Kinetic Energy and Work; Potential Energy;Conservation

of Energy.

Lecture 07

Thursday: 5 February 2004

WORK

•Work provides a means of determining the motion of an object when the force applied to it is known as a function of position.

•For example, the force exerted by a spring varies with position:

F=-kx

where k is the spring constant and x is the displacement from equilibrium.

WORK (Constant Force)

W

W Fd

F d

cos

WORK (Variable Force)

W F x dxx

x

i

f ( )

Work Energy Theorem

• Wnet is the work done by

• Fnet the net force acting on a body.

W F x dxnet netx

x

i

f ( )

Work Energy Theorem (continued)

W F dx

madx mdvdt

dx

mdxdt

dv m vdv

net netxx

xx

xx

vv

vv

i

f

i

f

i

f

i

f

i

f

Work Energy Theorem (continued)

W m vdv

mv

m v v

W mv mv

net vv

v

v

f i

net f i

i

f

i

f

212

2 2

12

2 12

2

2( )

Work Energy Theorem (concluded)

• Define Kinetic Energy

• Then,

• Wnet = Kf - Ki

• Wnet = K

K mv12

2

Recall Our Discussion of the Concept of Work

cosdFW

W

dF

•Work has no direction associated with it (it is a scalar).

•However, work can still be positive or negative.

•Work done by a force is positive if the force has a component (or is totally) in the direction of the displacement.

CONSERVATIVE FORCES•A force is conservative if the work it does on a particle that moves through a closed path is zero. Otherwise, the force is nonconservative.

•Conservative forces include: gravitational force and restoring force of spring.

• Nonconservative forces include: friction, pushes and pulls by a person .

F r d 0

Fg d

CONSERVATIVE FORCES

If a force is conservative, then the work it does on a particle that moves between two points is the

same for all paths connecting those points.

This is handy to know because it means

that we can indirectly calculate the work

done along a complicated path by calculating

the work done along a simple (for example, linear) path.

Work Done by Conservative Forces is

of Special Interest• The work “done” in the course of a motion, is

“undone” in if you move back.

This encourages us to define another kind of energy (as opposed to kinetic energy)- a “stored” energy associated with conservative forces.

• We call this new type of energy potential energy and define it as follows:

U = – Wc

Fg d

Potential Energy Associated with the Gravitational Force

ymgyymgU

dymg

dymgU

mgF

dyFU

dWU

if

y

y

y

y

y

y

y y

f

i

f

i

f

i

f

i

)(

)(

r

rsF

Potential Energy Associated with the Spring

Force

2212

21

2212

21

force spring

force, spring afor that deducecan weSo,

.

if

fi

kxkxU

kxkxW

We know (or should know) from our homework,

Tying Together What We Know about Work and

Energy U = – Wc

• Wnet = K

So, under the condition that there are only conservative forces present :

Wnet = Wc

In that case, K = – U

K + U = 0

The “Bottom Line”• Ei = Ef

• Ki + Ui = Kf + Uf

• The “Total Mechanical Energy” of a System is the sum of Kinetic and Potential energies. This is

what is “conserved” or constant.

Gravitational force: U= mgh Restoring force of a spring: U =1/2kx2

(KE=1/2mv2)

An Example

A 70 kg skate boarder is moving at 8 m/s on flat stretch of road. If the skate boarder now encounters a hill which makes an angle of 10o with the horizontal, how much further up the road will the he be able to go without additional pushing? Ignore Friction.

10oh

d

KEi+Ui=KEf +Uf (only conservative forces)so

KEi + 0 = 0+Uf (Ui=0 and KEf=0)

1/2mv2 = mgh

1/2v2 = gh

h = v2/(2g) = 82/(2*9.8) = 3.26 m

h/d = Sin 10o

d = 18.8 m