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## Physics 218: Mechanics

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### Physics 218: Mechanics * * Work-energy theorem: Mechanical energy is conserved! Examples Strategy: ... Good example: Friction (like on Roller Coasters) ... – PowerPoint PPT presentation

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Title: Physics 218: Mechanics

1
Physics 218 Mechanics
Instructor Dr. Tatiana Erukhimova
Lectures 22, 23, 24
2
Work-energy theorem
Mechanical energy is conserved!
3
Examples
Strategy write down the total mechanical energy,
E, E KE U at the initial and final
positions of a particle
4
Initial E1KE1U1
5
Final E2KE2U2
6
Then use
or
7
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8
Water Slide
Who hits the bottom with a faster speed?
9
Roller Coaster
• You are in a roller coaster car of mass M that
starts at the top, height H, with an initial
speed V00. Assume no friction.
• What is the speed at the bottom?
• How high will it go again?
• Would it go as high if there were friction?

H
10
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 gt y2).
• Assuming we dont know anything about the
friction or the path, how much work is done by
friction on this path?

11
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)

12
Law of Conservation of Energy
• Mechanical Energy NOT always conserved
• If youve 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! K1U1
K2U2EHeat

13
Total Energy is what is conserved! K1U1
K2U2EHeat
14
Force of gravity
Potential energy function
Spring
Potential energy function
15
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??
16
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17
A gun shoots a bullet at angle ? 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 will it go?
18
Quiz
A block of mass m is placed against a vertical
spring, spring constant k. The spring is
unstretched at y0.
A
If the spring is compressed an amount A and the
block released from rest, how high will it go?
19
Quiz
A block of mass m is attached to a vertical
spring, spring constant k. The spring is
unstretched at y0.
A
If the spring is compressed an amount A and the
block released from rest, how high will it go?
20
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
y0
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22
ENERGY DIAGRAMS
23
Potential Energy Diagrams
• For Conservative forces can draw energy diagrams
• Equilibrium points
• Motion will move around the equilibrium
• If placed there with no energy, will just stay
(no force)

24
Stable vs. Unstable Equilibrium Points
• The force is zero at both maxima and minima but
• If I put a ball with no velocity there would it
stay?
• What if it had a little bit of velocity?

25
• A particle moves along the x-axis while acted on
by a single conservative force parallel to the
x-axis. The force corresponds to the
potential-energy function graphed in the Figure.
The particle is released from rest at point A.
• What is the direction of the force on the
particle when it is at point A?
• At point B?

26
c) At what value of x is the kinetic energy of
the particle a maximum? d) What is the force on
the particle when it is at point C?
e) What is the largest value of x reached by the
particle during its motion? f) What value or
values of x correspond to points of stable
equilibrium? g) Of unstable equilibrium?
27
2 or 3D cases
If
or
then
28
Several dimensions U(x,y,z)
Partial derivative is taken assuming all other
arguments fixed
Compact notation using vector del, or nabla
29
http//reynolds.asu.edu/topo_gallery/topo_gallery.
htm
30
Given the potential energy function
find the x and y components of the corresponding
force.
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32
Have a great day! Reading Chapter 9 Hw
Chapter 9 problems and exercises