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ดร. พิภัทร พฤกษาโรจนกุล phiphat.phr@kmutt.ac.th
MEE 212 Monday 13:30 – 16:30 CB2405 ดร. พิภัทร พฤกษาโรจนกุล สอบกลางภาค, สอบครั้งที่ 2 และ สอบปลายภาค (ภาษาอังกฤษ) 18/11/61 ME ดร. พิภัทร
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Kinematics of a particle
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Objectives Concepts of position, displacement, velocity, and acceleration. Particle motion along a straight line Particle motion along a curved path using different coordinate systems. Analysis of dependent motion of two particles. • Principles of relative motion of two particles using translating axes. 18/11/61 ME ดร. พิภัทร
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Introduction to Dynamics
Statics Engineering Mechanics Dynamics พลวัต Kinematics Kinetics 18/11/61 ME ดร. พิภัทร
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Rectilinear Kinematics
Origin Define a fixed point in space Position Defined by a position vector r or an algebraic scalar s 18/11/61 ME ดร. พิภัทร
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Rectilinear Kinematics
Displacement Change in position Velocity 18/11/61 ME ดร. พิภัทร
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Rectilinear Kinematics
Acceleration 18/11/61 ME ดร. พิภัทร
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Constant Acceleration
18/11/61 ME ดร. พิภัทร
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EXAMPLE 1 The car moves in a straight line such that for a short time its velocity is defined by v = (0.9t t) m/s where t is in sec. Determine it position and acceleration when t = 3s. When t = 0, s = 0. 18/11/61 ME ดร. พิภัทร
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EXAMPLE 2 A small projectile is forced downward into a
fluid medium with an initial velocity of 60m/s. Due to the resistance of the fluid the projectile experiences a deceleration equal to a = (-0.4v3)m/s2, where v is in m/s2. Determine the projectile’s velocity and position 4s after it is fired. 18/11/61 ME ดร. พิภัทร
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Problem 12-6 A freight train travels at v=20(1-e-t) m/s, where t is the elapsed time in seconds. Determine the distance traveled in three seconds, and the acceleration at this time. 18/11/61 ME ดร. พิภัทร
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Problem 12-10 A particle is moving along a straight line such that its acceleration is defined as a=(-2v) m/s2, where v is in meters per second. If v=20 m/s when s=0 and t=0, determine the particle’s velocity as a function of position and the distance the particle moves before it stops. 18/11/61 ME ดร. พิภัทร
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Problem 12-28 The acceleration of a particle along a straight line is defined by a=(2t-9) m/s2, where t is in seconds. At t=0, s=1 m and v=10 m/s. When t=9 s, determine (a) the particle’s position, (b) the total distance traveled and (c) the velocity. 18/11/61 ME ดร. พิภัทร
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General Curvilinear Motion
Curvilinear motion occurs when the particle moves along a curved path Position. The position of the particle, measured from a fixed point O, is designated by the position vector r = r(t). 18/11/61 ME ดร. พิภัทร
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General Curvilinear Motion
Displacement. Suppose during a small time interval Δt the particle moves a distance Δs along the curve to a new position P`, defined by r` = r + Δr. The displacement Δr represents the change in the particle’s position. 18/11/61 ME ดร. พิภัทร
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General Curvilinear Motion
Velocity 18/11/61 ME ดร. พิภัทร
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General Curvilinear Motion
Acceleration. 18/11/61 ME ดร. พิภัทร
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Curvilinear Motion: Rectangular Components
Position. Position vector is defined by The magnitude of is always positive and defined as The direction of r is specified by the components of the unit vector 18/11/61 ME ดร. พิภัทร
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Curvilinear Motion: Rectangular Components
Velocity. The velocity has a magnitude defined as the positive value of and a direction that is specified by the components of the unit vector and is always tangent to the path. 18/11/61 ME ดร. พิภัทร
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Curvilinear Motion: Rectangular Components
Acceleration. The acceleration has a magnitude defined as the positive value of 18/11/61 ME ดร. พิภัทร
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Curvilinear Motion: Rectangular Components
The acceleration has a direction specified by the components of the unit vector Since a represents the time rate of change in velocity, a will not be tangent to the path. 18/11/61 ME ดร. พิภัทร
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Example 3 At any instant the horizontal position of the weather balloon is defined by x = (9t) m, where t is in second. If the equation of the path is y = x2/30, determine the distance of the balloon from the station at A, the magnitude and direction of the both the velocity and acceleration when t = 2 s. 18/11/61 ME ดร. พิภัทร
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Example 4 The motion of box B is defined by the position vector r = {0.5sin(2t)i + 0.5cos(2t)j – 0.2tk} m, where t is in seconds and the arguments for sine and cosine are in radians (π rad = 180°). Determine the location of box when t = 0.75 s and the magnitude of its velocity and acceleration at this instant. 18/11/61 ME ดร. พิภัทร
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Motion of Projectile Constant downward acceleration, no air resistance
Mathematical expressions, ↑ [=] +, → [=] + 18/11/61 ME ดร. พิภัทร
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Example 5 The chipping machine is designed to eject wood at chips vO = 7.5 m/s. If the tube is oriented at 30° from the horizontal, determine how high, h, the chips strike the pile if they land on the pile 6 m from the tube. 18/11/61 ME ดร. พิภัทร
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Curvilinear Motion: Normal and Tangential Components
When the path of motion of a particle is known, describe the path using n and t coordinates which act normal and tangent to the path Consider origin located at the particle Tangential direction ทิศทางสัมผัส Normal direction ทิศทางตั้งฉาก 18/11/61 ME ดร. พิภัทร
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Curvilinear Motion: Normal and Tangential Components
Velocity. Since the particle is moving, s is a function of time Particle’s velocity v has direction that is always tangent to the path and a magnitude that is determined by taking the time derivative of the path function s = s(t) 18/11/61 ME ดร. พิภัทร
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Curvilinear Motion: Normal and Tangential Components
Acceleration Acceleration of the particle is the time rate of change of velocity 18/11/61 ME ดร. พิภัทร
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Curvilinear Motion: Normal and Tangential Components
Acceleration Find 18/11/61 ME ดร. พิภัทร
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Curvilinear Motion: Normal and Tangential Components
18/11/61 ME ดร. พิภัทร
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Example 6 Race car C travels round the horizontal circular track that has a radius of 90 m. If the car increases its speed at a constant rate of 2.1 m/s2, starting from rest, determine the time needed for it to reach an acceleration of 2.4 m/s2. What is its speed at this instant? 18/11/61 ME ดร. พิภัทร
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Problem At a given instant the train engine at E has a speed of 20 m/s and an acceleration of 14 m/s2 acting in the direction shown. Determine the rate of increase in the train’s speed and the radius of curvature of the path. 18/11/61 ME ดร. พิภัทร
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Problem The two particles A and B start at the origin O and travel in opposite directions along the circular path at constant speeds vA=0.7 m/s and vB=1.5 m/s respectively. Determine at t= 2s, (a) the displacement along the path of each particle, (b) the position vector to each particle, and (c) the shortest distance between the particles. 18/11/61 ME ดร. พิภัทร
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Curvilinear Motion: Cylindrical Components
Fixed origin Radial direction Transverse direction 18/11/61 ME ดร. พิภัทร
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Curvilinear Motion: Cylindrical Components
Position 18/11/61 ME ดร. พิภัทร
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Curvilinear Motion: Cylindrical Components or Polar
Velocity 18/11/61 ME ดร. พิภัทร
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Curvilinear Motion: Cylindrical Components
Acceleration 18/11/61 ME ดร. พิภัทร
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Example 7 The rob OA is rotating in the horizontal plane such that θ = (t3) rad. At the same time, the collar B is sliding outwards along OA so that r = (100t2)mm. If in both cases, t is in seconds, determine the velocity and acceleration of the collar when t = 1s. 18/11/61 ME ดร. พิภัทร
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Example 8 The searchlight casts a spot of light along the face of a wall that is located 100m from the searchlight. Determine the magnitudes of the velocity and acceleration at which the spot appears to travel across the wall at the instant θ = 45°. The searchlight is rotating at a constant rate of 4 rad/s 18/11/61 ME ดร. พิภัทร
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Problem A truck is traveling along the horizontal circular curve of radius r=60 m with a constant speed v=20 m/s. Determine the truck’s radial and transverse components of acceleration. Determine angular rate of the radial line. 18/11/61 ME ดร. พิภัทร ME221
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Absolute Dependent Motion
Dependent motions of two particles are normally associated with systems of connected masses via inextensible cords and pulleys. 18/11/61 ME ดร. พิภัทร
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Absolute Dependent Motion
18/11/61 ME ดร. พิภัทร
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Example 9 Determine the speed of block A if block B has an upward speed of 2m/s. 18/11/61 ME ดร. พิภัทร
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Problem If the hydraulic cylinder at H draws rod BC in by 200 mm, determine how far the slider at A moves. 18/11/61 ME ดร. พิภัทร
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Example 10 A man at A s hoisting a safe S by walking to the right with a constant velocity vA = 0.5m/s. Determine the velocity and acceleration of the safe when it reaches the elevation at E. The rope is 30m long and passes over a small pulley at D. 18/11/61 ME ดร. พิภัทร
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Problem If block A is moving downward with a speed of 1.2 m/s while C is moving up at 0.6 m/s, determine the speed of block B. 18/11/61 ME ดร. พิภัทร
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Problem The cord is attached to the pin at C and passes over the two pulleys at A and D. The pulley at A is attached to the smooth collar that travels along the vertical rod. Determine the velocity of the end of the cord at B if at the instant SA = 1.2 m the collar is moving upwards at 1.5 m/s. 18/11/61 ME ดร. พิภัทร
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Relative Motion Analysis
The relative position of B with respect to A is given by The relative velocity and acceleration of B with respect to A are given by 18/11/61 ME ดร. พิภัทร
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Example 11 A train, traveling at a constant speed of 90km/h, crosses over a road. If automobile A is traveling t 67.5km/h along the road, determine the magnitude and direction of relative velocity of the train with respect to the automobile. 18/11/61 ME ดร. พิภัทร
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