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- 1. Faculty of Engineering ENG1040 Engineering Dynamics Kinematics of a Particle Dr Lau Ee Von – Sunway Lecture 4 ENG1040 – Engineering Dynamics
- 2. Lecture outline Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion • Concepts of position, displacement, velocity, and acceleration • Study particle motion along a straight line • Erratic motion: the graphical method • Projectile motion – two dimensional motion
- 3. Rectilinear Kinematics: Continuous Motion Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion RECTILINEAR KINEMATICS Defines a particle’s position, displacement, velocity, and acceleration at any instant in time.
- 4. Rectilinear Kinematics: Continuous Motion Lecture Outline Revision: kinematics Example: Kinematics POSITION • A particle’s position is defined from an origin. • We must always define a coordinate system to a problem. Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion DISPLACEMENT
- 5. Rectilinear Kinematics: Continuous Motion VELOCITY Lecture Outline Revision: kinematics • Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion SPEED
- 6. Rectilinear Kinematics: Continuous Motion Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion ACCELERATION
- 7. Rectilinear Kinematics: Continuous Motion ACCELERATION Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion +
- 8. Rectilinear Kinematics: Continuous Motion ACCELERATION Lecture Outline Revision: kinematics Example: Kinematics Example from Lecture 3: Coordinate system 2 vB 2 vA Erratic motion: Graphical method 2aC ( s B sA ) -9.81 m/s2 Example: Graphical method Projectile motion Example: Projectile motion 2 vC 2 vB 2aC ( sC -9.81 m/s2 sB ) +
- 9. Kinematics Lecture Outline by definition: Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion by rearranging: Example: Projectile motion 9 9
- 10. Kinematics Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method When do we use a dv dt and a v dv ? ds Example: Find the velocity if s = 2m when t = 0 s Given a = 20t Given a = 20s Example: Graphical method Projectile motion Example: Projectile motion 10
- 11. Kinetics/Kinematics problems... Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method Analysis procedure 1. Establish a coordinate system 2. Draw Free Body Diagram(s) • Graphical representation of all forces acting on the system. 3. Establish known & unknown quantities Projectile motion Example: Projectile motion 4. Apply Equation(s) of Motion in each direction 5. Evaluate kinematics to solve problem
- 12. Example 12.4 Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion A metallic particle travels downward through a fluid that extends from plate A and plate B under the influence of magnetic field. If particle is released from rest at midpoint C, s = 100 mm, and acceleration, a = (4s) m/s2, where s in meters, determine velocity when it reaches plate B and time need to travel from C to B.
- 13. Example 12.4 Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method Step 1: choose a coordinate system Projectile motion Example: Projectile motion
- 14. Example 12.4 Lecture Outline Revision: kinematics a v Example: Kinematics dv dt ds dt a dv v ds Erratic motion: Graphical method Example: Graphical method Step 2: employ kinematics Projectile motion Example: Projectile motion a = (4s) m/s2
- 15. Example 12.4 Lecture Outline Revision: kinematics a v Example: Kinematics dv dt ds dt a dv v ds Erratic motion: Graphical method Example: Graphical method Step 2: employ kinematics Projectile motion Example: Projectile motion a = (4s) m/s2 a dv v ds 4s dv v ds
- 16. Example 12.4 Lecture Outline Revision: kinematics a Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion dv dt v ds dt a dv v ds Step 2: employ kinematics a dv v ds s dv v ds 4s v 4sds sinitial vdv vinitial
- 17. Example 12.4 Lecture Outline Revision: kinematics a v Example: Kinematics Erratic motion: Graphical method Example: Projectile motion ds dt a dv v ds Step 2: employ kinematics Example: Graphical method Projectile motion dv dt s v 4sds sinitial 2s 2 s sinitial vdv vinitial 1 2 v 2 v vinitial
- 18. Example 12.4 Lecture Outline Revision: kinematics a v Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion dv dt ds dt a dv v ds Step 2: employ kinematics s v 4sds sinitial 2s 2 s 0.1 vdv vinitial 1 2 v 2 v 0 Only put in initial limits
- 19. Example 12.4 Step 2: employ kinematics Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method 1 2 v 2 v 2s 2 s Example: Graphical method Projectile motion Example: Projectile motion 2 2 a 2 0.1 0.01 Leave it in the general form of equation dv dt v ds dt a dv v ds Substitute sb = 200mm = 0.2m to find vb vb 0.346 m / s
- 20. Example 12.4 Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion A metallic particle travels downward through a fluid that extends from plate A and plate B under the influence of magnetic field. If particle is released from rest at midpoint C, s = 100 mm, and acceleration, a = (4s) m/s2, where s in meters, determine velocity when it reaches plate B and time need to travel from C to B.
- 21. Example 12.4 Time to reach plate B? Lecture Outline Revision: kinematics a Example: Kinematics Erratic motion: Graphical method a Example: Graphical method 2s Projectile motion Example: Projectile motion v dv v ds 2 s 4s 1 2 v 2 2 s 0.1 2 dv v ds dv dt v ds dt a dv v ds v 0 0.01 Use general form of equation!
- 22. Example 12.4 Lecture Outline Time to reach plate B? Revision: kinematics v Example: Kinematics a dv dt v ds dt a dv v ds ds v dt Erratic motion: Graphical method 2s Example: Graphical method s 0.1 Projectile motion Example: Projectile motion 2 s 2 0.01 t 2 0.01 ds s 2 0.1 t 0.5 0 0.5 dt 2 dt ln (0.2) 2 0.01 s 2 Only put in initial limits 2.303 Leave it in the general form of equation
- 23. Example 12.4 Lecture Outline Revision: kinematics Substitute sb = 200mm = 0.2m to find tb Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion t = 0.658s a dv dt v ds dt a dv v ds NOTE: Why can’t we use v u at s s0 ut 1 2 ? at 2 Acceleration is NOT a constant (a = 4s)
- 24. Faculty of Engineering ENG1040 Engineering Dynamics Erratic motion and graphical methods Dr Greg Sheard - Clayton Dr Lau Ee Von - Sunway Lecture 4 ENG1040 – Engineering Dynamics
- 25. Erratic motion and graphical methods Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion 25
- 26. Erratic motion and graphical methods Lecture Outline Revision: kinematics • When particle’s motion is erratic, it is described graphically using a series of curves • A graph is used to described the relationship with any 2 of the variables: Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion a, v, s, t • a We use dv dt v ds dt a dv v ds
- 27. Example 12.6 Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion A bicycle moves along a straight road such that it position is described by the graph as shown. Construct the v-t and a-t graphs for 0 ≤ t ≤ 30s.
- 28. Example 12.6 Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion A bicycle moves along a straight road such that it position is described by the graph as shown. Construct the v-t and a-t graphs for 0 ≤ t ≤ 30s. a dv dt ds v dt dv a v ds
- 29. Example 12.6 Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion Solution v-t Graph By differentiating the equations defining the s-t graph, we have ds v 0.6t 2 0 t 10s; s 0.3t dt ds 10s t 30s; s 6t 30 v 6 dt
- 30. Example 12.6 Lecture Outline Solution Revision: kinematics Example: Kinematics a-t Graph Erratic motion: Graphical method By differentiating the eqns defining the lines of the v-t graph, Example: Graphical method Example: Projectile motion 0.6t a 10 Projectile motion 0 t 10 s; v 6 a t 30 s; v dv dt dv dt 0.6 0 a dv v ds a dv dt
- 31. Example 12.7 Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion A test car starts from rest and travels along a straight track such that it accelerates at a constant rate for 10 s and then decelerates at a constant rate. Draw the v-t and s-t graphs and determine the time t’ needed to stop the car. How far has the car traveled?
- 32. Example 12.7 Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion A test car starts from rest and travels along a straight track such that it accelerates at a constant rate for 10 s and then decelerates at a constant rate. Draw the v-t and s-t graphs and determine the time t’ needed to stop the car. How far has the car traveled? a dv dt v ds dt a dv v ds
- 33. Example 12.7 Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Solution v-t Graph Using initial condition v = 0 when t = 0, v 0 t 10s a 10; 0 dv t 0 10 dt , v 10 t When t = 10s, v = 100m/s, 10s t t; a v 2; 100 v dv t 10 2 dt , 2t 120 Example: Projectile motion a dv dt
- 34. Example 12.7 Solution Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method s-t Graph. Using initial conditions s = 0 when t = 0, s 0 t 10s; v 10t; 0 t ds 0 5t 2 When t = 10s, s = 500m, 10s t 60s; v s 2t 120; ds 500 Projectile motion 10 t dt , s s t 10 2t 120 dt t 2 120 t 600 Example: Projectile motion v ds dt
- 35. Example 12.7 Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion A test car starts from rest and travels along a straight track such that it accelerates at a constant rate for 10 s and then decelerates at a constant rate. Draw the v-t and s-t graphs and determine the time t’ needed to stop the car. How far has the car traveled? a dv dt v ds dt a dv v ds
- 36. Example 12.7 Lecture Outline Revision: kinematics Time needed to stop the car? Solution Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion 10s t t; v 2t 120 When t = t’, v = 0 t’ = 60 s
- 37. Example 12.7 Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion A test car starts from rest and travels along a straight track such that it accelerates at a constant rate for 10 s and then decelerates at a constant rate. Draw the v-t and s-t graphs and determine the time t’ needed to stop the car. How far has the car traveled? a dv dt v ds dt a dv v ds
- 38. Example 12.7 Lecture Outline Revision: kinematics Total distance travelled? Solution Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion 10 s t 60 s; s t 2 120 t 600 When t = t’ = 60s, s = 3000m
- 39. Faculty of Engineering ENG1040 Engineering Dynamics Projectile motion Dr Greg Sheard – Clayton Dr Lau Ee Von – Sunway Lecture 4 ENG1040 – Engineering Dynamics
- 40. Projectile motion Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion 40
- 41. Projectile motion Lecture Outline Revision: kinematics Example: Kinematics We can resolve the velocity or acceleration to its x and y directions, and vice versa. a dv dt v ds dt a dv v ds Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion 41
- 42. Projectile motion Lecture Outline Some simplifications (for ENG1040) Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion • • • • • Projectile’s acceleration always acts vertically Projectile launched at (x0, y0) and path is defined in the x-y plane Fluid resistance is neglected Only force is its weight downwards ac = g = 9.81 m/s2 (constant downwards acceleration)
- 43. 12.6 Motion of Projectile Lecture Outline Revision: kinematics Horizontal Motion • Since ax = 0, Example: Kinematics • Erratic motion: Graphical method v Example: Graphical method x Projectile motion Example: Projectile motion We can use the constant acceleration equations v2 • v0 ac t ; 1 2 x0 v0t ac t ; 2 2 v0 2ac ( s s0 ); vx (v0 ) x x x0 (v0 ) x t vx (v0 ) x Horizontal component of velocity remain constant during the motion
- 44. 12.6 Motion of Projectile Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion Vertical Motion • Positive y axis is upward, thus ay = - g Once again, we can use the constant acceleration equations: v v0 ac t ; y v 2 y0 2 0 v 1 2 v0t ac t ; 2 2ac ( y y0 ); vy y 2 vy (v0 ) y gt 1 2 y0 (v0 ) y t gt 2 (v0 ) 2 2 g ( y y0 ) y
- 45. 12.6 Motion of Projectile Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion PROCEDURE FOR ANALYSIS 1. Establish a coordinate system 2. Sketch the trajectory of the particle 3. Specify 3 unknowns and data between any two points on the path 4. Employ the equations of motion 5. Acceleration of gravity always acts downwards 6. Express the particle initial and final velocities in the x, y components Note: Positive and negative position, velocity and acceleration components always act in accordance with their associated coordinate directions
- 46. Example Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion The chipping machine is designed to eject wood chips vO = 7.5 m/s. If the tube is oriented at 30° from the horizontal, determine how high, h, a chip is when it is 6 metres away (horizontally) from the tube.
- 47. Example Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method The chipping machine is designed to eject wood chips at vO = 7.5 m/s. If the tube is oriented at 30° from the horizontal, determine how high a chip is when it is 6 metres away (horizontally) from the tube. Step 1: Establish a coordinate system: Example: Graphical method Projectile motion Example: Projectile motion y x
- 48. Example Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion The chipping machine is designed to eject wood chips at vO = 7.5 m/s. If the tube is oriented at 30° from the horizontal, determine how high a chip is when it is 6 metres away (horizontally) from the tube. Step 2: Determine the vertical and horizontal components of initial velocity (vO ) x (7.5 cos30 ) (vO ) y (7.5 sin 30 ) 3.75m / s 6.5m / s
- 49. Example Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion The chipping machine is designed to eject wood chips at vO = 7.5 m/s. If the tube is oriented at 30° from the horizontal, determine how high a chip is when it is 6 metres away (horizontally) from the tube. Step 3: Apply the (relevant) equations of motion yA yO (v0 ) y tOA 1 2 gtOA 2 1 equation, 2 unknowns Vertical motion vy (v0 ) y y y0 (v0 ) y t 2 vy (v0 ) 2 y gt 1 2 gt 2 2 g ( y y0 ) Remember: these are just simplified constant acceleration equations
- 50. Example Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method The chipping machine is designed to eject wood chips at vO = 7.5 m/s. If the tube is oriented at 30° from the horizontal, determine how high a chip is when it is 6 metres away (horizontally) from the tube. Step 3: Apply the (relevant) equations of motion Example: Graphical method vx xA Projectile motion Example: Projectile motion Horizontal motion x0 (v0 ) x tOA tOA 0.923s (v0 ) x x x0 (v0 ) x t vx (v0 ) x Remember: these are just simplified constant acceleration equations
- 51. Example Lecture Outline Revision: kinematics Example: Kinematics Erratic motion: Graphical method Example: Graphical method Projectile motion Example: Projectile motion The chipping machine is designed to eject wood chips at vO = 7.5 m/s. If the tube is oriented at 30° from the horizontal, determine how high a chip is when it is 6 metres away (horizontally) from the tube. Step 3: Apply the (relevant) equations of motion tOA yA 0.9231 s yO h 1.38m (v0 ) y tOA Vertical motion vy 1 2 gtOA 2 (v0 ) y y y0 (v0 ) y t 2 vy (v0 ) 2 y gt 1 2 gt 2 2 g ( y y0 ) Remember: these are just simplified constant acceleration equations
- 52. Conclusions • We have considered the rectilinear equations for kinematics in three situations: • 1 dimensional motion – rectilinear, continuous motion • 1 dimensional motion – erratic motion • The ‘Graphical’ method • 2 dimensional motion • Projectile motion 52

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