

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