Chatper 8 - Potential energy and
conservation of energy
Chapter 8 - Potential
energy and conservation
of energy
Objectives (Ch 8)
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
External Forces
David J. Starling
Penn State Hazleton
Fall 2013
Objectives (Ch 8)
Objectives for Chapter 8
(a) Recognize potential energy as a property of a system, not an
isolated object, and that it is definable only for a
conservative force.
(b) Calculate the gravitational potential energy, elastic potential
energy and kinetic energy of a system.
(c) Determine the role of dissipative forces in changing the
energy of a system.
(d) Interpret one-dimensional potential energy diagrams and
determine the force, turning points, kinetic energy, speed,
stable and unstable equilibria.
(e) Use conservation of energy to analyze closed systems and
determine the energy transfer in/out of systems that are not
closed.
Chapter 8 - Potential
energy and conservation
of energy
Objectives (Ch 8)
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
External Forces
Objectives (Ch 8)
Chapter 8 - Potential
energy and conservation
of energy
Objectives (Ch 8)
A mountain climber pulls a supply pack up the side of a mountain
at constant speed. Which one of the following statements
concerning this situation is false?
(a) The net work done by all the forces acting on the pack is
zero joules.
(b) The work done on the pack by the normal force of the
mountain is zero joules.
(c) The work done on the pack by gravity is zero joules.
(d) The gravitational potential energy of the pack is increasing.
(e) The climber does "positive" work in pulling the pack up the
mountain.
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
External Forces
Conservative vs. Non-conservative Forces
Chapter 8 - Potential
energy and conservation
of energy
Work W is how energy is transferred to or from a
Objectives (Ch 8)
system.
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
External Forces
Conservative vs. Non-conservative Forces
Forces can be split into two categories known as
conservative and non-conservative.
Chapter 8 - Potential
energy and conservation
of energy
Objectives (Ch 8)
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
External Forces
In one case, energy is “lost.”
Conservative vs. Non-conservative Forces
Consider:
I
Two or more objects (e.g., earth + box)
I
A force between them (e.g., mg)
I
One object moves and work W1 is done (lift box up)
I
The object returns and more work is done W2 (set box
Chapter 8 - Potential
energy and conservation
of energy
Objectives (Ch 8)
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
down)
If W1 = −W2 is always true, no net work was
done and that force is conservative.
External Forces
Conservative vs. Non-conservative Forces
Chapter 8 - Potential
energy and conservation
of energy
Objectives (Ch 8)
The net work done by a conservative force on a
Conservative vs.
Non-conservative Forces
particle moving around any closed path is zero.
Work and Potential
Energy
Conservation of Energy
External Forces
Equivalently: The net work done by a conservative force on
a particle moving from point a to point b is independent of
the path.
Conservative vs. Non-conservative Forces
Chapter 8 - Potential
energy and conservation
of energy
Objectives (Ch 8)
Conservative: Gravity, Spring, Electric Force
Non-conservative: Friction, Air Drag
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
External Forces
Conservative vs. Non-conservative Forces
Chapter 8 - Potential
energy and conservation
of energy
Objectives (Ch 8)
Example 1: A 2.0 kg block of slippery cheese slides down a
frictionless track from point a to point b, 2.0 m to the right
and 0.80 m down. How much work is done on the cheese by
the gravitational force during the slide?
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
External Forces
Work and Potential Energy
Potential Energy is a form of energy associated
with a conservative force between a system of
objects.
Chapter 8 - Potential
energy and conservation
of energy
Objectives (Ch 8)
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
External Forces
The spring force is conservative; it stores potential energy
and then releases it.
Work and Potential Energy
Potential Energy U is a form of energy
associated with a conservative force between a
system of objects.
Chapter 8 - Potential
energy and conservation
of energy
Objectives (Ch 8)
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
External Forces
The gravitational force is conservative; it stores potential
energy and then releases it when the object is dropped.
Chapter 8 - Potential
energy and conservation
of energy
Work and Potential Energy
Objectives (Ch 8)
For a conservative force, the change in Potential
Conservative vs.
Non-conservative Forces
Energy ∆U is defined as minus the work done by
Work and Potential
Energy
that conservative force.
Conservation of Energy
External Forces
Z
xf
∆U = −W = −
F(x)dx
xi
Example: If you lift an object, gravity does negative work,
so ∆U > 0.
Work and Potential Energy
Chapter 8 - Potential
energy and conservation
of energy
Objectives (Ch 8)
Example 2: A particle of mass m moves along the y-axis.
As the particle moves from y = yi to yf , the gravitational
force does work. Find the change in potential energy.
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
External Forces
Work and Potential Energy
Chapter 8 - Potential
energy and conservation
of energy
Objectives (Ch 8)
Conservative vs.
Non-conservative Forces
The change in gravitational potential energy of an
Work and Potential
Energy
object near Earth’s surface is
Conservation of Energy
External Forces
∆U = mg(∆y).
Note: only the change is important!
Work and Potential Energy
Example 3: A block of mass m is attached to a spring with
spring constant k. As the block moves from x = xi to xf
along the x-axis, the conservative force from the spring is
Fx = −kx. Find the change in potential energy in the
block-spring system.
Chapter 8 - Potential
energy and conservation
of energy
Objectives (Ch 8)
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
External Forces
Work and Potential Energy
Chapter 8 - Potential
energy and conservation
of energy
Objectives (Ch 8)
Conservative vs.
Non-conservative Forces
The change in spring potential energy is
Work and Potential
Energy
Conservation of Energy
1
1
∆U = kxf2 − kxi2 .
2
2
Note again: only the change is important!
External Forces
Work and Potential Energy
Example 4: A 2.0 kg sloth hangs from a tree. Find the
gravitational potential energy of the sloth using the
reference point
Chapter 8 - Potential
energy and conservation
of energy
Objectives (Ch 8)
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
External Forces
(a) at the ground.
(b) at the balcony floor.
(c) at the limb.
(d) at 1.0 m above the limb.
Work and Potential Energy
Chapter 8 - Potential
energy and conservation
of energy
Potential energy and conservative forces are
Objectives (Ch 8)
related through a derivative/integral (by
definition).
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
Z
∆U = −W = −
F(x) = −
F(x)dx ≈ −F(x)∆x
∆U
dU
→ F(x) = −
∆x
dx
External Forces
Conservation of Energy
Chapter 8 - Potential
energy and conservation
of energy
The mechanical energy of a system is the sum of
Objectives (Ch 8)
its kinetic and potential energies.
Emec = K + U
∆Emec = ∆K + ∆U
If a system has only conservative forces:
∆U = −W
∆K = W (last chapter)
∆U = −∆K
∆K + ∆U = 0
∆Emec = 0
Mechanical energy is conserved but can transform from one
type (K or U) to another.
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
External Forces
Conservation of Energy
Chapter 8 - Potential
energy and conservation
of energy
A pendulum is a good example of conservation of
Objectives (Ch 8)
mechanical energy.
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
External Forces
Conservation of Energy
Chapter 8 - Potential
energy and conservation
of energy
Objectives (Ch 8)
Example 5: A child of mass m is released from rest at the
top of a 8.5 m high water slide. Assuming the slide is
frictionless, what is the speed of the child at the bottom?
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
External Forces
Conservation of Energy
Chapter 8 - Potential
energy and conservation
of energy
Objectives (Ch 8)
A roller coaster car travels down a hill and is moving at 18
m/s as it passes through a section of straight, horizontal
track. The car then travels up another hill that has a
maximum height of 15 m. If frictional effects are ignored,
determine whether the car can reach the top of the hill. If it
does reach the top, what is the speed of the car at the top?
(a) No, the car doesn’t make it up the hill.
(b) Yes, the car just barely makes it to the top and stops.
(c) Yes, and the car is moving at 5.4 m/s at the top.
(d) Yes, and the car is moving at 9.0 m/s at the top.
(e) Yes, and the car is moving at 18 m/s at the top.
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
External Forces
Conservation of Energy
Chapter 8 - Potential
energy and conservation
of energy
An object subjected to a conservative force may
Objectives (Ch 8)
have the following potential energy curve.
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
External Forces
Conservation of Energy
Chapter 8 - Potential
energy and conservation
of energy
What happens if we place an object at rest at each
Objectives (Ch 8)
of the 5 points shown?
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
External Forces
Conservation of Energy
Chapter 8 - Potential
energy and conservation
of energy
Objectives (Ch 8)
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
External Forces
I
x1 : falls to the right (unstable)
I
x2 : sits in place and is restored if displaced (stable
equilibrium)
I
x3 : sits in place and is falls if displaced (unstable
equilibrium)
I
x4 : stable equilibrium
I
x2 : unstable equilibrium
Conservation of Energy
Example 6: A 2.0 kg particle moves in one dimension and
is acted on by a conservative force. The potential energy of
the system is plotted below. At x = 7.00 m, the particle has
a velocity of −4.00 m/s.
Chapter 8 - Potential
energy and conservation
of energy
Objectives (Ch 8)
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
External Forces
(a) What is the particle’s speed at x = 4.5 m?
(b) Where is the particle’s turning point?
(c) Find the force on the particle from 1.9 < x < 4.0 m.
Conservation of Energy
Chapter 8 - Potential
energy and conservation
of energy
Objectives (Ch 8)
The total energy in a system includes mechanical
energy, thermal energy and internal energy.
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
E = K + U + Eth + Eint
The total energy in an isolated system is conserved:
∆E = ∆K + ∆U + ∆Eth + ∆Eint = 0.
External Forces
Conservation of Energy
Internal energy can be transferred to kinetic
energy, which can the be transferred to thermal
energy.
Chapter 8 - Potential
energy and conservation
of energy
Objectives (Ch 8)
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
External Forces
Conservation of Energy
Chapter 8 - Potential
energy and conservation
of energy
Objectives (Ch 8)
A 65 kg hiker eats a 250 calorie granola bar. Assuming the
body converts this snack with an efficiency of 25%, what
change of altitude could this hiker achieve by hiking up the
side of a mountain before completely using the energy in the
snack? (one food calorie is equal to 4186 joules)
(a) 270 m
(b) 410 m
(c) 650 m
(d) 880 m
(e) 1600 m
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
External Forces
Chapter 8 - Potential
energy and conservation
of energy
External Forces
An external force can supply energy to a system.
Objectives (Ch 8)
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
External Forces
The lifting force supplies energy:
Wlift = ∆K + ∆U = ∆Emech
(postive work!)
External Forces
An external force can supply energy to a system.
Chapter 8 - Potential
energy and conservation
of energy
Objectives (Ch 8)
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
External Forces
The pushing force supplies energy and friction sucks it
away:
Wlift + Wfriction = ∆K + ∆U = ∆Emech
External Forces
Chapter 8 - Potential
energy and conservation
of energy
Objectives (Ch 8)
Example 7: A 14 kg crate is pushed along a concrete floor
with a force of 40 N for 0.50 m. The speed decreases from
0.60 m/s to 0.20 m/s.
Conservative vs.
Non-conservative Forces
Work and Potential
Energy
Conservation of Energy
(a) How much work is done by the pushing force?
(b) What is the increase in thermal energy of the crate and
floor during the push?
External Forces