Liceo Scientifico Isaac Newton

Physics course Potential Energy and

mechanical energy conservation

Professor

Serenella Iacino

Read by

Cinzia Cetraro

1. Gravitational Potential energy

2. Elastic potential energy

Potential energy

Gravitational potential energy represents the work done by gravity

fig.1

h P→m

W = P ∙ s = P ∙ h ∙ cos 0° = mgh

The Potential energy is indicated by the symbol U or only ( E P ).

→→

fig.2

m

hA

A

B

Bh

A B BAP→

W = m g h - m g h = U - U

P→

mA

B

h=40m

fig.3

Let’s make an example:

W = U - U

m g h - m g h = U - U

m g 40 – 0 = 25000 from which

m g = = 625 N (this is the weight).

BAAB

BA B A

25000

40

Gravitational potential energy depends only on the height h

h

CB

A

θ

s→

s→

h

CB

A

θ

s→

• the route ACB

• the vertical route AB

W = P s= P AB = P h cos0°=mghAB

∙→→

∙→ →

= P AC cos(90°- )+ P CBcos90°=

hsenθ

θsen

→

= mg = mgh

W = W + W = P AC + P CB =ACB AC CB ∙∙→ → →

θfig.5

fig.4

N→

N→

P→

P→

s→

s→

Conservative force

vif

v >-1

2m v

f2 1

2m v

i2

W = > 0 from which

vif

v <-1

2m v

f2 1

2m v

i2

W = < 0 from which

fig.6

m g h =1

2v

A

2m 1

2m g h = v

B

2m

N→

P→

s→

N→

P→

s→

A B

-1

2m v

f2 1

2m v

i2= 0

P→W = 0

fig.7

Elastic potential energy of a compressed spring

U = 1

2K x

2

which represents the work done by the elastic force to pull the spring back towards its original length.

We can observe that the work depends only on the compression x and so on the initial and final positions of the spring, therefore the elastic force is a conservative force.However not all forces are conservative.

Non conservative forces:

fig.8

s→ s→

s→

s→D C

A B

→Fa

→Fa

→Fa

→Fa

W = W + W + W + W = =DACDBCAB

- Fa

s- Fa

s - Fa

s - 4 Fa

s- Fa

s

Friction

Mechanical Energy

E = U + K

the work – energy theorem:sum

= -1

2m v

f2 1

2m v

i2

W = K - Kf i

the difference in potential energy:

It is conserved only in systems where conservative forces are involved.

Wconservative force = U - U

i f

U - Ui f

K - Kf i

= U + Kf f

U + Ki i

=from which we have

Einitial = E

final

highest point - highest gravitational potential energy

If there is no friction, the Roller Coaster is a demonstration of Energy Conservation.

Mechanical energy remains constant.

fig.9

Spring and energy conservation

m

v→

fig.10

v→

fig.11

When the object compresses the spring, its kinetic energy decreases and is transformed into elastic potential energy.

m

When the motion is reversed, the potential energy decreases while the kinetic energy increases and when the object leaves the spring, the kinetic energy returns to its initial value.

The pinball machine:

s→

fig.12

N→

P→

To fire the ball of mass m, suppose we compresse the spring, having a constant equal to K, by length x.Ignoring friction, we want to know what is the launch velocity of the ball.

U + Kf f

U + Ki i

=1

2K x

2 1

2v

f

2m+ 0 = 0 +

v f=

m

2K x m s

Water Park:

h h

v 2

v 1

two children, two slides, no friction, same height h,

1

2v

1

2m+ 0 = 0 +m g h

1

2v

2

2m+ 0 = 0 +m g h

U + Kf f

U + Ki i

=

from which v =2

2 g hv 1= 2 g h and

Conservative and non conservative forces:

sum=W Wcons Wnon cons+

sumW K - Kf i=

K - Kf i=Wcons Wnon cons+

Wcons= U - Ui f Wnon cons+U - Ui f K - Kf i=

Wnon cons U + Kf f U + Ki i= -

E initial-E finalWnon cons =

Law of energy conservation is no longer valid.

THE ENDenergy