Download The work-energy theorem

Lecture PowerPoint
Physics for Scientists and
Engineers, 3rd edition
Fishbane
Gasiorowicz
Thornton
© 2005 Pearson Prentice Hall
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Chapter 6
Work and Kinetic Energy
Main Points of Chapter 6
• Kinetic energy and the work-energy theorem
• Work in more than one dimension
• Variable forces
• Conservative and nonconservative forces
• Power
• Kinetic energy at very high speeds
6-1 Kinetic Energy, Work, and the
Work–Energy Theorem
Definition of kinetic energy:
(6-2)
Q. Why do we define this?
A. It turns out to make our
calculations easier!
6-1 Kinetic Energy, Work, and the
Work–Energy Theorem
The work-energy theorem: the net
work done on an object is equal to
its change in kinetic energy
(6-6)
6-1 Kinetic Energy, Work, and the
Work–Energy Theorem
Q. So, why does defining kinetic
energy make calculations easier?
A. Kinetic energy is a scalar and
can be simply added. Work is a
scalar too but it’s a dot product –
you need more information to
calculate it.
6-1 Kinetic Energy, Work, and the
Work–Energy Theorem
• Work may be positive or negative.
• If force has a component in the
direction of the velocity, work is
positive.
6-1 Kinetic Energy, Work, and the
Work–Energy Theorem
• Work may be positive or negative.
• If force has a component
opposite to the direction of the
velocity, work is negative.
6-1 Kinetic Energy, Work, and the
Work–Energy Theorem
Units of work and energy:
1 joule (J) = 1
6-1 Kinetic Energy, Work, and the
Work–Energy Theorem
Work Done by Individual Forces
For each force,
Therefore, one can find the net work done
in two ways:
1. Find the vector sum of the forces and
multiply it by the displacement, or
2. Find the work done by each individual
force and add them.
6-2 Constant Forces in More than
One Dimension
• Each Cartesian direction
can be treated separately
(x, y, z)
• Now the angle between
the force and the
displacement needs to be
used
6-2 Constant Forces in More than
One Dimension
Component equations for FΔr:
(6-8a)
(6-8b)
Combining,
(6-9,11)
6-2 Constant Forces in More than
One Dimension
This leads to the general
expression for work done by a
constant force:
(6-12)
6-3 Forces that Vary with Position
Variable forces in one dimension:
• Divide displacement into
small enough pieces that the
force is almost constant
• Do the usual calculation of
work for each piece
• Add them all up
6-3 Forces that Vary with Position
If you know the force as a function of
displacement, you can make the pieces
infinitesimally small and integrate.
(6-14)
6-3 Forces that Vary with Position
Work Done by a Spring
Spring force:
(6-15)
• k is the spring constant
• restoring force – always in the
direction of x = 0
• force varies with x
6-3 Forces that Vary with Position
Work Done by a Spring
Integrating gives the work done to
stretch the spring from 0 to L:
6-3 Forces that Vary with Position
Forces that Vary in both
Magnitude and Direction
• Now need to combine both the
previous results
• magnitude variation means
integration
• direction variation means dot
product
6-3 Forces that Vary with Position
So we need to calculate the work
at every point on the path:
(6-16)
Now you see why using the work-energy
theorem is so much easier!
6-3 Forces that Vary with Position
No Work Is Done in Uniform Circular Motion
Remember that:
(6-16)
At every point on a circle:
Therefore, no work!
6-4 Conservative and Nonconservative
Forces
• A conservative force is
one where the work done
does not depend on the
path taken.
• Gravity is a conservative
force.
6-4 Conservative and Nonconservative
Forces
• A nonconservative force is one where the
work done depends on the path taken.
• Friction is a nonconservative force.
6-5 Power
Power P is the rate at which work is done:
(6-18)
This result can be generalized:
(6-20)
6-5 Power
The units of power are watts:
1 W = 1 J/s
In the English system, power
is measured in horsepower:
1 hp = 550 ft-lb/s = 746 W
The kilowatt-hour, power
multiplied by time, is a
measure of energy use:
1 kWh = 3.6 x 106 J
6-6 Kinetic Energy at Very High Speeds
• When objects approach the
speed of light, many strange
things happen.
• One is that the kinetic energy can
become very large.
(6-21)
• The speed of light c = 3 x 108 m/s
• As v approaches c, K approaches
infinity
Summary of Chapter 6
• Definition of kinetic energy:
• Work-energy
theorem:
• For more than one dimension, each
coordinate must be calculated separately.
• Work done by variable forces is found by
integrating.
(6-6)
Summary of Chapter 6, cont.
• Conservative force: work does not
depend on path
• Nonconservative force: work depends on
path
• Power is the rate at which work is done