Engineering Mechanics: Dynamics - Inside Minesinside.mines.edu/UserFiles/File/FEexam/10- Energy...

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Engineering Mechanics: Dynamics Engineering Mechanics: Dynamics Work of a Force Method of work and energy: directly relates force, mass, velocity and displacement. 13 - 1 Work of the force is dz F dy F dx F ds F r d F dU z y x + + = = = α cos r r

Transcript of Engineering Mechanics: Dynamics - Inside Minesinside.mines.edu/UserFiles/File/FEexam/10- Energy...

Page 1: Engineering Mechanics: Dynamics - Inside Minesinside.mines.edu/UserFiles/File/FEexam/10- Energy Momentum of... · Engineering Mechanics: Dynamics Work of a Force • Method of work

Engineering Mechanics: DynamicsEngineering Mechanics: Dynamics

Work of a Force

• Method of work and energy: directly relates force, mass, velocity

and displacement.

13 - 1

• Work of the force is

dzFdyFdxF

dsF

rdFdU

zyx ++=

=

•=

αcos

rr

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Engineering Mechanics: DynamicsEngineering Mechanics: Dynamics

Work of a Force

• Work of the force of gravity,

dyW

dzFdyFdxFdU

y

zyx

−=

++=

13 - 2

( ) yWyyW

dyWU

y

y

∆−=−−=

−= ∫→

12

21

2

1

• Work of the weight is positive when ∆y < 0,

i.e., when the weight moves down.

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Engineering Mechanics: DynamicsEngineering Mechanics: Dynamics

Work of a Force

• Work of the force exerted by spring,

222

dxkxdxFdU

x

−=−=

13 - 3

222

1212

121

2

1

kxkxdxkxU

x

x

−=−= ∫→

• Work of the force exerted by spring is positive

when x2 < x1, i.e., when the spring is returning to

its undeformed position.

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Engineering Mechanics: DynamicsEngineering Mechanics: Dynamics

Particle Kinetic Energy: Principle of Work & Energy

dvmvdsF

ds

dvmv

dt

ds

ds

dvm

dt

dvmmaF tt

=

==

==

• Consider a particle of mass m acted upon by force.

The component Fn does no work.Fr

13 - 4

dvmvdsF t =

• Integrating from A1 to A2 ,

energykineticmvTTTU

mvmvdvvmdsF

v

v

s

s

t

==−=

−==

∫∫

2

21

1221

212

1222

12

1

2

1

• The work of the force is equal to the change in

kinetic energy of the particle.

Fr

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Engineering Mechanics: DynamicsEngineering Mechanics: Dynamics

Power and Efficiency

• rate at which work is done.

vF

dt

rdF

dt

dU

Power

rr

rr

•=

•==

=

• Dimensions of power are work/time or force*velocity.

Units for power are

13 - 5

W746s

lbft550 hp 1or

s

mN 1

s

J1 (watt) W 1 =

⋅=⋅==

inputpower

outputpower

input work

koutput wor

efficiency

=

=

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Engineering Mechanics: DynamicsEngineering Mechanics: Dynamics

Potential Energy

2121 yWyWU −=→

• Work of the force of gravity ,Wr

• Work is independent of path followed; depends

only on the initial and final values of Wy.

=

= WyVg

potential energy of the body with respect

13 - 6

= potential energy of the body with respect

to force of gravity.

( ) ( )2121 gg VVU −=→

• Choice of datum from which the elevation y is

measured is arbitrary.

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Engineering Mechanics: DynamicsEngineering Mechanics: Dynamics

Potential Energy

• Work of the force exerted by a spring depends

only on the initial and final deflections of the

spring,

222

1212

121 kxkxU −=→

• The potential energy of the body with respect

to the elastic force,

13 - 7

to the elastic force,

( ) ( )2121

221

ee

e

VVU

kxV

−=

=

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Engineering Mechanics: DynamicsEngineering Mechanics: Dynamics

Conservative Forces• Concept of potential energy can be applied if the

work of the force is independent of the path

followed by its point of application.

( ) ( )22211121 ,,,, zyxVzyxVU −=→

Such forces are described as conservative forces.

• For any conservative force applied on a closed path,

13 - 8

• For any conservative force applied on a closed path,

0=•∫ rdFrr

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Engineering Mechanics: DynamicsEngineering Mechanics: Dynamics

Conservation of Energy

• Work of a conservative force, 2121 VVU −=→

• Concept of work and energy,

1221 TTU −=→

• Follows that

2211 +=+ VTVT

13 - 9

constant

2211

=+=

+=+

VTE

VTVT

• When a particle moves under the action of conservative forces, the total

mechanical energy is constant.

• Friction forces are not conservative. Total mechanical energy of a system

involving friction decreases.

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Engineering Mechanics: DynamicsEngineering Mechanics: Dynamics

Principle of Impulse and Momentum

• From Newton’s second law,

( ) == vmvmdt

dF

rrrlinear momentum

( )vmddtFrr

=

• Method of impulse and momentum: directly relates force, mass, velocity,

and time.

13 - 10

2211

21 force theof impulse 2

1

vmvm

FdtF

t

trr

rr

=+

==

→∫

Imp

Imp

• The final momentum of the particle can be obtained by adding vectorially its

initial momentum and the impulse of the force during the time interval.

12

2

1

vmvmdtF

t

t

rrr−=∫

• Nonimpulsive forces are forces

for which is small and

therefore, may be neglected. For

example, the impulse of the

gravity force on the ball

tF∆r

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Engineering Mechanics: DynamicsEngineering Mechanics: Dynamics

Impulsive Motion

• Force acting on a particle during a very short

time interval that is large enough to cause a

significant change in momentum is called an

impulsive force.

• When impulsive forces act on a particle,

21 vmtFvmrrr

=∆+∑

13 - 11

• When a baseball is struck by a bat, contact

occurs over a short time interval but force is

large enough to change sense of ball motion.

• Nonimpulsive forces are forces for which

is small and therefore, may be

neglected.tF∆

r

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Engineering Mechanics: DynamicsEngineering Mechanics: Dynamics

Direct Central Impact• Bodies moving in the same straight line,

vA > vB .

• Upon impact the bodies are in contact and

are moving at a common velocity.

• The bodies either regain their original

shape or remain permanently deformed.

13 - 12

• Wish to determine the final velocities of the

two bodies.

• The total momentum of the two body

system is preserved,

BBAABBAA vmvmvmvm ’+’=+

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Engineering Mechanics: DynamicsEngineering Mechanics: Dynamics

Direct Central Impact

• The second relation between the final

velocities.

e = Coefficient of Restitution

( )BAAB vvevv −=′−′

• A second relation between the final velocities is required.

13 - 13

e = Coefficient of Restitution

0 ≤ e ≥ 1

• Perfectly plastic impact, e = 0: vvv AB ′=′=′ ( )vmmvmvm BABBAA ′+=+

• Perfectly elastic impact, e = 1:

Total energy and total are momentum

conserved.

BAAB vvvv −=′−′