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King Fahd University of Petroleum & Minerals Mechanical Engineering Dynamics ME 201 BY Dr. Meyassar N. Al-Haddad Lecture # 2

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Objective l To introduce the concepts of position, displacement, velocity, and acceleration. l To study particle motion along a straight line.

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Rectilinear Kinematics Section 12.2 l Rectilinear : Straight line motion l Kinematics : Study the geometry of the motion dealing with s, v, a. l Rectilinear Kinematics : To identify at any given instant, the particle’s position, velocity, and acceleration. (All objects such as rockets, projectiles, or vehicles will be considered as particles “has negligible size and shape” particles : has mass but negligible size and shape

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Position l Position : Location of a particle at any given instant with respect to the origin r : Displacement ( Vector ) s : Distance ( Scalar )

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Distance & Displacement l Displacement : defined as the change in position. l r : Displacement ( 3 km ) l s : Distance ( 8 km ) Total length l For straight-line Distance = Displacement s = r s r Vector is direction oriented r positive (left ) r negative (right)

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Velocity & Speed l Velocity : Displacement per unit time l Average velocity : V = r t l Speed : Distance per unit time l Average speed : l sp s T t ( Always positive scalar ) l Speed refers to the magnitude of velocity l Average velocity : avg = s / t

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Velocity (con.) l Instantaneous velocity : For straight-line r = s

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Problem l A particle moves along a straight line such that its position is defined by s = (t 3 – 3 t 2 + 2 ) m. Determine the velocity of the particle when t = 4 s. At t = 4 s, the velocity = 3 (4) 2 – 6(4) = 24 m/s

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Acceleration l Acceleration : The rate of change in velocity {(m/s)/s} l Average acceleration : l Instantaneous acceleration : l If v ‘ > v “ Acceleration “ l If v ‘ < v “ Deceleration”

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Problem l A particle moves along a straight line such that its position is defined by s = (t 3 – 3 t 2 + 2 ) m. Determine the acceleration of the particle when t = 4 s. l At t = 4 a(4) = 6(4) - 6 = 18 m/s 2

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Problem l A particle moves along a straight line such that its position is defined by s = (t 3 – 12 t 2 + 36 t -20 ) cm. Describe the motion of P during the time interval [0,9] t02469 s-2012-4-2061 v360-12063 a-24-1201230 Total time = 9 seconds Total distance = (32+32+81)= 145 meter Displacement = form -20 to 61 = 81 meter Average Velocity = 81/9= 9 m/s to the right Speed = 9 m/s Average speed = 145/9 = 16.1 m/s Average acceleration = 27/9= 3 m/s 2 to the right

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Relation involving s, v, and a No time t Acceleration Velocity Position s

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Problem 12.18 l A car starts from rest and moves along a straight line with an acceleration of a = ( 3 s -1/3 ) m/s 2. where s is in meters. Determine the car’s acceleration when t = 4 s. Rest t = 0, v = 0

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For constant acceleration a = a c

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Velocity as a Function of Time

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Position as a Function of Time

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velocity as a Function of Position

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Free Fall l Ali and Omar are standing at the top of a cliff of height H. Both throw a ball with initial speed v 0, Ali straight down and Omar straight up. The speed of the balls when they hit the ground are v A and v O respectively. Which of the following is true: (a) v A v O v0v0v0v0 v0v0v0v0 OmarAli H vAvAvAvA vOvOvOvO

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Free fall… Free fall… l Since the motion up and back down is symmetric, intuition should tell you that v = v 0 çWe can prove that your intuition is correct: v0v0v0v0 Omar H v = v 0 Equation: This looks just like Omar threw the ball down with speed v 0, so the speed at the bottom should be the same as Ali’s ball. y = 0

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Free fall… l We can also just use the equation directly: Ali : v0v0v0v0 v0v0v0v0 AliOmar y = 0 Omar: same !! y = H

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Summary l Time dependent acceleration l Constant acceleration This applies to a freely falling object:

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Gravity facts: l g does not depend on the nature of the material! çGalileo (1564-1642) figured this out without fancy clocks & rulers! l Nominally, g = 9.81 m/s 2 çAt the equatorg = 9.78 m/s 2 çAt the North poleg = 9.83 m/s 2 l More on gravity in a few lectures!

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Kinematics Kinematics is the branch of physics that describes the motion of points, bodies (objects) and systems of bodies (groups of objects) without.

Kinematics Kinematics is the branch of physics that describes the motion of points, bodies (objects) and systems of bodies (groups of objects) without.

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