1. Coordinates

2. Cartesian Coordinates  Cartesian coordinates form a linear system. Perpendicular axesPerpendicular axes Right-handed systemRight-handed system Values add linearlyValues add linearly  Coordinates are ( x, y, z ). Alternate: ( x 1, x 2, x 3 )Alternate: ( x 1, x 2, x 3 )  Standard unit vectors are directed along the axes. Notation with a hat – i, j, kNotation with a hat – i, j, k Alternate: vector labeled eAlternate: vector labeled e x=x 1 y=x 2 z=x 3

3. Cylindrical Coordinates  Cylindrical polar coordinates are an alternate perpendicular system. Coordinates are ( , z )Coordinates are ( , z ) Alternate: ( q 1, q 2, q 3 )Alternate: ( q 1, q 2, q 3 )  Coordinates transformations go between systems. Cylindrical to CartesianCylindrical to Cartesian x=x 1 y=x 2 z=q 3  =q 2  =q 1 z=x 3

4. Plane Polar Coordinates  Plane polar coordinates are a 2-dimensional system. Coordinates are ( r,  )Coordinates are ( r,  )  Unit vectors can be defined at the point r. Axial component rAxial component r Transverse component Transverse component  x y r 

5. Newton Transformed  Newton’s second law is well- defined in Cartesian coordinates.  The second law is not the same in polar coordinates. Extra terms Extra forces?  Is there a way to write Newton’s law independently of the coordinate system?

6. Constraints  Some systems have constraints on the motion. Move along a wireMove along a wire Swing on a pendulumSwing on a pendulum Stay on a surfaceStay on a surface Roll without slippingRoll without slipping  Constraints remove coordinates from the motion or link coordinates together. Problem  A plane pendulum swings freely from a fixed point. Write the constraint on the Cartesian coordinates.  The pendulum length L is constant. Origin at the pivot L 2 = x 2 + y 2 is the constraint

7. Holonomic Constraints  A holonomic constraint is a function of only the coordinates and time.  (x 1, x 2, x 3, t) = 0  (x 1, x 2, x 3, t) = 0 Fixed constraint without timeFixed constraint without time Moving constraint with timeMoving constraint with time  Non-holonomic constraints include terms like velocity or acceleration. Rolling ball velocity is related to radius and angular speedRolling ball velocity is related to radius and angular speed

8.  The constraint for the block is moving but scleronomic. Scleronomic Constraints  Scleronomic constraints are time-independent. Static constraintsStatic constraints Dynamic constraints if time is not explicit.Dynamic constraints if time is not explicit.  Rheonomic constraints are time-dependent. Explicit dependency on timeExplicit dependency on time m  M X x

9. Degrees of Freedom  Each particle begins with three degrees of freedom ( f ). Motion in 3 dimensionsMotion in 3 dimensions N free particles: f = 3NN free particles: f = 3N  The constraints reduce the number of degrees of freedom f. Number of constraints kNumber of constraints k f = 3N – kf = 3N – k

10. Rigid Body  A rigid body has no more than 6 degrees of freedom. For three masses rigidly attached, f = 6.For three masses rigidly attached, f = 6. Assume N masses have f = 6, so k = 3N – 6.Assume N masses have f = 6, so k = 3N – 6. Add one mass, three rigid attachments constrain it in space to all others.Add one mass, three rigid attachments constrain it in space to all others. For N+1 masses, k’ = 3N – 6 + 3.For N+1 masses, k’ = 3N – 6 + 3. f = 3(N+1) – k’ = 6.f = 3(N+1) – k’ = 6. next