(Work and Energy)

- 1 - Diponegoro UniversityMechanical Engineering Dept.

DYNAMICSDYNAMICSDYNAMICS-- Chapter #6: Work and EnergyChapter #6: Work and Energy--

Dr. Achmad WidodoDr. Achmad Widodo

Mechanical Engineering DepartmentMechanical Engineering DepartmentDiponegoro UniversityDiponegoro University


Work and Energy RelationsWork and Energy Relations

Work of Forces and CouplesWork of Forces and Couples

The work done by a force F has been treated in Chapter #3 and is given by

In Fig. 6/11, we see immediately that during the translation the work done by one of force cancels that done by the other force , so that the netnet work done is

The work done by a couple M which acts on a rigid body during its motion is given by

∫ ⋅= rF dU ∫= dsFU cosα

∫= θdMU

θθ MdbdFdU == )(



Kinetic Energy

1. Translation.1. Translation. The translating rigid body (Fig. 6/12a) has a mass m and all of its particle have a common velocity v, so the kinetic energy of mass mi of the body is

2. Fixed2. Fixed--axis rotation. axis rotation. The rigid body in Fig. 6/12b rotate with angular velocity ω about the fixed axis through O. The kinetic energy is given

2

21 mvT =2

21

iii vmT =

2

21 ωOIT =2)(

21 ωiii rmT =


3. General plane motion. 3. General plane motion. For this case, the formulation of kinetic energy is given by

In the case of instantaneous center zero velocity, we can express

Kinetic Energy


∑ ∑ ++== )cos2(21

21 2222 θωρωρ iiiii vvmvmT

∑ ∑ == 0cos iiii ymvmv ωθρω

∑ ∑+= 222

21

21

iii mmvT ρω

2

21 ωCIT =



Potential energy and the work energy equation

The kinetic energy is defined as the total work which must be done on the particle to bring it from a state of rest to a velocity v

In other form (work-kinetic energy relation):

Alternatively, the work-energy relation may be expressed as the initial kinetic energy T1 plus the work done U1-2 equals the final kinetic energy T2, or

)(21 2vmT =

TTTU Δ=−=− 1221

2211 TUT =+ −


Elastic Potential EnergyPotential Energy

The work which is done on the spring to deform it is stored in the spring is called its elastic potential energy Ve.This energy is recoverable in the form of work done by the spring on the body attached to its movable end during the release of the deformation of the spring.

The change of elastic potential energy is

∫ ==x

e kxdxkxV0

2

21

)(21 2

122 xxkVe −=Δ



Work-Energy Equation

With elastic member included in the system, we modify the work-energy equation to account for the potential-energy terms.

VTU Δ+Δ=−21'

222111 ' VTUVT +=++ −



Potential energy and the work energy equation


• The last equation represents a major advantage of the method of work-energy is that avoids the necessity of computing acceleration and leads directly to the velocity changes as functions of the forces which do work.• The work-energy equation involves only those forces which do work and thus give rise to changes in magnitude of the velocities. • Application of work-energy method requires isolation of the particle or system under consideration e.g. drawing of free-body diagram that showing all externally applied forces.


• The capacity of a machine is measured by the time rate at which it can do work or deliver the energy.• The total work or energy output is not a measure of this capacity since a motor, no matter how small, can deliver a large amount of energy if given sufficient time.• On the other hand, a large and powerful machine is required to deliver a large amount of energy in a short period of time.• Thus, the capacity of a machine is rated by its power, which is defined as the time rate of dong work.


vFrF⋅=

⋅==

dtd

dtdUP

Power



Power

For the couple M acting on the body, the power developed by the couple at a given instant is the rate at which it is doing work

If the force F and the couple M act simultaneously, the total power is

Power developed from the total mechanical energy

ωθ Mdt

Mddt

dUP ===

ωMP +⋅= vF

dVdTdU +=

)(' VTdtdVT

dtdUP +=+== &&

ωωα

ωω

ω

MIm

Im

Imdtd

dtdTT

+⋅=+⋅=

+⋅+⋅=

⎟⎠⎞

⎜⎝⎛ +⋅==

vRva

avva

vv

)(

)(21

21

21 2

&

&













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