Download Potential Energy and Work Conservative/Non

Work and Energy
Conservative/Non-conservative
Forces
Important Concepts:
• The total amount of energy in the Universe is
conserved. This is a wonderful property of our
Universe: Energy can be transformed from one
type to another and transferred from one object to
another, but the total amount is always the same.
This is the principle of conservation of energy. No
exception to this rule has ever been found.
• The total amount of energy within any system is
conserved, as long as no energy enters or leaves
the system. It remains constant as long as the same
amount of energy is added and removed.
• Kinetic energy, K, is
associated with the
state of motion of an
object. When an
object is stationary, its
kinetic energy is zero.
1 2
K  mv
2
Potential energy, U, is energy associated with an
object’s position relative to another object or
the arrangement of a system of objects.
• Gravitational potential energy, UG, is the potential energy of
an object or system due to interactions of gravitational fields.
U G  mgh
GmM
UG  
r
*Why is it negative?
near the
surface
“farther afield”
Because U is 0 at infinity and gets larger and larger negative as
a mass moves closer to Earth.
Want a more complicated answer? F=-U, and the gravitational force is negative.
• Elastic potential
energy is described for
a system such as a
spring that has a
defined spring
constant, k, that is
extended a distance x
from a position of
equilibrium.
Fs  kx
1 2
U s  kx
2
• Electric potential
energy is described for
a charged particle q
near another charged
particle, Q:
kqQ
UE 
r
Mechanical energy includes kinetic
energy and potential energy.
• In the case of a falling object (neglecting air
friction), the gravitational potential energy is
converted to kinetic energy as the object falls—
maintaining constant total mechanical energy.
• If we do include air friction, the total mechanical
energy is not constant as the object falls, because
some of the mechanical energy has been converted
to thermal energy in the molecules of the object
and the air (which means a temperature increase
in both the object and the air).
The work done by any force on an
object or system is:
W  F s
•Work is a scalar quantity, measured in Joules, that is
positive if the force and displacement are in the same
direction and negative if the force and displacement are
in opposite directions.
•Positive work done on a system increases the
mechanical energy of the system.
•Negative work done on a system decreases the
mechanical energy of the system.
Example: As a ball rolls across a floor, its
gravitational potential energy does not
change, but its kinetic energy decreases as the
ball rolls across the floor. Explain this in
terms of conservation of energy.
The friction force on the ball as it rolls is in a direction
opposite the direction that the ball rolls. Thus, the dot
product of the friction force and the displacement is
negative. The friction force does negative work on the
ball, eventually decreases its kinetic energy to zero as it
converts that mechanical energy to thermal energy.
A box of mass m is pushed by a force F to the top of a
ramp of length d and height h. Determine the work done
by the force in pushing the box to the top of the ramp (a)
neglecting friction, and (b) including friction, with
coefficient .
h
d
F
First, add the forces on the box. Then use d and h to
determine the angle .
N
F
mg

Construct the components for mg.
N
F
mgcos
mg
mgsin

Method 1: Energy method for determining work with no friction
Work = Change in Potential Energy
W = U = mgh
N
F
mgcos
mg
mgsin

Method 2: Force method for determining work with
no friction.
W = Fs where F = mgsin and s = d
W = mgsind BUT dsin = ______
SO: W = _______
N
F
mgcos
mg
mgsin

Now we consider the same situation with friction.
N
Ff
F
mgcos
mg
mgsin

Method 1: Energy method with friction
W = Change in Potential Energy + Loss to Thermal Energy due to Friction
W = mgh + Ffd
W = mgh + mgcos
Ff
N
F
mgcos
mg
mgsin

Method 2: Force method of determining work with friction
W= F s where F = Ff + mgsin and s = d
W = mgcos d + mgsin d
W = ____ + _______
Ff
N
F
mgcos
mg
mgsin

A Combination Pendulum
Motion
A pendulum of length L is released from angle .
When it swings to vertical, it hits a rod that is
perpendicular to the plane of the swing (i.e., it
projects out of the page) and positioned at ½ L.

Find the angle to which the pendulum will swing
after hitting the bar.

?


Consider the initial potential energy of the pendulum
bob and the final kinetic energy of the bob.


mgL(1 - cos) = mg(1/2 L)(1 - cos )
1 - cos = (1/2) (1 - cos )
½ = cos - ½ cos 

