mechanic machine

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Engineering Mechanics :DYNAMICS

JJ205

PN NORHAYATI BINTI AHMADPOLITECHNIC IBRAHIM SULTAN

Department of Mechanical Engineering

CHAPTER 6

KINEMATICS OF PARTICLES

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OBJECTIVES:

Understand concepts of kinematics * Explain the concepts of kinematics of particles * Understand rectilinear motion of particles * Describe position, velocity and acceleration motion of particles * Determine motion of a particles * Describe uniform rectilinear motion

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Rectilinear motionPosition, velocity, and acceleration of a particle as it moves along a straight line

Moving in a negativedirection from the origin

Moving in a positivedirection from the origin

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Position of a particle is defined by positive or negative distance of particle from a fixed origin on the line.

Displacement of a particle is defined as the change in its position during the interval time Δt .

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Velocity v of the particle is equal to the time derivative of the position coordinate s

Instantaneous velocity may be positive or negative. Magnitude of velocity is referred to as particle speed.

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Acceleration a is obtained by differentiating v withrespect to t

We can also express a as

tdt

dva

ttv

dt

xd

dt

dv

t

va

t

612

312e.g.

lim

2

2

2

0

• From the definition of a derivative,

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• The velocity v and acceleration a are represented by algebraic numbers which can be positive or negative.

• A positive value for v indicates that the particle moves in the positive direction, and a negative value that it moves in the negative direction.

• A positive value for a, however, may mean that the particle is truly accelerated (i.e., moves faster) in the positive direction, or that it is decelerated (i.e., moves more slowly) in the negative direction.

• A negative value for a is subject to a similar interpretation.

Example 1

A particle moves on a straight line with the relationship s=t3-9t2+15t where s is in meters and t is in seconds. Determine the displacement, velocity and acceleration when t=2s. Determine also the distance travelled for the duration 0 ≤ t ≤ 6 s.

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Example 2

A particle moves on a straight line with the relationship v = 3t2 -12t where v is in m/s and t is in seconds and s=10m when t=0. determine the displacement, velocity and acceleration when t=2s. Determine also the distance travelled for the duration 0 ≤ t ≤ 6 s.

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• Consider particle with motion given by

326 ttx

Example

2312 ttdt

dxv

tdt

xd

dt

dva 612

2

2

• at t = 0, x = 0, v = 0, a = 12 m/s2

• at t = 2 s, x = 16 m, v = vmax = 12 m/s, a = 0

• at t = 4 s, x = xmax = 32 m, v = 0, a = -12 m/s2

• at t = 6 s, x = 0, v = -36 m/s, a = 24 m/s2

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Graphical interpretations

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Analytical integration

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Example

A particle moves on a straight line with the velocity profile shown. Determine

a) a when t=7s,b) a when t=20s,c) s when t=10s,d) distance travelled for the duration 0 ≤ t ≤ 25 se) s when t=12s.

Given that when t = 0, s = -50m

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A particle moves along the x-axis with an initial velocity vx=50 m/s at the origin when t = 0. For the first 4 seconds it has no acceleration, and there after it is acted on by a retarding force which gives it a constant acceleration ax= −10 m/s2. Calculate the velocity and the x-coordinate of the particle for the conditions of t =8 s and t = 12 s and find the maximum positive x-coordinate reached by the particle.

Example