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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• 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)

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