Kinetics of a particle: Work & Energy

Objectives

• To develop the principle of work and energy and apply it to solve problems that involve force, velocity, and displacement.

• To study problems that involve power and efficiency.

• To introduce the concept of a conservative force and apply the theorem of conservation of energy to solve kinetic problems.

18/06/54 ME212 ดร. พิภัทร 2

Work of a Force

• A force F does work on a particle only when the particle undergoes a displacement in the direction of the force.

18/06/54 ME212 ดร. พิภัทร 3

cosdsFdU

rdFdU

Work of a Force

• If the particle undergoes a finite displacement along its path from r1 to r2 or s1 to s2, the work is determined by integration.

18/06/54 ME212 ดร. พิภัทร 4

2

1

2

1

cos21

s

s

r

rdsFrdFU

Work of a Weight

18/06/54 ME212 ดร. พิภัทร 5

yW

yyWdyW

zdydxdW

dU

y

y

s

s

12

21

2

1

2

1

kjij

rF

Work of a Spring Force

• Work done on a spring

18/06/54 ME212 ดร. พิภัทร 6

21

22

21

2

1

2

1

2

1

2

1

ksks

dsksdsFUs

s

s

ss

Force and Displacement are in the same direction.

Work of a Spring Force

• Work done on a particle or body

18/06/54 ME212 ดร. พิภัทร 7

2

1

2

2212

1

2

1ksksU

Force and Displacement are in the different direction.

Example 1

The 10-kg block rest on a smooth incline. If the spring is originally stretched 0.5 m, determine the total work done by all forces acting on the block when a horizontal force P = 400 N pushes the block up the plane s = 2 m.

18/06/54 ME212 ดร. พิภัทร 8

Principle of Work and Energy

• Consider a particle P, which at the instant considered located on the path as measured from an inertial coordinate system.

• For the particle in the

tangential direction, ∑Ft = mat

18/06/54 ME212 ดร. พิภัทร 9

21

22

2

1

2

12

1

2

1

2

1

mvmvdsF

dvmvdsF

s

st

v

v

s

st

Principle of Work and Energy

18/06/54 ME212 ดร. พิภัทร 10

21

22

2

1

2

12

1

2

1

2

1

mvmvdsF

dvmvdsF

s

st

v

v

s

st

21

2221

2

1

2

1mvmvU

2211 TUT

Problem 14-2

The 100-N crate has a velocity of vA=4 m/s when it is at A. Determine its velocity after it slides s=2 m down the plane. The coefficient of kinetic friction between the crate and the plane is μk=0.2.

18/06/54 ME212 ดร. พิภัทร 11

Example 2

The block A and B have a mass of 10-kg and 100-kg respectively. Determine the distance B travels from the point where it is released from rest to the point its speed become 2 m/s.

18/06/54 ME212 ดร. พิภัทร 12

Example 3

The platform P is tied down so that the 0.4-m-long cords keep

a 1-m-long spring compressed 0.6-m when nothing is on the platform. If a 2-kg platform is placed on the platform and released from rest after the platform is pushed down 0.1 m, determine the max height h the block rises in the air, measure from the ground.

18/06/54 ME212 ดร. พิภัทร 13

Power

18/06/54 ME212 ดร. พิภัทร 14

Power

• It is defined as the amount of work performed per unit of time.

• The power generated by a machine or engine that performs an amount of work dU within a time interval dt is

dt

d

dt

d

dt

dUP

rF

rF

vPF

v

Efficiency

18/06/54 ME212 ดร. พิภัทร 15

Efficiency

• It is defined as the ratio of the output of useful power produced by the machine to the input of power supplied to the machine

inputpower

outputpower

inputenergy

outputenergy

Example 4

18/06/54 ME212 ดร. พิภัทร 16

The motor M of the hoist operates with an efficiency of ε = 0.85. Determine the power that must be supplied to the motor to lift the 375-N crate C at the instant point Pon the cable has an acceleration of 1.2m/s2, and a velocity of 0.6 m/s

Conservative Force

• Work done is independent of the path

• Examples:

– Weight of a particle

– Elastic force of a spring

18/06/54 ME212 ดร. พิภัทร 17

Potential Energy

• Amount of work done by a conservative force from moving from a given position to datum.

• Capacity of work stored in a particle.

18/06/54 ME212 ดร. พิภัทร 18

Potential Energy

18/06/54 ME212 ดร. พิภัทร 19

WyVg 2

2

1ksVe

Conservation of Energy

• If a particle is subject to both conservative and non-conservative forces:

• No non-conservative forces:

18/06/54 ME212 ดร. พิภัทร 20

2,2,2.211,1,1 )( egnonconseg VVTUVVT

22.2111 )( VTUVT noncons

2211 VTVT