Chapter 10 Work and Energy

Units of Chapter 10 Work Done by a Constant Force Work Done by a Variable Force The Work–Energy Theorem: Kinetic Energy Potential Energy Conservation of Energy Power

10.1 Work Done by a Constant Force Definition of work: The work done by a constant force acting on an object is equal to the product of the magnitudes of the displacement and the component of the force parallel to that displacement.

10.1 Work Done by a Constant Force In (a), there is a force but no displacement: no work is done. In (b), the force is parallel to the displacement, and in (c) the force is at an angle to the displacement.

Read the following five statements and determine whether or not they represent examples of work. A teacher applies a force to a wall and becomes exhausted. A book falls off a table and free falls to the ground. A waiter carries a tray full of meals above his head by one arm straight across the room at constant speed. (Careful! This is a very difficult question which will be discussed in more detail later.) A rocket accelerates through space. Mathematically, work can be expressed by what equation?

No. This is not an example of work. The wall is not displaced No. This is not an example of work. The wall is not displaced. A force must cause a displacement in order for work to be done. Yes. This is an example of work. There is a force (gravity) which acts on the book which causes it to be displaced in a downward direction (i.e., "fall"). No. This is not an example of work. There is a force (the waiter pushes up on the tray) and there is a displacement (the tray is moved horizontally across the room). Yet the force does not cause the displacement. To cause a displacement, there must be a component of force in the direction of the displacement.

Yes. This is an example of work Yes. This is an example of work. There is a force (the expelled gases push on the rocket) which causes the rocket to be displaced through space.

10.1 Work Done by a Constant Force If the force is at an angle to the displacement, as in (c), a more general form for the work must be used: Unit of work: newton • meter (N • m) 1 N • m is called 1 joule.

10.1 Work Done by a Constant Force If the force (or a component) is in the direction of motion, the work done is positive. If the force (or a component) is opposite to the direction of motion, the work done is negative.

10.1 Work Done by a Constant Force If there is more than one force acting on an object, it is useful to define the net work: The total, or net, work is defined as the work done by all the forces acting on the object, or the scalar sum of all those quantities of work.

10.2 Work Done by a Variable Force The force exerted by a spring varies linearly with the displacement:

10.2 Work Done by a Variable Force A plot of force versus displacement allows us to calculate the work done:

10.3 The Work–Energy Theorem: Kinetic Energy The net force acting on an object causes the object to accelerate, changing its velocity:

10.3 The Work–Energy Theorem: Kinetic Energy We can use this relation to calculate the work done:

10.3 The Work–Energy Theorem: Kinetic Energy Kinetic energy is therefore defined: The net work on an object changes its kinetic energy.

10.3 The Work–Energy Theorem: Kinetic Energy This relationship is called the work–energy theorem.

10.4 Potential Energy Potential energy may be thought of as stored work, such as in a compressed spring or an object at some height above the ground. Work done also changes the potential energy (U) of an object.

10.4 Potential Energy We can, therefore, define the potential energy of a spring; note that, as the displacement is squared, this expression is applicable for both compressed and stretched springs.

10.4 Potential Energy Gravitational potential energy:

5.4 Potential Energy Only changes in potential energy are physically significant; therefore, the point where U = 0 may be chosen for convenience.

5.5 Conservation of Energy We observe that, once all forms of energy are accounted for, the total energy of an isolated system does not change. This is the law of conservation of energy: The total energy of an isolated system is always conserved. We define a conservative force: A force is said to be conservative if the work done by it in moving an object is independent of the object’s path.

5.5 Conservation of Energy So, what types of forces are conservative? Gravity is one; the work done by gravity depends only on the difference between the initial and final height, and not on the path between them. Similarly, a nonconservative force: A force is said to be nonconservative if the work done by it in moving an object does depend on the object’s path. The quintessential nonconservative force is friction.

5.5 Conservation of Energy Another way of describing a conservative force: A force is conservative if the work done by it in moving an object through a round trip is zero. We define the total mechanical energy:

5.5 Conservation of Energy For a conservative force: Many kinematics problems are much easier to solve using energy conservation.

5.5 Conservation of Energy All three of these balls have the same initial kinetic energy; as the change in potential energy is also the same for all three, their speeds just before they hit the bottom are the same as well.

5.5 Conservation of Energy In a conservative system, the total mechanical energy does not change, but the split between kinetic and potential energy does.

5.5 Conservation of Energy If a nonconservative force or forces are present, the work done by the net nonconservative force is equal to the change in the total mechanical energy.

5.6 Power The average power is the total amount of work done divided by the time taken to do the work. If the force is constant and parallel to the displacement,

5.6 Power Mechanical efficiency: The efficiency of any real system is always less than 100%.

5.6 Power

Review of Chapter 5 Work done by a constant force is the displacement times the component of force in the direction of the displacement. Kinetic energy is the energy of motion. Work–energy theorem: the net work done on an object is equal to the change in its kinetic energy. Potential energy is the energy of position or configuration.

Review of Chapter 5 The total energy of the universe, or of an isolated system, is conserved. Total mechanical energy is the sum of kinetic and potential energy. It is conserved in a conservative system. The net work done by nonconservative forces is equal to the change in the total mechanical energy. Power is the rate at which work is done.