CHAPTER 8: Potential Energy and Conservation of Energy
Matthew Bardeen, Alex Shtam, Sebastian Herrera, Hyun Nam Kim,
Kevin Rudd
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*Denotes problem that may be on the Exam |
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Potential Energy:
- A force is Conservative when the net work done to a particle moving it along
a closed path, and back to point zero, is Zero
- Potential Energy
: When a conservative force acts does work W on a
particle, the change D U in potential energy is : DU
= -W
- Gravitational Potential Energy
: If a particle moves from height yi
to height yf , the change D
U in gravitational potential energy is : DU = mg(yf
- yi) = mg D y
If yi = 0 and the initial Potential Energy Ui
= 0 then gravitational potential energy U when the particle is at height y is: U
= mgy
- Elastic Potential Energy
is the energy associated with state of
compression or extension of an elastic object. If a spring exerts a force F =
-kx then elastic potential energy U at position x is:
U(x) = kx2
Mechanical Energy:
- Mechanical Energy E is the sum of kinetic energy K and potential energy U
E = K + U
- Conservation of Mechanical Energy:
mechanical energy is always constant
if the work done to a system is only done by conservative forces:
DE = DK
+ DU = 0
Reading a Potential Energy Curve:
- Finding the force analytically
Given the change in potential energy we can solve for the force by:
U(x)=-W ==> -F(x)dx=-W ==> F(x)=-W/dx ==> F(x)=-dU(x)/dx.
The potential energy curve
Given the graph of the potential energy curve, taking the slope at various
points gives you the curve of the force that acts on that particle.
Turning points
The turning points on a potential energy curve occurs when the potential
energy is equal to the total energy.(in other words, the kinetic energy is zero
so the particle is momentarily at rest.)
Equilibrium points
Neutral equilibrium: K=0, the sum of the forces =0.
(ex. a marble on a table)
Unstable equilibrium: K=0. If the sum of the forces is not zero then the
particle will continue to move in the direction of the applied force.
(ex. a marble on a bowling ball)
Stable equilibrium:
(ex. a marble in a bowl)
***This section is pretty straightforward, the hardest parts
of these problems is understanding what they are asking you for. So when doing
the problems look in section 8-5 and see if there is another way of stating the
problem so that it is obvious to you what they are asking for.
Work Done By Nonconservative Forces:
When a nonconservative force does work on an object, the
mechanical energy of the system changes. We will examine two types of
nonconservative forces: an applied force, and a kinetic frictional
force.
Work Done by an Applied Force
If we have a body with only two forces acting on it, those being an applied
force, and the other being its weight, then from the Work-Kinetic Energy theorem
we can say that
Wapp + Wg = D
K
Wg = - D U thus:
Wapp = D E
Work is then an energy transferred to or from a system via a force.
Work Done by a Kinetic Frictional Force
Let’s say we have a block sliding a distance d down a
non-frictionless surface. We know that in this case all of its kinetic energy
is dissipated by kinetic frictional force (fk). We know that D
K = F· d = Fdcosq , in
this case frictional force is acting in the direction opposite to the
displacement thus
D E = -fkd
(dissipated mechanical energy)
- Without additional information we cannot determine Wf since
it equals the portion of the dissipated energy that was transferred
from the block to the floor.
Conservation of Energy:
- In an Isolated System, energy can be transferred from one type to another,
but the total energy of the system remains the same.
D ETOT
= D K + D
U + D EINT = 0
- Energy can not magically appear or disappear
- New Average Power Formula P = D E
/ D T
Tips and Tricks when Solving Kinetic and Potential energy probloms:
- If energy is conserved with in a system, that means the total energy
within the system is always the same.
- When the object is not moving, there is no kinetic energy. For example, in
a problem where an object is dropped onto a spring, the potential energy
stored in the spring once the object stops moving is equal to the potential
energy of the object before it is dropped.
- In Non-conservative systems, the work done due to outside forces is equal
to the total energy change. Friction is an example of one of these forces.
- Sometimes you'll find problems that have a combination of both
frictionless surfaces and surfaces that have friction. It helps to envision
these as two separate problems, one of a system where energy is conserved,
and one of a system where energy is not conserved.
Links:
This link gives a VERY simple definition / example of a conservation of
energy problem.
This link shows the different Potential Energy Equations and for they are
related to specific conservative forces (litte messy)
Very good definition of Conservation of Mechanical Energy. Also has 4
sample problems
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