Catatan Perkuliahan ( Lecture Notes )

MCS220801 - KINEMATIKA DAN DINAMIKA

Topik 6: Dinamika Benda Titik

Kerja dan Energi

Pengelola dan pengajar : Wahyu Nirbito, DR., Ir., MSME

Departemen Teknik Mesin

Fakultas Teknik

Universitas Indonesia

Depok, 2013

Topik 6: Dinamika Benda Titik

Kerja dan Energi

Chapter 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.

Work of a Force.

A force does work when it undergoes a displacement along its line of action.

If the force varies with the displacement, then

Graphically, this represents the area under the F-s disgram

If the force is constant, then for a displacement ∆s in the direction of the force, U

= F∆s

A typical example of this case is the work of weight, U = W∆y.

∆y is the vertical displacement.

A spring force, F = ks, depend upon the elongation or compression s of the spring

The work is determined by integration to be

Principle of Work and Energy

The Principle of Work and Energy.

If the equation of motion in the tangential direction, ∑Ft = mat, is combined with

the kinematics equation, at ds = v dv, we obtain the Principle of Work and Energy

Initial kinetic energy of the particle plus the work done by all forces that act up on

the particle as it moves equals to final kinetic energy

dsFU

2

1

2

22

1

2

1ksksU

2211 TUT

The principle of work and energy is useful for solving problems that involve

force, velocity, and displacement.

For application, the free-body diagram of the particle should be drawn in order to

identify the forces that do work.

PROCEDURE FOR ANALYSIS :

Work (Free-Body Diagram)

• Establish the initial coordinate system and draw a FBD of the particle to account

for all the forces that do work on the particle as it moves along its path

Principle of Work and Energy

• Apply the principle of work and energy

• The kinetic energy at the initial and final points is always positive since it

involves the speed squared

• A force does work when it moves through a displacement in the direction of the

force

• Work is always positive when the force component is in the same direction as its

displacement, otherwise, it is negative

• Forces that are functions of displacement must be integrated to obtain the work

• Graphically, the work is equal to the area under the force-displacement curve

• The work of a weight is the product of the weight magnitude and the vertical

displacement

• It is positive when the weight moves downwards

• The work of the spring is in the form of

where k is the spring stiffness and s is the stretch or compression of the spring

Power and Efficiency.

Power is the time-rate of doing work. It is defined by P = dU/dt, or P = F.v

Efficiency is the ratio of power output to power input.

PROCEDURE FOR ANALYSIS :

The power supplied to a body can be computed using the following procedure.

• Determine the external force F acting on the body which causes motion.

• If the body is accelerating, it may be necessary to draw its free-body diagram and

apply the equation of motion (∑F = ma) to determine F.

• Once F and v is found, power can be determined with the formula (P = F.v = Fv

cos θ)

• In some problems the power may be found by calculating the work done by F per

unit of time

2211 TUT

2

2

1ksU s

inputpower

outputpower

dtdUP

tUPavg

/

/

Conservative of Energy

PROCEDURE FOR ANALYSIS :

• The conservation of energy is used to solve problem involving velocity,

displacement and conservative force systems.

• It is easier to apply than the principle of work and energy because the energy

equation requires specifying the particle’s kinetic and potential energies at only

two points along the path.

Potential Energy.

• Draw two diagrams showing the particle located at its initial and final points along

the path

• If the particle is subjected to a vertical displacement, establish the fixed

horizontal datum from which to measure the particle’s gravitational potential

energy.

• Data pertaining to the elevation y of the particle from the datum and the extension

or compression s of any connecting springs can be determined from the geometry

associated with the two diagrams.

• Recall Vg = Wy, where y is positive upward from the datum and negative

downward from the datum, is always positive.

Conservation of Energy

• Apply the equation

• When determining the kinetic energy, the particle’s speed v must

always be measured from an inertial reference frame.

2

2

1ksVe

2211 VTVT 2

2

1mvT