Topic 5 Basic Concepts in Dynamics

5.0 BASIC CONCEPTS IN DYNAMICS

SYNOPSIS This topic deals with fundamental and application of dynamics. It also covers the general aspects, the importance of dynamics, relationships between linear and angular velocity as well as linear and angular acceleration.

LEARNING OUTCOMES Upon completion of this course, students should be able to:1.

Apply the principles of statics and dynamics to solve engineering problems.

2.

Study the theory of engineering mechanics to solve related engineering problems in group.

5.0 BASIC CONCEPTS IN DYNAMICS 5.1

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INTRODUCTION

DYNAMICS, is concern about the accelerated motion of a body. The subjected of dynamics will be presented in two parts: kinematics and kinetics. Kinematics is treats only the geometric aspect of motion while kinetics is dealing with the analysis of the forces causing the motion. To develop these principles, the dynamics of a particle will be discussed first, followed by topics in kinematics and kinetics in the following chapter.

The principles of dynamics developed when it was possible to make an accurate measurement of time. Galileo Galilei was one of the major contributors to this field. His work consisted of experiments using pendulums and falling bodies. Another contributor to this field is none other than Sir Isaac Newton. His formulation of the three fundamental laws of motion and the law of universal gravitational attraction has been used by other physicist to develop important techniques for the application of dynamics.

The principles of dynamics are often used in solving problem in engineering or in designing process. Typically the structural design of any vehicle, such as an automobile or airplane or airplane, requires consideration of the motion to which it is subjected. This is also true for many mechanical devices, such as motors, pumps, movable tools and many more. Furthermore, predictions of the motions of artificial satellites, projectiles and spacecraft are based on theory of dynamics. With further advances in technology, there will be an even greater need for knowing how to apply the principles of this subjected.

5.2

LINEAR MOTIONS

Linear motion (also called rectilinear motion) is motion along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: uniform linear motion, with constant velocity or zero acceleration; non uniform linear motion, with variable velocity or non-zero acceleration. The motion of a particle (a point-like object) along a line can be described by its position x, which varies with t (time). Linear motion is the most basic of all motion. According to Newton's first law of motion, objects that experience no net force will continue to move in a straight line with a constant velocity until they are subject to a net force. Under everyday circumstances, external forces such as gravity and friction can cause an object to change the direction of its motion, so that its motion cannot be described as linear. One may compare linear motion to general motion. In general motion, a particle's position and velocity are described by vectors, which have a magnitude and direction. In linear motion, the directions of all the vectors describing the system are equal and constant: MECHANICAL ENGINEERING DEPARTMENT PSAS

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objects move along the same axis and do not change direction. The analysis of such systems may therefore be simplified by neglecting the direction components of the vectors involved and dealing only with the magnitude. An example of linear motion is that of a ball thrown straight up and falling back straight down. We will discuss the following terms that are required in analytical work in linear motion.

Distance and Displacement: Distance and displacement are two quantities that may seem to mean the same thing yet have distinctly different definitions and meanings. Displacement is a vector quantity that refers to the object's overall change in position. Displacement is a measurement of change in position of the particle in motion. Its magnitude and direction are measured by the length and direction of the straight line joining initial and final positions of the particle. Obviously, the length of the straight line between the positions is the shortest distance between the points. Distance is a scalar quantity that refers to "how much ground an object has covered" during its motion. Speed and Velocity: Speed is the rate of change of distance with time, and Velocity is the rate of change of displacement with time. Speed is the first derivative of distance with respect to time, and Velocity is the first derivative of displacement with respect to time. The average speed during the course of a motion is often computed using the following formula:

In contrast, the average velocity is often computed using this formula

Acceleration: It is the rate of change of velocity with respect to time. The term acceleration is used in general for an increase the magnitude of velocity wiht respect to time. a decrease in velocity is called deceleration.

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Acceleration shows the change in velocity in a unit time. Velocity is measured in meters per second, m/s, so acceleration is measured in (m/s)/s, or m/s2, which can be both positive and negative.

Uniform acceleration Uniform or constant acceleration is a type of motion in which the velocity of an object changes by an equal amount in every equal time period. Simple formulae that relate the following quantities: displacement, initial velocity, final velocity, acceleration, and time:

Where s = displacement u = initial velocity v = final velocity a = constant acceleration t = time

Name Distance / displacement Speed / velocity Acceleration

Symbol s u or v a

SI Unit Meter (m) Meter per second (m/s) Meter per second square (m/s2)

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5.0 BASIC CONCEPTS IN DYNAMICS 5.3

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ANGULAR MOTIONS

In physics, angular motion (also called circular motion) is rotation along a circular path or a circular orbit. It can be uniform, that is, with constant angular rate of rotation (and thus constant speed), or non-uniform, that is, with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. The equations describing angular motion of an object do not take size or geometry into account, rather, the motion of a point mass in a plane is assumed. In practice, the center of mass of a body can be considered to undergo angular motion.

Examples of angular motion include: an artificial satellite orbiting the Earth in geosynchronous orbit, a stone which is tied to a rope and is being swung in circles (cf. hammer throw), a racecar turning through a curve in a race track, an electron moving perpendicular to a uniform magnetic field, and a gear turning inside a mechanism.

Angular motion is accelerated even if the angular rate of rotation is constant, because the object's velocity vector is constantly changing direction. Such change in direction of velocity involves acceleration of the moving object by a centripetal force, which pulls the moving object toward the center of the circular orbit. Without this acceleration, the object would move in a straight line, according to Newton's laws of motion.

Angular displacement Angular displacement of a body is the angle in radians (degrees, revolutions) through which a point or line has been rotated in a specified sense about a specified axis.

In the example illustrated above, a particle on object P at a fixed distances r from the origin, O, rotating counterclockwise. It becomes important to then represent the position of particle P in terms of its polar coordinates (r, θ). In this particular example, the value of θ is changing, while the value of the radius remains the same. As the particle moves along the circle, it travels an arc length s, which becomes related to the angular position through the relationship: MECHANICAL ENGINEERING DEPARTMENT PSAS

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Angular displacement is measured in radians rather than degrees. This is because it provides a very simple relationship between distance traveled around the circle and the distance r from the centre.

For example if an object rotates 360 degrees around a circle radius r the angular displacement is given by the distance traveled the circumference which is 2πr divided by the radius in:

which easily simplifies to θ = 2π. Therefore 1 revolution is 2π radians.

Angular velocity In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per second, revolutions per minute, degrees per hour, etc. It is sometimes also called the rotational velocity and its magnitude the rotational speed, typically measured in cycles or rotations per unit time (e.g. revolutions per minute). Angular velocity is usually represented by the symbol omega (ω, rarely Ω).

The direction of the angular velocity vector is perpendicular to the plane of rotation, in a direction which is usually specified by the right-hand rule. ω P r θ O

The angular velocity of a particle is measured around or relative to a point, called the origin. As shown in the diagram (with angles θ in radians), if particle P is moving in a circle with fixed radius r, then the angular velocity of particle P is given by:

where dθ is the changed in angular displacement and dt is the time interval taken by the particle moves from origin to the current position. MECHANICAL ENGINEERING DEPARTMENT PSAS

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Angular acceleration Angular acceleration is the rate of change of angular velocity over time. In SI units, it is measured in radians per second squared (rad/s2), and is usually denoted by the Greek letter alpha (α). We can express angular acceleration in mathematical model as:

where ω is the angular velocity and r is the distance from the origin of the coordinate system that defines θ and ω to the point of interest.

5.4

THE RELATIONSHIPS OF LINEAR AND ANGULAR MOTIONS

Relationship between linear speed and angular speed If a point P move round a circle of radius r with constant linear speed, v, (see figure below) then the angular speed, ω, will be constant at

Where t is the time to move from Q to P along the arc QP of the curve. However, arc length QP is rθ when θ is measured in radians. Hence linear speed v is

Substituting Equation 1 into Equation 2 leads to this relationship for circular motion:

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Relationship between linear acceleration and angular acceleration

Known that angular acceleration is:

And angular velocity is:

Thus; ( )

As r is a constant this can be written (

)

and as ( )is linear acceleration a,

Name Angular displacement Angular velocity Angular acceleration

Symbol θ ω α

SI Unit Radian (rad) Radian per second (rad/s) Radian per second square (rad/s2)

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