Transcript Mechanical energy is conserved!

Physics 218: Mechanics
Instructor: Dr. Tatiana Erukhimova
Lecture 15
Quiz
This is a one-dimensional problem. Suppose a
particle is attracted to the origin with a force
k
Fx   3
x
Find the potential function.
Work-energy theorem:
total
1 2
W
W
con
1 2
W
nc
1 2
 U 2  U1   W
nc
1 2
nc
1 2
W
nc
1 2
If W
 KE2  KE1
 KE2  KE1
 KE2  U 2  KE1  U1
 0, KE2  U 2  KE1  U1  const
Mechanical energy is conserved!
Examples
Strategy: write down the
total mechanical energy, E,
E = KE + U
at the initial and final
positions of a particle:
Initial E1=KE1+U1…
Final E2=KE2+U2
Then use
nc
1 2
If W
 0, KE2  U 2  KE1  U1
or
nc
1 2
W
 KE2  U 2  KE1  U1
H
Water Slide
Who hits the bottom with a faster speed?
Roller Coaster
You are in a roller coaster car of mass M that
starts at the top, height H, with an initial
speed V0=0. Assume no friction.
a) What is the speed at the bottom?
b) How high will it go again?
c) Would it go as high if there were friction?
H
Roller Coaster with Friction
A roller coaster of mass m starts at rest at height
y1 and falls down the path with friction, then
back up until it hits height y2 (y1 > y2).
Assuming we don’t know anything about the
friction or the path, how much work is done by
friction on this path?
Conservative Forces
If there are only conservative forces in the
problem, then there is conservation of
mechanical energy
• Conservative: Can go back and forth along any
path and the potential energy and kinetic energy
keep turning into one another
– Good examples: Gravity and Springs
• Non-Conservative: As you move along a path,
the potential energy or kinetic energy is turned
into heat, light, sound etc… Mechanical energy
is lost.
– Good example: Friction (like on Roller Coasters)
Law of Conservation of Energy
• Mechanical Energy NOT always
conserved
• If you’ve ever watched a roller
coaster, you see that the friction turns
the energy into heating the rails,
sparks, noise, wind etc.
• Energy = Kinetic Energy + Potential
Energy + Heat + Others…
–Total Energy is what is
conserved! K1+U1 = K2+U2+EHeat…
Total Energy is
what is conserved!
K1+U1=
K2+U2+EHeat…
A gun shoots a bullet at angle θ with the x axis
with a velocity of magnitude Vm. What is
magnitude of the velocity when the bullet
returns to the ground? How high it will go?
Spring problem revisited
A block of mass M is on a horizontal surface and is attached to
a spring, spring constant k. If the spring is compressed an
amount A and the block released from rest, how far from
unstretched position will it go before stopping if there is no
friction between the block and the surface?
How will this answer change is the block is not attached
to the spring??
Block of mass m has a spring connected to the
bottom. You release it from a given height H
and want to know how close the block will get
to the floor. The spring has spring constant k
and natural length L.
H
y=0