¾ Energy Conservation (에너지 보존)

Chapter 8. Potential Energy Chapter 8. Potential Energy and Conservation of Energy and Conservation of Energy

¾ Potential Energy (U)

z 중력 U, 탄성 U, 전자기 U, …..

¾ Conservative Force ( 보존력)

¾ Energy Conservation (에너지 보존)

Physics, Page 2

¾ Work: Transfer of Energy by Force

• W

= |F| |S| cosθ

¾ Kinetic Energy (Energy of Motion)

• K = 1/2 mv

¾ Work-Kinetic Energy Theorem:

• ΣW = ΔK

Review

θ

Question Question

Imagine that you are comparing three different ways of having a ball move down through the same height. In which case does the ball reach the bottom with the highest speed?

1. Dropping

2. Slide on ramp (no friction) 3. Swinging down

4. All the same

ball A is the only one without any tension or friction acting on it

In all three cases, the work done by the gravitational force is the same since the change in vertical

distance is the same

1 2 3

correct

it has the most distance to accelerate

has tension and gravity as forces, causing the greater acceleration

Physics, Page 4

Question Question

Imagine that you are comparing three different ways of having a ball move down through the same height. In which case does the ball get to the bottom first?

1. Dropping

2. Slide on ramp (no friction) 3. Swinging down

4. All the same 1 2 3

correct

It has the most direct path.

They each have only gravitational force acting on them so they fall at the same speed. Since they are at the same height, they will hit at the same time.

Question Question

Which of the following statements correctly define a Conservative Force (보존력):

1. A force is conservative when the work it does on a moving object is independent of the path of the motion between the object's initial and final positions.

2. A force is conservative when it does no net work on an object moving around a closed path, starting and finishing at the same point.

correct 3. Both of the above statements are correct.

4. Neither of the above statements is correct.

Physics, Page 6

Suppose the initial kinetic and potential energies of a system are 75J and 250J respectively, and that the final kinetic and potential energies of the same system are 300J and -25J respectively. How much work was done on the system by non-conservative forces?

1. 0J 2. 50J 3. -50J 4. 225J 5. -225J

W_nc = E_f - E_i

=(KE_f + PE_f) - (KE_i + PE_i)

=(300J -25J) - (75J + 250J)

=275J - 325J

= -50J correct

The change in kinetic energy plus the change in potential energy equals the work done

on the system by non-conservative forces

Question

Question

Galileo

Galileo ’ ’ s Pendulum ACT s Pendulum ACT

How high will the pendulum swing on the other side now?

A) h

> h

B) h

= h

C) h

< h

h₁ h₂

m

Conservation of Energy 0 = ΔK + Δ U

K

+ U

= K

+U

0 + mgh

= 0 + mgh

h

= h

Physics, Page 8

- Conservative force : Work is independent of the path.

- Non-conservative force : Work is dependent of the path.

보존력과 보존력과 비보존력 비보존력

W = 0 (Conservative force) (Non-conservative force)

Potential energy Potential energy

어떤 힘이 보존력 이면, 그 힘에 대한 potential energy의 정의 가능 !

Conservative force(보존력) : 중력, 탄성력, 전기력

(Non-conservative force : 쓸림힘, 끌림힘) => 확인

Physics, Page 10

Potential energy

Potential energy 구하기 구하기

Physics, Page 12

일과 일과 potential energy potential energy

ΔU > 0

W (< 0)

W (> 0)

ΔU < 0

ΔU > 0 W < 0

ΔU < 0 W > 0

ΔK < 0

ΔK > 0 평형점

(a) 수축 시

(b) 이완 시

Physics, Page 14

역학에너지

역학에너지 보존 보존

E_m = U + K = 일정

Physics, Page 16

(보기문제 8-3)

수직힘은

수직힘,

(보기문제)

Physics, Page 18

E = U + K = 5 J

E = U + K = 4 J

(X₂, X₃, X₄, 0~X₀, X₅~) (X₂, X₄)

(X₃)

Potential energy

Potential energy 곡선읽기 곡선읽기

(X₁)

되돌이점 (x₁)

Physics, Page 20

외부 외부 힘이 힘이 계에 계에 한 한 일 일

(보기문제)

30^o N)

Physics, Page 22

Energy is Conserved ! Energy is Conserved !

• Energy is “Conserved” meaning it can not be created nor destroyed.

–Can change the form –Can be transferred

• Total Energy does not change with time.

에너지 에너지 보존 보존

마찰력마찰력 (비(비 보존력보존력)이)이 있는있는경우경우

Physics, Page 24

(보기문제)

h₂

Δ

Skiing Example (no friction)

A skier goes down a 78 meter high hill with a variety of slopes. What is the maximum speed should can obtain if she starts from rest at the top?

Conservation of energy:

K

+ U

= K

+ U

½ m v

+ m g y

= ½ m v

+ m g y

0 + g y

= ½ v

+ g y

v

= 2 g (y

-y

)

v

= sqrt( 2 g (y

-y

))

0 = K

-K

+ U

- U

Physics, Page 26

Pendulum Example

As the pendulum falls, the work done by the string is 1) Positive 2) Zero 3) Negative

How fast is the ball moving at the bottom of the path?

W = F d cos θ. But θ = 90 degrees so Work is zero.

h

Conservation of Energy 0 ⁼ Δ ^{K +} Δ ^U

0 = K

- K

+ U

- U

K

+ U

= K

+U

0 + mgh = ½ m v

+ 0 v

= sqrt(2 g h)

30

“Why mass doesn't matter when you drop something?”

θ

h

iˆ jˆ

Total Energy

mgh mv

U K

E = +

=

+

At y = 0

mgh mv

mv

E =

=

+

j v

i v

v

=

cos θ ˆ +

sin θ ˆ θ

= cos v v

:

iˆ

gt sin

v v

:

jˆ

=

θ −

g

gh sin

v sin

t v

gt t

sin v

h y

2 0

2 2

0 0

2 2

1 0

+ θ +

= θ

=

−

⋅ θ +

=

gh sin

v

v

= −

θ + 2

θ +

+ θ

= +

= v

v

v

sin

2 gh v

cos

v

(보기문제)

v at y = 0?

v_o

에너지 보존법칙 이용

등가속운동 법칙 이용

Physics, Page 28

m₂

m₁ R θ

Total Energy of m₁

E

= K

+ U

= K

+ U

( ^K

^{− K}

) ( ^{= − U}

^{− U}

) ^⇒

^m

^v

^{= − m}

^gR ( ^{− 1 + cos θ}

)

( ) ^{m g} (

)

R m v a

m g

m T

F θ 0 2

1 cos θ

2 1 1

1

= = = −

−

=

=

(

)

g m

T =

3 − 2 cos θ g

m T >

If , it lift up m

In this case, m

= 2m

( ) ^{m g m g}

g

m

3 − 2 cos θ

>

= 2

(보기문제)

What is the minimum of the initial angle θ_ο for lifting m₂ (= 2m₁) at θ = 0?

2

cos θ

<

Therefore, If θ

> 60 ° , it lifts up m

(a) No Friction

E

0 30 sin

2 2

1 2 1

2 2

1

+

=

=

=

+

=

f

i i

mv mgd mgd

mgy mv

o

gl v

=

(b) Friction coefficient μ_k

H U

K U

K

E

=

+

=

+

+

Heat Loss

H = f_k·d = μ_kmg cos30°·d

mgcosθ

mgd mv

mgd

μ

2

1

= +

( ) ^gd

v = 1 − 3 μ

(보기문제)

v

Physics, Page 30

Summary Summary

¾ Conservative Forces

• Work is independent of path

• Define Potential Energy U

– U

= m g y – U

= ½ k x

¾ Work – Energy Theorem

internal

E U

K E

W = Δ

= Δ + Δ + Δ