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Catatan Perkuliahan ( Lecture Notes )
ENME 600008 - KINEMATIKA DAN DINAMIKA
Topik 5: Dinamika Benda Titik
Gaya dan Percepatan
Pengelola dan pengajar : Wahyu Nirbito, Ir., MSME, Dr.
Departemen Teknik Mesin
Fakultas Teknik
Universitas Indonesia
Depok, 2013
Topik 5: Dinamika Benda Titik
Gaya dan Percepatan
Chapter Objectives :
To state Newton’s Laws of Motion and Gravitational attraction and to define mass
and weight.
To analyze the accelerated motion of a particle using the equation of motion with
different coordinate systems.
To investigate central-force motion and apply it to problems in space mechanics.
Kinetic.
Study of the relationship between forces and the acceleration they cause
Based on the Newton’s second law
∑F = ma
Mass m is proportionality constant between the resultant force acting on the
particle and the acceleration caused by this resultant
Mass represents the quantity of matter contained within the particle
Measures the change in its motion
The Equation of Motion
PROCEDURE FOR ANALYSIS :
Free-Body Diagram
• Select the inertial coordinate system
• Once the coordinates are established, draw the particle’s free body diagram
(FBD). It provides a graphical representation that accounts for all forces (ΣF)
which acts on the particle, and thereby makes it possible to resolve these forces
into their x, y, z components.
• The direction and sense of the particle’s acceleration a should also be established.
If the senses of the components is unknown, assume that they all are in the same
direction as the positive inertial coordinate axes.
• The acceleration may be represented as the ma vector on the kinetic diagram.
• Identify the unknowns in the problem.
Equation of Motion
• If the forces can be resolved directly from the FBD, apply the equations of motion
in their scalar component form.
• If the geometry of the problem appears complicated, Cartesian vector analysis can
be used for the solution.
Kinematics
• If the velocity or position of the particle is to be found, it will be necessary to
apply the proper kinematics equations once the particle’s acceleration us
determined from ΣF = ma
• If acceleration is a function of time, use a = dv/dt and v = ds/dt, which integrated,
yield the particle’s velocity and position.
• If acceleration is a function of displacement, integrate a ds = v dv to obtain the
velocity as a function of position.
• If acceleration is constant, use
• In all cases, make sure the positive inertial coordinate directions used for writing
the kinematic equations are the same as those used for writing the equations of
motion, otherwise, simultaneous solution of the equations will result in errors.
• If the solution for an unknown vector component yields a negative scalar, it
indicates that the component acts in the direction opposite to that which is
assumed.
Inertial Coordinate Systems
Important to measure the acceleration from
an inertial coordinate system when applying
equations of motion
Has axes that are fixed or translate with
constant velocity
Various types of inertial coordinate systems
can be used to apply ∑F = ma in component
form
Rectangular axes x, y and z are used to
describe rectilinear motion along each axes
Normal and tangential n, t axes are used
when the path is known
an is always directed in the +n direction
an indicates the change in the velocity
magnitude
Cylindrical coordinates are useful when
angular motion of the radial coordinate r is specified or when the path can
conveniently be described with these coordinates
For some problem, the direction of the forces on the FBD require coordinate angle
ψ between the extended radial coordinate and the tangent to the curve
Equations of Motion: Normal and Tangential Coordinates
PROCEDURE FOR ANALYSIS :
Free-Body Diagram
• Establish the inertial t, n, b coordinate system at the particle and draw the
particle’s free-body diagram.
• The particle’s normal acceleration an, always acts in the positive n direction.
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2
1tatvss c
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2 2 ssavv c
tavv c 0
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ddr
r
/tan
• If the tangential acceleration at, is unknown, assume it acts in the positive t
direction
• Identify the unknowns in the problem
Equation of Motion
• Apply the equations of motion,
Kinematics
• Formulate the tangential and normal components of acceleration; i.e. at =dv/dt or
at = v dv/ds and an = v2/ρ
• If the path is defined as y = f(x), the radius of curvature at the point where the
particle is located can be obtains from
Equations of Motion: Cylindrical Coordinates
PROCEDURE FOR ANALYSIS :
Free-Body Diagram
• Establish the r, θ, z inertial coordinate system and draw the particle’s free body
diagram.
• Assume that ar, aθ, az act in the positive directions of r, θ, z if they are unknown.
• Identify all the unknowns in the problem.
Equations of Motion
• Apply the equations of motion
Kinematics
• Determine r and the time derivatives and then evaluate the
acceleration components
• If any of the acceleration components is computed as a negative quantity, it
indicates that is acts in it negative coordinate direction.
• Use chain rule when taking the time derivatives of r = f(θ)
0b
nn
tt
F
maF
maF
222/32
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zz
rr
maF
maF
maF
zrr ,,,,
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