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Physics 218 Mechanics

Instructor Dr. Tatiana Erukhimova

Lectures 22, 23, 24

Work-energy theorem

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 E1KE1U1

Final E2KE2U2

Then use

or

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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 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

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?

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 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

Total Energy is what is conserved! K1U1

K2U2EHeat

Force of gravity

Potential energy function

Spring

Potential energy function

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??

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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?

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?

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?

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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ENERGY DIAGRAMS

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)

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?

- 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?

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?

2 or 3D cases

If

or

then

Several dimensions U(x,y,z)

Partial derivative is taken assuming all other

arguments fixed

Compact notation using vector del, or nabla

http//reynolds.asu.edu/topo_gallery/topo_gallery.

htm

Given the potential energy function

find the x and y components of the corresponding

force.

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Have a great day! Reading Chapter 9 Hw

Chapter 9 problems and exercises