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Chapters 7, 8 Energy

Explore the essential concepts of energy, including kinetic energy, work, power, and conservation laws. Understand how to calculate kinetic energy, work done, and power in different scenarios. Learn about conservative forces, potential energy, and the principles governing energy transformations. Enhance your understanding of energy fundamentals through practical problem-solving exercises.

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Chapters 7, 8 Energy

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  1. Chapters 7, 8 Energy

  2. Energy • What is energy? • Energy - is a fundamental, basic notion in physics • Energy is a scalar, describing state of an object or a system • Description of a system in ‘energy language’ is equivalent to a description in ‘force language’ • Energy approach is more general and more effective than the force approach • Equations of motion of an object (system) can be derived from the energy equations

  3. Scalar product of two vectors • The result of the scalar (dot) multiplication of two vectors is a scalar • Scalar products of unit vectors

  4. Scalar product of two vectors • The result of the scalar (dot) multiplication of two vectors is a scalar • Scalar product via unit vectors

  5. Some calculus • In 1D case

  6. Some calculus • In 1D case • In 3D case, similar derivations yield • K – kinetic energy

  7. James Prescott Joule (1818 - 1889) • Kinetic energy • K = mv2/2 • SI unit: kg*m2/s2 = J (Joule) • Kinetic energy describes object’s ‘state of motion’ • Kinetic energy is a scalar

  8. Chapter 7 Problem 31 A 3.00-kg object has a velocity of (6.00^i – 2.00^j) m/s. (a) What is its kinetic energy at this moment? (b) What is the net work done on the object if its velocity changes to (800^i + 4.00^j) m/s?

  9. Work–kinetic energy theorem • Wnet – work (net) • Work is a scalar • Work is equal to the change in kinetic energy, i.e. work is required to produce a change in kinetic energy • Work is done on the object by a force

  10. Work: graphical representation • 1D case: Graphically - work is the area under the curve F(x)

  11. Net work vs. net force • We can consider a system, with several forces acting on it • Each force acting on the system, considered separately, produces its own work • Since

  12. Work done by a constant force • If a force is constant • If the displacement and the constant force are not parallel

  13. Work done by a spring force • Hooke’s law in 1D • From the work–kinetic energy theorem

  14. Work done by the gravitational force • Gravity force is ~ constant near the surface of the Earth • If the displacement is vertically up • In this case the gravity force does a negative work (against the direction of motion)

  15. Lifting an object • We apply a force F to lift an object • Force F does a positive work Wa • The net work done • If in the initial and final states the object is at rest, then the net work done is zero, and the work done by the force F is

  16. James Watt (1736-1819) • Power • Average power • Instantaneous power – the rate of doing work • SI unit: J/s = kg*m2/s3 = W (Watt)

  17. Power of a constant force • In the case of a constant force

  18. Chapter 8 Problem 32 A 650-kg elevator starts from rest. It moves upward for 3.00 s with constant acceleration until it reaches its cruising speed of 1.75 in/s. (a) What is the average power of the elevator motor during this time interval? (b) How does this power compare with the motor power when the elevator moves at its cruising speed?

  19. Conservative forces • The net work done by a conservative force on a particle moving around any closed path is zero • The net work done by a conservative force on a particle moving between two points does not depend on the path taken by the particle

  20. Conservative forces: examples • Gravity force • Spring force

  21. Potential energy • For conservative forces we introduce a definition of potential energy U • The change in potential energy of an object is being defined as being equal to the negative of the work done by conservative forces on the object • Potential energy is associated with the arrangement of the system subject to conservative forces

  22. Potential energy • For 1D case • A conservative force is associated with a potential energy • There is a freedom in defining a potential energy: adding or subtracting a constant does not change the force • In 3D

  23. Chapter 7 Problem 44 A single conservative force acting on a particle varies F = (– Ax + Bx2) ^i N, where A and B are constants and x is in meters. (a) Calculate the potential energy function U(x) associated with this force, taking U = 0 at x = 0. (b) Find the change in potential energy and the change in kinetic energy of the system as the particle moves from x = 2.00 m to x = 3.00 m.

  24. Gravitational potential energy • For an upward direction the y axis

  25. Elastic potential energy • For a spring obeying the Hooke’s law

  26. Internal energy • The energy associated with an object’s temperature is called its internal energy, Eint • In this example, the friction does work and increases the internal energy of the surface

  27. Conservation of mechanical energy • Mechanical energy of an object is • When a conservative force does work on the object • In an isolated system, where only conservative forces cause energy changes, the kinetic and potential energies can change, but the mechanical energy cannot change

  28. Work done by an external force • Work is transferred to or from the system by means of an external force acting on that system • The total energy of a system can change only by amounts of energy that are transferred to or from the system • Power of energy transfer, average and intantaneous

  29. Conservation of mechanical energy: pendulum

  30. Potential energy curve

  31. Neutral equilibrium Unstable equilibrium Stable equilibrium Potential energy curve: equilibrium points

  32. Chapter 8 Problem 55 A 10.0-kg block is released from point A. The track is frictionless except for the portion between points B and C, which has a length of 6.00 m. The block travels down the track, hits a spring of force constant 2250 N/m, and compresses the spring 0.300 m from its equilibrium position before coming to rest momentarily. Determine the coefficient of kinetic friction between block and the rough surface between B and C.

  33. Answers to the even-numbered problems Chapter 7 Problem 2: (a) 3.28 × 10−2 J (b) - 3.28 × 10−2 J

  34. Answers to the even-numbered problems Chapter 7 Problem 10: 16.0

  35. Answers to the even-numbered problems Chapter 7 Problem 46: (7−9x2y)ˆi−3x3ˆj

  36. Answers to the even-numbered problems Chapter 8 Problem 14: (a) 0.791 m/s (b) 0.531 m/s

  37. Answers to the even-numbered problems Chapter 8 Problem 28: 8.01 W

  38. Answers to the even-numbered problems Chapter 8 Problem 34: 194 m

  39. Answers to the even-numbered problems Chapter 8 Problem 50: (a) 0.588 J (b) 0.588 J (c) 2.42 m/s (d) UC = 0.392 J, KC = 0.196 J

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