Mechanics 105

Work done by a constant forceScalar productWork done by a varying forceKinetic energy, Work-kinetic energy

theoremNonisolated systemsKinetic frictionPower

Energy and energy transfer (chapter six)

Mechanics 105

Systems

In discussion of work and energy, it is important that we are clear about the objects we are considering. A system is an object, group of objects or region of space with well defined boundaries.

Mechanics 105

Work done by a constant forceEffect of force acting on an object over some

distance.

The component of the force in the direction of the displacement, multiplied by the magnitude of the displacement (or the component of the displacement in the direction of the force times the magnitude of the force.)

Result is a scalar quantityUnits: N·mJ b(Joule)

cosrFW

Mechanics 105

Scalar products

product of two vectors:

where is the angle between the two vectors

In terms of components:

rFrFW

BABA

cos e.g.

cos

zzyyxx BABABABA

Mechanics 105

Properties of scalar product Commutative

Distributive

for the unit vectors

ABBA

CABACBA

)(

0ˆˆˆˆˆˆ

1ˆˆˆˆˆˆ

ikkjji

kkjjii

Mechanics 105

Work done by a varying forceIncrement of work: force applied over

small displacement

Total work: sum of increments

Taking the limit for infinitesimally small displacements

rFW

i

ii rrFWW

)(

i

r

r

iir

f

i

rdrFrrFWW

)()(lim

0

Mechanics 105Example: work done pushing a block up a

frictionless inclined plane at constant velocity

F

x

N

gm

F

xmgFdxW

mgF

mgFF

f

i

x

x

x

sin

sin

0sin

Mechanics 105

Question

What do Winnie the Pooh and Attila the Hun have in common?

Mechanics 105

Question

What do Winnie the Pooh and Attila the Hun have in common?

Answer: Same middle name.

Mechanics 105

Kinetic energy

A net force acting on an object in the x-direction will do an amount of work

Using Newton’s

2nd law

F

xFWf

i

x

x

22

2

1

2

1if

v

v

x

x

x

x

x

x

mvmv

vddt

dxmxd

dt

dx

dx

dvm

xddt

dvmdxmaW

f

i

f

i

f

i

f

i

Mechanics 105

Kinetic energy

From this expression we define the kinetic energy

and the expression for net work becomes

This is the work-kinetic energy theorem

2

2

1mvK

KKKmvmvW ifif 22

2

1

2

1

Mechanics 105

ExampleA block slides down a frictionless plane – what is the

velocity at the bottom?

Forces perpendicular to the plane do no work

x

N

gm

mghW

x

xmgxFW

mgFx

hblock ofheight initialsin

sin

sin

Mechanics 105

Web quiz

Mechanics 105

Web quiz

Mechanics 105Example – work done by a spring

Force exerted by spring (Hooke’s law)

where x is the displacement of the spring from its unstretched

position

The work done by the spring is

kxxFs )(

22

2

1

2

1)( fi

x

x

s kxkxdxxFWf

i

Mechanics 105Example – work done by a spring

A spring (k=100 N/m) is slowly stretched by 2 cm from its unstretched length. What is the work done by the force of the spring?

The result is negative since the force is applied in the opposite direction to the displacement.

What would be the work done by the spring if it were compressed by the same amount?

What is the work done by the force stretching or compressing the spring?

JmmN

kxkxW fi

02.0)02.0)(/100(2

10

2

1

2

1

2

22

Mechanics 105

Example – speed of a block on a spring The same spring in the last example is put in contact with a block (mass m=1.00 kg), compressed 2.00 cm and then let go. How fast is the block moving when it loses contact with the spring?

v

JmmNkxkxW fi2222 1000.200.0)0200.0)(/.100(

2

1

2

1

2

1

Work done by the spring (now positive – spring is returning to unstretched length)

Must equal the change in kinetic energy

smv

JvkgmvmvK if

/200.0

1000.200.0)00.1(2

1

2

1

2

1 2222

Mechanics 105

Example –block dropped onto a spring Same spring, same mass, now the mass is dropped from a height 1.00 m above the uncompressed spring. How far down does the spring compress?

)00.1( dmmgWg

Work done by gravity on the block is

The work done on the block by the spring is

md

dmNdmNJ

dmNdmmgWW sg

551.0

0)/(0.50)/80.9(80.9

0))(/.100(2

1)00.1(

2

2

1.00m + d

222 ))(/.100(2

100.0

2

1

2

1dmNkxkxW fis

Since the change in kinetic energy is zero, the

total work done must also be zero

Mechanics 105

Nonisolated systems Work can be thought of as the transfer of energy

between a system and its environment Forms of energy of a system other than kinetic: internal

(thermal) Ways of energy transfer other than work (mechanical

waves, heat, matter transfer, electrical transmission, EM radiation)

Mechanics 105

Kinetic friction

The new form of the work-KE theorem:

looks the same, but note that now the application point of the force (friction) is changing

For the large system of both objects

WxfK k

xfE

xfEKE

k

k

int

intint0

Mechanics 105Example

A block slides down an inclined plane with coefficient of kinetic friction k – what is the velocity at the bottom?

x

f

kx

y

kkx

mvK

xmgxFW

mgNmgNF

NmgfmgF

2

1

)cos(sin

cos0cos

sinsin

N

gm

kf

Mechanics 105

Power

power = rate of energy transferaverage power

instantaneous power

more general definition: Units: Watt (=J/s)hp=550 ft·lb = 746 W

t

WP

vFdt

rdF

dt

dWP

dt

dEP

Mechanics 105

Example

How much power is needed to accelerate a 1.00103 kg car from 0-60.0 mph in 5.00 seconds, ignoring air resistance and friction? How much is needed to keep it moving at 60.0 mph if friction and air resistance equal 1.00 102 N?

Assuming a constant acceleration (60 mph = 26.8 m/s)

Ws

smkg

t

mv

t

K

t

WP

423

2

1018.700.5

)/8.26)(1000.1(500.0

21

Mechanics 105

To keep it moving at a constant velocity, the magnitude of the applied force must equal that from the air resistance and friction

WsmNvFP 2680)/8.26)(1000.1( 2