Exploring Mechanical Energy and Work in Physics

1. Chapters 10 -11 Energy, Work, and Conservation

3. Some Energy Considerations • Energy can be transformed from one form to another. • The total amount of energy in the Universe never changes. • This is also true for isolated system. • Energy transformations are essential to the study of physics, chemistry, biology, geology, astronomy. • Can be used in place of Newton’s laws to solve certain problems more simply

4. Mechanical Energy • The focus of this chapter is on mechanical energy. • “Mechanical energy is the energy that is possessed by an object due to its motion or due to its position.” Mechanical Energy Potential Energy (PE) Kinetic Energy (KE)

5. Force, Energy, and Work • With this relationship we can find expressions for different forms of PE and discuss the principle of conservation of energy and apply it to solving problems. • The connection between work and energy can be used in place of Newton’s 2nd law to solve for certain problems involving force. Force Work Energy

6. What is work? • “Work” is the link that ties force and energy together. • In physics, work is defined differently than in everyday life. • “The work, W, done by a constant force on an object is defined as the product of the component of the force along the direction of displacement and the magnitude of the displacement.”

7. Work: General Case W = (F cos)x • F is the magnitude of the force • Δ x is the magnitude of the object’s displacement • q is the angle between

8. Work: Specific Case W = F x • This equation applies when the force is in the same direction as the displacement. • are in the same direction.

9. Sample Problem A man cleaning his apartment pulls the canister of a vacuum cleaner with a force of magnitude F = 55.0 N. The force makes an angle of 30.0o with the horizontal. The canister is displaced 3.00 m to the right. Calculate the work done by the 55.0 N force on the vacuum cleaner.

10. More About Work • This definition of work gives no information about: • The time it took for the displacement to occur. • The velocity or acceleration of the object. • These value can be obtained indirectly from work and then use of PE, KE, and the conservation of energy. • More on this later.

11. NO Displacement = NO Work!! • As long as an object does not change its position, no work is being done. • There are two other cases when no work is done. Can you think of them?

12. Force & Displacement Perpendicular = NO Work!! • If displacement is horizontal and the applied force is vertical, the angle between them will be 90o. • cos 90° = 0 • Whenever the displacement and the applied force are perpendicular to each other no work is done.

13. No Applied Force = NO Work!! • No applied force means no work is done. • ΣF = 0 N, W = 0 J

14. Multiple Forces and 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. ∆x WT = (ΣFcosθ) ∆x

15. A Quick Summary of Work • Work doesn’t happen by itself. • A net force must be acting on an object and causing a displacement for work to be done. • Work is done by something in the environment, on the object of interest. • The forces discussed here are constant. • The forces could be variable. • Work is related to energy as well. • This will be covered in the next section.

16. Work and Energy • In the previous section we looked at the link between work and force. • W = (F cos)x • In this section we will look at the relationship between work and energy. • We can use this relationship as an alternative to solving problems involving Newton’s 2nd law. • From which we can solve for velocity, displacement, and acceleration.

17. Work Causes Displacement • Work does not occur unless displacement occurs. • Displacement does not occur unless speed/velocity is involved. • Thus, work and speed/velocity are also related. • Using work, we can determine the speed of an object if we know its displacement and the net forces acting on it.

18. Determining an object’s speed given the net force acting on it and the work done to it.

19. Kinetic Energy • Energy associated with the motion of an object. • Scalar quantity with the same dimensions as work. • SI Units: J • Work is related to kinetic energy. • Wnet = ∆KE = KEf– KEi

20. 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. • Speed will increase if work is positive. • Speed will decrease if work is negative.

21. Work and Kinetic Energy • An object’s kinetic energy can also be thought of as the amount of work a moving object could do in coming to rest. • The moving hammer has kinetic energy and can do work on the nail.

22. Work and Kinetic Energy • Work can also be used to impart KE to an object. • The golf ball at rest is given a velocity and thus KE because of the work done by the club.

23. Sample Problem A car of mass 1.40x103 kg being towed has a net forward force of 4.50x103 N applied to it. The car starts from rest and travels down a horizontal highway. Find the speed of the car after it has traveled 100.0 m. (Ignore kinetic energy losses caused by friction and air resistance.)

24. Potential Energy • Potential energy (PE)is associated with the position of the object within some system. • Potential energy is a property of the system, not the object. • A system is a collection of objects interacting via forces or processes that are internal to the system. • PE again is only associated with conservative forces.

25. 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. • The potential energy is for the earth-object system.

26. Work and Gravitational Potential Energy • An expression for PE can be derived using work. • The expression for PE shows that it is determined by the mass of the object, the gravitational acceleration at the Earth’s surface, and the vertical location of the object. • This expression for PE is only valid for object near the Earth’s surface.

27. Work and Gravitational Potential Energy • PE = mgy • This equation relates work to PE. • The work done by PE depends only on the height of the object. • A “zero” point must be established. • More on this in a moment.

28. Reference Levels for Gravitational Potential Energy • A location where the gravitational potential energy is zero must be chosen for each problem. • The choice is arbitrary since the change in the potential energy is the important quantity. • Choose a convenient location for the zero reference height. • Often the Earth’s surface. • May be some other point suggested by the problem. • Once the position is chosen, it must remain fixed for the entire problem.

29. Example of Reference Levels • At location A, the desk may be the convenient reference level. • At location B, the floor could be used. • At location C, the ground would be the most logical reference level. • The choice is arbitrary, though.

30. Sample Problem How much work does a bricklayer do to carry 30.2 kg of bricks from the ground up to the height of 11.1 m.? What is the potential energy of the bricks when the bricklayer reaches this height?

31. Conservation • Conservation in general: • To say a physical quantity is conserved is to say that the numerical value of the quantity remains constant throughout any physical process although the quantities may change form. • The form of a physical quantity may change but the overall numeric value (or magnitude) will not. • Conservation plays an important role in physics and in other sciences.

32. Conservation of Mechanical Energy • In conservation of mechanical energy, the total mechanical energy remains constant. • In any isolated system of objects interacting only through conservative forces, the total mechanical energy of the system remains constant. • Total mechanical energy is the sum of the kinetic and potential energies in the system. • Other types of potential energy functions can be added to modify this equation

33. Problem Solving with Conservation of Energy • Define the system • Verify that only conservative forces are present. • Select the location of zero gravitational potential energy, where y = 0. • Do not change this location while solving the problem. • Identify two points the object of interest moves between. • One point should be where information is given. • The other point should be where you want to find out something. • Apply the conservation of energy equation to the system. • Immediately substitute zero values, then do the algebra before substituting the other values. • Solve for the unknown: • Typically a speed or a position • Substitute known values • Calculate result

34. Sample Problem A sled and its rider together weigh 800 N. They move down a frictionless hill through a vertical distance of 10.0 m. Use conservation of mechanical energy to find the speed of the sled at the bottom of the hill, assuming the rider pushes with an initial speed of 5.00 m/s. Neglect air resistance.

35. Potential Energy Stored in a Spring • Spring force is also a conservative force. • Thus a PE expression can be derived for the spring force and the work a spring does is not path dependent. • Let’s 1st look at the spring force. • Involves the spring constant, k • Hooke’s Law gives the force • F = - k x • F is the restoring force • F is in the opposite direction of x • k depends on how the spring was formed, the material it is made from, thickness of the wire, etc.

36. Potential Energy in a Spring • Elastic Potential Energy • Related to the work required to compress a spring from its equilibrium position to some final, arbitrary, position x. • We can relate this PE to kinetic energy in the same way we related gravitational PE to kinetic energy.

37. Spring Potential Energy • A) The spring is in equilibrium, neither stretched or compressed. • B) The spring is compressed, storing potential energy. • C) The block is released and the potential energy is transformed to kinetic energy of the block.

38. Conservation of Mechanical Energy Including a Spring • The PE of the spring is added to both sides of the conservation of mechanical energy equation to give: • The same problem-solving strategies apply to these problems as well.

39. Sample Problem A 0.50 kg block rests on a horizontal, frictionless surface as seen in the figure. The block is pressed against a light spring having a spring constant of k=80.0 N/m. The spring is compressed a distance of 2.0 cm and released. Consider the system to be the spring, block, and Earth and find the speed of the block when it is at the bottom of the incline.

40. Energy Transfers • By Work • By applying a force. • Produces a displacement of the system. • Heat • The process of transferring heat by collisions between atoms or molecules. • For example, when a spoon rests in a cup of coffee, the spoon becomes hot because some of the KE of the molecules in the coffee is transferred to the molecules of the spoon as internal energy.

41. More Energy Transfers • Mechanical Waves • A disturbance propagates through a medium. • Examples include sound, water, seismic. • Electrical transmission • Transfer by means of electrical current. • This is how energy enters any electrical device.

42. More Energy Transfers • Electromagnetic radiation • Any form of electromagnetic waves. • Light, microwaves, radio waves. • For example • Cooking something in your microwave oven. • Light energy traveling from the Sun to the Earth.

43. Collisions • Momentum is conserved in any collision. • Total KE is not always conserved. • Some KE is converted to other forms of energy (i.e. internal energy, sound energy, etc.) or is used to do the work needed to deform an object. • Two broad categories of collisions: • Elastic collisions • Inelastic collisions: Perfectly inelastic and inelastic • Elastic and perfectly inelastic collisions represent ideal cases of collisions. • Most real world cases fit somewhere between these two extremes.

44. Inelastic Collisions • Momentum is conserved. • Kinetic energy is not conserved. • Two categories: inelastic and perfectly inelastic. • Perfectly inelastic collisions occur when the objects stick together. • Not all of the KE is necessarily lost. • Colliding objects have different initial velocities but the same final velocities. • Use:

45. Perfectly Inelastic Collisions • When two objects stick together after the collision, they have undergone a perfectly inelastic collision. • Conservation of momentum becomes:

46. Elastic Collisions • Mostly occurs at the atomic and subatomic levels. • Macroscopic collisions can only approximate elastic collisions. • Both momentum and kinetic energy are conserved for both objects. • No KE is lost as sound, heat, light, etc. • Typically have two unknowns. • Solve the equations simultaneously

47. Sample problem Two billiard balls move towards one another as in the figure. The balls have identical masses, and assume that the collision between them is perfect elastic. If the initial velocities of the balls are 30 cm/s and -20cm/s, what is the velocity of each ball after the collision?

48. Summary of Types of Collisions In an elastic collision, both momentum and kinetic energy are conserved. In an inelastic collision, momentum is conserved but kinetic energy is not. In a perfectly inelastic collision, momentum is conserved, kinetic energy is not, and the two objects stick together after the collision, so their final velocities are the same.

49. Final Notes About Conservation of Energy • We can neither create nor destroy energy. • The total energy of the universe must always remain the same. • Another way of saying energy is conserved. • If the total energy of the system does not remain constant, the energy must have crossed the boundary by some mechanism. • Applies to areas other than physics.

50. Power • Often also interested in the rate at which the energy transfer takes place. • Power is defined as this rate of energy transfer. • SI units for power are Watts (W).