Lecture 16 ME 231: Dynamics

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Clementine Hardy
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1 Kinematics of Particles (Ch. 2) Review Lecture 16

2 Question of the Day What is the most important concept in Chapter 2? Time Derivative of a Vector 2

3 Outline for Today Question of the day Where are we in the course? Categories of motion Choice of coordinate systems Inertial versus moving coordinate systems Degrees of freedom Velocity and acceleration Exam 1 breakdown (kinematics of particles) 3

4 Where are we in the course? Concept: What is dynamics? Chapters 1, 2, 6 Chapters 3, 5, 7, 8 Kinematics Kinetics Dynamics Relationship among position, velocity, and acceleration Relationship among forces and acceleration 4

A push or pull exerted on a body, characterized by: magnitude direction point

5 Where are we in the course? Calculation: How do we use dynamics? Newton s 2 nd Law Force. A push or pull exerted on a body, characterized by: magnitude direction point of application F = m a Mass. Measure of the resistance of a body to linear acceleration. Acceleration. Velocity rate of change with respect to time 5

6 Outline for Today Question of the day Where are we in the course? Categories of motion Choice of coordinate systems Inertial versus moving coordinate systems Degrees of freedom Velocity and acceleration Exam 1 breakdown (kinematics of particles) 6

7 Categories of Motion Geometry of a problem identifies the category. Lecture 2: rectilinear (1D) Lecture 3: curvilinear (2D) Lecture 6: space (3D) Many of our motion problems involve curvilinear (2D), or plane motion. 7

8 Curvilinear (2D) Time Derivative: Exercise A particle moving in two-dimensions has a position vector (r) as a function of time (t) with coordinates given by x(t) = t 2 4t + 20, where r is measured in inches and t is in seconds. Determine the velocity (v) and the acceleration (a). y(t) = 3 sin(2t) 8

9 Outline for Today Question of the day Where are we in the course? Categories of motion Choice of coordinate systems Inertial versus moving coordinate systems Degrees of freedom Velocity and acceleration Exam 1 breakdown (kinematics of particles) 9

10 Choice of Coordinate Systems Motion of a problem identifies the coordinate system. Rectangular (x, y, z) Polar (r,, z) Spherical (R,, ) Normal and Tangential (n, t) Many of our motion problems involve 2D rectangular coordinates (x, y). 10

11 Inertial Versus Moving Coordinate Systems Moving coordinate systems are measured with respect to an inertial coordinate system whose motion is negligible. 11

12 Inertial Versus Moving Coordinate Systems r r r A A/ v r r r A A A/ a v r r r A A A A/ Absolute position of is defined in an inertial coordinate system X-Y Attach a set of translating (non-rotating) axes x-y to particle and define the position of A Define position of A relative to ( A/ ) in x-y 12

13 Vector Representation: Exercise r r r A A/ v r r r A A A/ a v r r r A A A A/ Train A travels with constant speed v A = 120 km/h. Anticipating the need to stop, car decreases its speed of 90 km/h at the rate of 3 m/s 2. Determine the velocity and acceleration of the train relative to the car. 13

14 Outline for Today Question of the day Where are we in the course? Categories of motion Choice of coordinate systems Inertial versus moving coordinate systems Degrees of freedom Velocity and acceleration Exam 1 breakdown (kinematics of particles) 14

15 Degrees of Freedom Simple system of two interconnected particles With L, r 2, r 1, and b are constant Horizontal motion (x) of A is twice the vertical motion (y) of Only one variable (x or y) is needed to specify the positions of all parts of the system Constraint Equations L x r2 2y r1 b 2 0 x 2y 0 va 2v 0 x 2y 0 a 2 A a 15

16 Degrees of Freedom Position of lower cylinder depends on two variables (y A and y ) Constraint Equations L L A y y A 0 y 2 0 y 2 2yD constant y y y C A y D A y D constant C D 0 y 2y 0 y 2 y C C y y D D 0 y A 2y 4y 0 y 2 y 4y A C C 16

17 Degrees of Freedom: Exercise How many degrees of freedom are necessary to specify the position of all parts of the system of interconnected particles? 17

18 Outline for Today Question of the day Where are we in the course? Categories of motion Choice of coordinate systems Inertial versus moving coordinate systems Degrees of freedom Velocity and acceleration Exam 1 breakdown (kinematics of particles) 18

19 Velocity and Acceleration Lecture Velocity Acceleration 2. Rectilinear v lim t 0 s t ds dt a lim t0 v t dv dt 2 d s 2 dt 3. Curvilinear v r x i y j a v r x i y j 4. Normal & Tangential v v e t a 2 v e n v e t 5. Polar Coordinates v r er r e r r 2 a e r 2 e r r 7. Relative Motion v r r r A A A/ a v r r r A A A A/ 19

20 Outline for Today Question of the day Where are we in the course? Categories of motion Choice of coordinate systems Inertial versus moving coordinate systems Degrees of freedom Velocity and acceleration Exam 1 breakdown (kinematics of particles) 20

21 Exam 1 reakdown (kinematics of particles) 100 number of points Rectilinear Motion Curvilinear Motion Normal & Tangential 5. Polar Coordinates 0 6. Space Motion Relative Motoin 4 8. Constrained Motion lecture 21

22 Final Course Grades (thru HW #5 LAST YEAR) 25 number of students <60% 60-69% 70-79% 80-89% % >100% calculated grade 22

23 Final Course Grades (thru HW #5 THIS YEAR) number of students <60% 60-69% 70-79% 80-89% % >100% calculated grade 23

24 For Next Time Complete Homework #6 due on Tuesday (10/2) Review Chapter 6 24

EQUATIONS OF MOTION: RECTANGULAR COORDINATES

EQUATIONS OF MOTION: RECTANGULAR COORDINATES Today s Objectives: Students will be able to: 1. Apply Newton s second law to determine forces and accelerations for particles in rectilinear motion. In-Class