Download Lecture 13

Physics I
95.141
LECTURE 13
10/20/10
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Circular Motion Problem
• (A) What is the centripetal acceleration/Force of/on the
bullet? (5pts)
• (B) Where does this Force come from? (95 pts)
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Circular Motion Problem
• R~1300m
• mbullet=.03kg
• vbullet=300m/s
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
40m
0.15m
Lecture 12 Review
• Translational Kinetic Energy
KE 
1
mv 2
2
• Work Energy Theorem
– The net work done on an object corresponds to the
change in translational kinetic energy of that object
(as long as this energy does not go into internal
energy…compressed spring, for instance)
Wnet   W  KE 
1
1
mv 22  mv 12
2
2
• Conservative vs. Non-Conservative Forces
– The work done by a conservative force to move an
object from point A to B depends only on the position
of A and B, not path or velocity.
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Potential Energy
• In the last class we defined Energy as the ability to do
work.
• In particular, we discussed Translational Kinetic Energy,
the energy associated with motion.
• However, there are numerous other types of Energy
– We know batteries can do work
– We know a coiled spring can do work
– We know a mass at some height, attached to a pulley can do
work
• All of these are examples of systems that have the
potential to do work, and we can associate with them a
potential energy.
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Gravitational Potential Energy
• Say we start with a mass, and raise it, at
constant velocity, to a height h.
• How much work do we do?
• How much work does gravity do?
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Gravitational Potential Energy
• The block now has the potential to do work….
• Say we drop the brick, at y=0, we can find the
Kinetic Energy of the brick by the work-energy
theorem:
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Gravitational Potential Energy
• At y=0, the block can do an amount of work
equal to it’s kinetic energy
• Imagine the brick being used to drive a stake
into the ground:
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Gravitational Potential Energy
• Summary
– Raising the brick gives it the potential to do work, that
potential energy is given by:
– As the brick falls, its potential energy is converted into
kinetic energy
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Gravitational Potential Energy
• We assign the letter U to the gravitational
potential energy
• The change in gravitational potential energy is
then:
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Gravitational Potential Energy
• The gravitational potential energy is associated
with the Force between the Earth and the object.
• How do we determine what y is?
• It is the change in Potential Energy that we are
usually concerned with
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Example Problem
• A 1000kg roller coaster moves from point 1 to points 2 and 3.
• What is the potential energy of the roller coaster at points 2 and 3
relative to point 1?
• What is the change in potential energy from points 2 to 3?
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Example Problem
• A 1000kg roller coaster moves from point 1 to points 2 and 3.
• What is the potential energy of the roller coaster at points 2 and 3
relative to point 3?
• What is the change in potential energy from points 2 to 3?
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
General Potential Energy
• Gravitational potential energy is defined as:
– The negative of the work done by gravity when the
object moves from height y1 to y2.
• In general, we can define the change in potential
energy associated with a particular Force F as
the negative of the work done by that Force.
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Elastic Potential Energy
• What is the potential energy of a spring
compressed from equilibrium by a distance x?
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Relating Force and Potential Energy
• Say we are given a Force as a function of position
• We can then write the change in potential energy
associated with this (conservative) Force as:
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Example
• Suppose we are given the potential energy as:
U ( x)  Ax 2e bx
• What is the Force F as a function of x for this
potential energy?
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
3D Example
• In 3D
• So if

U ˆ U ˆ U ˆ
F ( x , y, z )  
i
j
k
x
y
z
z
U ( x , y, z )  3 xy  4
x
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Conservation of Energy
• For a conservative system (only conservative
forces do work) where energy is transformed
between kinetic and potential
• Work Energy Principle
• Relation between potential energy and work:
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Conservation of Energy
• Combining the work-energy principle and our
definition of potential energy, we see that:
• We can define the total mechanical energy of
the system as:
• We can then see that the total energy of the
system
• Is constant!
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Conservation of Energy
• As long as no non-conservative forces do work,
the total mechanical energy of the system is a
conserved quantity!
• Principle of conservation of mechanical energy:
– If only conservative forces are doing work, the total
mechanical energy of a system neither increases or
decreases in any process. It is conserved.
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Power of Energy Conservation
• Imagine dropping a mass m from a height h
above the ground.
• Solve for speed of the mass at the ground using
our equations of kinetic motion
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Power of Energy Conservation
• Imagine dropping a mass m from a height h
above the ground.
• Now solve for speed of the mass at the ground
using energy conservation
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Power of Energy Conservation
• But now, imagine sliding a mass m released
from rest on the frictionless track shown below.
• Solve for speed of the mass at the ground using
our equations of kinetic motion
h
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Power of Energy Conservation
• But now, imagine sliding a mass m released
from rest on the frictionless track shown below.
• Solve for speed of the mass at the ground using
conservation of energy
h
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Power of Energy Conservation
• We can even solve this if the mass is given an
initial velocity vo.
• Solve for speed of the mass at the ground using
conservation of energy
h
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Example Problem
• A 2 kg mass, starting at rest, slides down the frictionless
track shown below and into a spring with spring constant
k=250N/m. How far is the spring compressed by the
mass?
2m
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Example Problem (Easier)
• A 2 kg mass, starting at rest, slides down the frictionless
track shown below and into a spring with spring constant
k=250N/m. How far is the spring compressed by the
mass?
2m
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Example Problem
• A 2 kg mass, with an initial velocity of 5m/s, slides down
the frictionless track shown below and into a spring with
spring constant k=250N/m. How far is the spring
compressed by the mass?
2m
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Spring Energy
•
•
•
•
What is spring constant of catapult?
What is energy stored in spring?
what is Kinetic Energy of Watermelon?
What is velocity of watermelon?
• Assume
– MassWoman=65kg
– MassMelon=2kg
– Δx=1.5m
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Spring Energy
• What is spring constant of catapult?
1.5m
θ=30°
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Spring Energy
• Need to know Force!
• Free body diagram…
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
• With Force, we can
now find k!
Spring Energy
• What is energy stored in spring?
• What is Kinetic Energy of Watermelon?
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
Spring Energy
• What is velocity of watermelon?
• Assume
– MassMelon=2kg
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
What does Energy vs Time look like?
K
US
E
95.141, F2010, Lecture 13
Department of Physics and Applied Physics
What Did We Learn Today?
• Potential Energy
• Conservation of Mechanical Energy
• Concept of Energy Conservation is a powerful
way to approach what might seem like complex
problems!
95.141, F2010, Lecture 13
Department of Physics and Applied Physics