General Physics I: Day 16Elastic Potential Energy & Nonconservative Forces

Conservative Forces & Potential Energy

Conservative forces can “store” energy. Technically defined in terms of “path-independent” work.

ALL forces conserve energy. Conservative forces also conserve mechanical energy.

Potential energy:

• Only associated with conservative forces

• Potential energy is associated with a system

• Depends on an arbitrary (usually) reference point

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3WarmUp: Gravity vs. FrictionA trunk of mass m is lifted along a curved path of length L to a height h. Another trunk with twice the mass is slid across a level floor () along a curved path also having length L. Which is greater, the work done against friction or the work done against gravity?

~51% → More work is done against friction.

~27% → More work is done against gravity.

~16% → The work done against friction is the same as the work done against gravity.

~6% → Cannot be determined from the given information.

Calculating Potential Energy

For conservative forces, potential energy equals how much work must be done against that force to achieve a certain configuration.

For gravity, to move an object from yi to yf:

Thus, gravitational potential energy:

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for a force done against force done by forceU W W

against grav.W mg y against i grav.o r, if =0 y W mgy

gU mgy

Cons. of Mech. Energy (Gravity only)

We define mechanical energy:

For an isolated system with only conservative forces, the total mechanical energy cannot change!

This means that any potential energy gained comes from kinetic energy lost, and vice versa.

or

or

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.mechE U K

U K

mech., initial mech., finalE E

i i f fU K U K

Applying Conservation of Mech. Energy

If you decide to use conservation of energy (or of mechanical energy) you must

• Choose your system!

• Decide on initial and final situations

–One should be as simple as possible, or should be one you know a lot about

–The other should be what you are trying to learn about

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Warm-Up: Work & PE

You throw a ball straight up in the air. On its way up is the work done by gravity positive or negative? Is the change in potential energy positive or negative during this same period?

~10% → Positive W, negative ΔUg

~0% → Positive W, positive ΔUg

~65% → Negative W, positive ΔUg

~15% → Negative W, negative ΔUg

~15% → Didn’t answer both parts… lost a point.

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Warm-Up: Work & PE

“Potential energy is positive, the higher something is the more kinetic energy that can be produced The work is negative, due to the fact that the force of gravity is opposing the current motion of the ball”

Some missed the word “change”!

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Warm-Up: Work & PE

“The work done by gravity is negative, that is because the work is done in the opposite direction of motion. The change in potential energy is positive, because the gravitational potential energy of the ball increases as the height increases.”

“The change in potential energy is positive but since work= -change in potential energy. Work done by gravity is negative.”

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Two marbles, one twice as heavy as the other, are dropped to the ground from the roof of a building. Just before hitting the ground, the heavier marble has

A) as much kinetic energy as the lighter one.

B) twice as much kinetic energy as the lighter one.

C) half as much kinetic energy as the lighter one.

D) four times as much kinetic energy as the lighter one.

E) Cannot be determined from what is given.

A coconut is thrown at a speed of 8 m/s from the top of a coconut tree and lands on the ground. Rank the following directions in order of which will result in the largest speed when the coconut hits the ground.

1) Almost straight up 2) 45° above horizontal

3) Horizontal 4) 45° below horizontal

5) Straight down

A) 1 > 2 > 3 > 4 > 5 B) 1 = 5 > 2 = 4 > 3

C) 1 = 2 = 3 = 4 = 5 D) 5 > 4 > 3 > 2 > 1

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Energy: The easy way!

A block slides down a frictionless ramp, starting out 1.5 meters above the ground (height).

How fast is it going at the bottom of the ramp?

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Two balls will be released from rest at the top of the apparatus shown. Ball 1 travels on the straight track, while ball 2 travels on the bent track. If we measure the speed of each ball as it leaves the track, what will we find?

A) Ball 1 will have a larger final speed.

B) Ball 2 will have a larger final speed.

C) They will have the same final speed.

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Two balls will be released from rest at the top of the apparatus shown. Ball 1 travels on the straight track, while ball 2 travels on the bent track. If they are released at the same time, which ball will reach the end first?

A) Ball 1 will win the race!

B) Ball 2 will win the race!

C) They will tie!

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Sample Problem

A Hot Wheels car will roll down a track as shown on the board. Assume there is no friction in the motion. From what height should the car be released so that it is going 0.8 m/s when it reaches point B which is 11 cm above the bottom of the track?

An aside: On a homework question you have to figure out how fast something must go to make it through a loop-the-loop…

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16Worked-Example: Spring Ball Launch

A vertical spring with a spring constant k = 150 N/m is compressed down 1.5 m. A 2-kg ball is placed on the compressed spring and released from rest. What height does the ball reach after it is released?

17Worked-Example: Spring Ball Launch

18Worked-Example: Spring Ball Launch

19Worked-Example: Spring Ball Launch

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Elastic Potential Energy

Since

Where x is measured from the relaxed position.

x

F

21by spr. 2W kx

F=kx

for a force done by that forceU W 212springU kx

Difficult/Interesting

“I thought it was interesting that unlike gravitational potential energy, for elastic potential energy we can't just choose x to be 0. It has to be at the point where the spring is neither compressing or stretching.”

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Cons. of Mech. Energy (Grav. + Elast.)

We define mechanical energy:

For an isolated system with only conservative forces, the total mechanical energy cannot change!

This means that any potential energy gained comes from kinetic energy lost, and vice versa.

or

or

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.mechE U K

U K

mech., initial mech., finalE E

i i f fU K U K

Sample Problem

A block (m = 4.5 kg) slides, from rest, down a 12 meter frictionless ramp, which is angled at 30° above the horizontal. The block then slides along a frictionless level surface until it encounters a spring

(k = 220 N/m) whose other end is attached to a wall. What is the maximum distance that the spring will be compressed by?

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A 2.0 kg block is moving at 10 m/s along a horizontal frictionless table. It encounters and compresses a spring whose other end is attached to a wall. How much potential energy is stored in the spring at the moment when the block stops moving?

A) 10 J B) 20 J

C) 100 J D) Not enough information

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Coming up…Thursday (10/16) → 7.4 – 7.5

Chapter 7 Homework due Sunday by 11:59 PM

Warm-Up due Wednesday by 10:00 PM

In-progress grades are posted!