Chapter 6 Work and Energy

1. Chapter 6 Work and 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. Forms of energy • Mechanical • Chemical • Electromagnetic • Nuclear • Energy can be transformed from one form to another

4. James Prescott Joule (1818 - 1889) Work • Work is a scalar • Work is done on the object by a force • Work (W) done by a constant force on an object: product of the component of the force along the direction of displacement and the magnitude of the displacement • SI unit: kg*m2/s2 = J (Joule)

5. Work • If there are multiple forces acting on an object, the total work done is the algebraic sum of the amount of work done by each force • Work is positive (negative) if the force and the displacement are in the same (opposite) direction • The work done by a force is zero when the force is perpendicular to the displacement: cos 90° = 0

6. Work • If there are multiple forces acting on an object, the total work done is the algebraic sum of the amount of work done by each force • Work is positive (negative) if the force and the displacement are in the same (opposite) direction) • The work done by a force is zero when the force is perpendicular to the displacement: cos 90° = 0

7. Kinetic energy • Kinetic energy is a scalar • Kinetic energy describes object’s ‘state of motion’ • Kinetic energy: • SI unit: kg*m2/s2 = J (Joule) • Work is related to kinetic energy

8. Work-kinetic energy theorem • When work is done by a net force on an object and the only change in the object is its speed, the work done is equal to the change in the object’s kinetic energy • Work is equal to the change in kinetic energy, i.e. work is required to produce a change in kinetic energy • Speed will increase (decrease) if work is positive (negative) • Kinetic energy can also be thought of as the amount of work the moving object could do in coming to rest

9. Work-kinetic energy theorem

10. Chapter 6 Problem 26 Under the influence of its drive force, a snowmobile is moving at a constant velocity along a horizontal patch of snow. When the drive force is shut off, the snowmobile coasts to a halt. The snowmobile and its rider have a mass of 136 kg. Under the influence of a drive force of 205 N, it is moving at a constant velocity whose magnitude is 5.50 m/s. The drive force is then shut off. Find (a) the distance in which the snowmobile coasts to a halt and (b) the time required to do so.

11. 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)

12. 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

13. 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, only on the initial and final positions of the object

14. Conservative forces: examples • Gravity force • Spring force • Electromagnetic force

15. 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 • Potential energy is a property of the system, not the object

16. Potential energy • A conservative force is associated with a potential energy and for every conservative force a potential energy function can be found • There is a freedom in defining a potential energy: adding or subtracting a constant does not change the force

17. Gravitational potential energy • Gravitational potential energy is the energy associated with the relative position of an object in space near the Earth’s surface • Objects interact with the earth through the gravitational force • Actually the potential energy is for the earth-object system

18. Spring force • Spring in the relaxed state • Spring force (restoring force) acts to restore the relaxed state from a deformed state

19. Robert Hooke(1635 – 1703) Hooke’s law • For relatively small deformations • Spring force is proportional to the deformation and opposite in direction • k – spring constant • Spring force is a variable force • Hooke’s law can be applied not to springs only, but to all elastic materials and objects

20. Elastic potential energy • Hooke’s law in 1D • For a spring obeying the Hooke’s law • Elastic potential energy related to the work required to compress a spring from its equilibrium position to some final, arbitrary, position x

21. 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

22. Conservation of mechanical energy: pendulum

23. Chapter 6 Problem 43 A skateboarder moving at 5.4 m/s along a horizontal section of a track that is slanted upward by 48° above the horizontal at its end, which is 0.40 m above the ground. When she leaves the track, she follows the characteristic path of projectile motion. Ignoring friction and air resistance, find the maximum height H to which she rises above the end of the track.

24. Nonconservative forces • A force is nonconservative if the work it does on an object depends on the path taken by the object between its final and starting points. • Examples of nonconservative forces: kinetic friction, air drag, propulsive forces • For friction force, the work required is less on the blue path than on the red path

25. Nonconservative forces • The friction force is transformed from the kinetic energy of the object into a type of energy associated with temperature • The objects are warmer than they were before the movement

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. 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 • Energy is conserved: we can neither create nor destroy energy • If nonconservative forces are present, then the full work-energy theorem must be used instead of the equation for conservation of mechanical energy

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

29. Chapter 6 Problem 70 A 1900-kg car experiences a combined force of air resistance and friction that has the same magnitude whether the car goes up or down a hill at 27 m/s. Going up a hill, the car’s engine produces 47 hp more power to sustain the constant velocity than it does going down the same hill. At what angle is the hill inclined above the horizontal?

30. Work done by varying forces • 1D case: work done by a variable force acting on an object that undergoes a displacement is equal to the area under the graph of Fx(x)

31. Work done by a spring force • Hooke’s law in 1D • The work is also equal to the area under the curve • In this case, the “curve” is a triangle

32. Questions?