WORK

The work dW done on a particle displaced along differential path dr, by an object exerting force F is defined as

A B

F

dr

dW d F r

The SI unit of work is 1J = 1N·1m

We can define work in an integral form: path

dW rF

For the work done on an object, one must specify on which point associated with the object the work is done (the center of mass, the point of the force application ... ).

kinetic energy

Kmv

2

2

A particle with mass m, moving with speed v has kinetic energy K of

Note. Kinetic energy is used to describe the motion of an object. When an object is approximated by a particle, the kinetic energy defined above is called the translational kinetic energy of the object.

Work-Energy Theorem I

2

mddK

2v

vdvm dt

dtm v

vd rda

m rdF

net netdW

In an inertial reference frame, the work dW done by all the forces exerted on the particle (the net force) is equal to the change in the kinetic energy dK of the particle

dW = dK

work & energyIn an integral form: W = K

Power

The power of a force is defined as the rate at which work is done by that force.

P tdW

dt( )

The SI unit of work is 1W = 1J/1s

inverse relation: 2

1

t

tdttPW

relation to force: vF

P

Conservative Interaction

If the work done by a “force” on an object moving between two positions is independent of the path of the motion, the force is called a conservative force.

A

B

All other forces are nonconservative.

(Theorem)The work done by a conservative force around a loop (the object returns to its initial position) is zero.

conservative interactions : gravitational, elastic, electrostatic.

Potential Energy

If a force exerted on a particle is conservative, the change in potential energy dU from one position to another is defined by the work dW performed by that force

dU - dW (or U = -W )

This definition assigns potential energy only with accuracy to a constant.

r

gU

Gravitational Potential Energym

Wh

rr

oWU ref,g

r

ro

rdg

m

hy

y

o

0

mgdy mgh

The gravitational potential energy Ug of a particle with

mass m, placed at a position with a vertical component different by h from the reference location is

 Ug = mgh

h

Ug

dr

Elastic Potential Energy

0 x

x

F = -kx x

sU

The elastic potential energy that an ideal spring has by virtue of being stretched or compressed is

U kxs

2

2

x

Us

sW x

0

'dx'kx2

kx2

bungee

Gravitational Potential Energy

Mm

rdrF

GU path

2ˆ

r

MmG rdr

r

r2

0

drr

MmG

1

r

1GMm

Where choose the reference?

The gravitational potential energy of a particle with mass m, placed at distance r from another particle with mass M is

r

MmGUG r

r

MmG

What about the reference at the

surface?

)h(oh

R

10GMm 2

2

R

1

hR

1

hR

GMm

2 mgh

Mechanical Energy

The sum of the kinetic and the potential energy of a particle is called the total mechanical energy of the particle.

E K + U

motionrelatedenergy

positionrelatedenergy

Energy Conservation

• All concepts of energy are defined in such a way that energy can neither be created nor destroyed, but can be converted from one form to another.

• The total energy of an isolated system is always constant.

E

Work-Energy Theorem II

UK cnet WW ncW

If some forces exerted on a particle are conservative, the work Wnc, done by all forces not included in the potential energy, is equal to the change in the mechanical energy E of the particle.

bungee

EWnc

Potential Energy and the Force The conservative force is opposite to the gradient of the potential energy caused by this force

z

U,

y

U,

x

UU

F

dzz

Udy

y

Udx

x

UdUdUdUdU zyx

dzFdyFdxFdWdU zyx rdF

because

Fdr

and

x

y

z

Example. Gravitational potential energy at the surface

m

W = - mg

z

mgzz,y,xU

mg,0,0z

U,

y

U,

x

U

Wx

y

z

Example. Gravitational potential energy

x

)z,y,x(r

GMmz,y,xU

y

z

r

z,y,x

r

MmGz,y,x 2 F

ii x

UF

222

i zyx

GMm

x

U

23

222

i

zyx2

x2GMm

r

x

r

MmG i

2

r̂r

MmG 2

F r

222 zyx

GMm

Example. What should be the initial speed of an object which is supposed to escape the gravitational field of the Earth?

M m

vprobevEarth

in the reference frame of the Earth:

00R

MmG

2

mv2esc

s

km2.11

R

GM2vesc

conservation of the mechanical energy

s

m1045.1

R

Gm2v 7

esc

00R

MmG

2

Mv2esc

in the reference frame of the object:

What is wrong