1. 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)
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2. Kinematics of a particle

3. 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.
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4. Introduction to Dynamics
Statics
Engineering
Mechanics
Dynamics พลวัต
Kinematics
Kinetics
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5. Rectilinear Kinematics
Origin
Define a fixed point in space
Position
Defined by a position vector r
or an algebraic scalar s
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6. Rectilinear Kinematics
Displacement
Change in position
Velocity
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7. Rectilinear Kinematics
Acceleration
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8. Constant Acceleration
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9. 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.
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10. 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.
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11. 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.
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12. 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.
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ME ดร. พิภัทร

13. 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.
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ME ดร. พิภัทร

14. 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).
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15. 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.
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16. General Curvilinear Motion
Velocity
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17. General Curvilinear Motion
Acceleration.
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18. 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
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19. 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.
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20. Curvilinear Motion: Rectangular Components
Acceleration.
The acceleration has a magnitude defined as the positive value of
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21. 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.
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ME ดร. พิภัทร

22. 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.
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ME ดร. พิภัทร

23. 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.
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ME ดร. พิภัทร

24. Motion of Projectile Constant downward acceleration, no air resistance
Mathematical expressions, ↑ [=] +, → [=] +
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25. 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.
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ME ดร. พิภัทร

26. 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
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27. 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)
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28. Curvilinear Motion: Normal and Tangential Components
Acceleration
Acceleration of the particle is the time rate of change of velocity
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29. Curvilinear Motion: Normal and Tangential Components
Acceleration
Find
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30. Curvilinear Motion: Normal and Tangential Components
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31. 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?
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32. 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.
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ME ดร. พิภัทร

33. 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.
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ME ดร. พิภัทร

34. Curvilinear Motion: Cylindrical Components
Fixed origin
Radial direction
Transverse direction
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35. Curvilinear Motion: Cylindrical Components
Position
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36. Curvilinear Motion: Cylindrical Components or Polar
Velocity
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37. Curvilinear Motion: Cylindrical Components
Acceleration
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38. 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.
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ME ดร. พิภัทร

39. 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
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ME ดร. พิภัทร

40. 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.
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ME ดร. พิภัทร
ME221

41. Absolute Dependent Motion
Dependent motions of two particles are normally associated with systems of connected masses via inextensible cords and pulleys.
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42. Absolute Dependent Motion
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43. Example 9
Determine the speed of block A if block B has an upward speed of 2m/s.
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44. Problem
If the hydraulic cylinder at H draws rod BC in by 200 mm, determine how far the slider at A moves.
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45. 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.
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46. 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.
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47. 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.
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48. 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
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49. 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.
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ME ดร. พิภัทร