Physics 170 Lecture 19. Chapter 12 - Kinematics Sections 8-10

Transcription

1 Phys 170 Lecture 0 1 Physics 170 Lecture 19 Chapter 1 - Kinematics Sections 8-10 Velocity & Acceleration in Polar / Cylinical Coordinates Pulley Problems

2 Phys 170 Lecture 0 Polar Coordinates Polar coordinates have some origin O, and point P at the end of vector r is described by r and θ. Define a radial unit vector û r at P pointing away from the origin. The absolute direction of this unit vector depends on P, which depends on θ. We also define a θ-unit vector û θ in the +θ (CCW) direction. We can describe the point P using these vectors as r r û r.

3 Phys 170 Lecture 0 3 Velocity in Polar Coordinates Velocity is the time-derivative of position, so v d r d ( ) r û r ûr + r d ûr By the same argument as for tnb coordinates, the time-derivative of û r is dû r û θ dθ So v dθ ûr + r û θ r û r + r θ û θ

4 Acceleration in Polar Coordinates Acceleration is the derivative of velocity, so a d v d dθ ûr + r ûθ Then using the product rule a d r d ûr + ûr + dθ ûθ + r d θ dθ ûθ + r The argument for û θ is like the others, but the direction flips: dû θ û r dθ Plug in the time-derivatives of both unit vectors: a d r ûr + dθ ûθ + dθ ûθ + r d θ ûθ + r dθ dθ ( ) Phys 170 Lecture 0 4 û r d ûθ

5 Acceleration in Polar Coordinates Get rid of the brackets a d r ûr + dθ ûθ + dθ ûθ + r d θ dθ ûθ r dθ ûr Combine terms and 3, and the square in term 5 a d r ûr + dθ ûθ + r d θ dθ ûθ r Change to dot notation, and combine components a ( r rθ )û r + ( r θ + r θ )û θ Every term has one factor of r, and two dots (either a second derivative, or the product of two first derivs). Phys 170 Lecture 0 5 û r

6 Phys 170 Lecture 0 6 Theta is in Radians!!! To use these formulas, the angular velocity θ dθ and the angular acceleration θ d θ must be in radians not degrees! The conversion is π radians 360, or 1 radian 180 /π 57.3

7 Phys 170 Lecture 0 7 Cylinical Coordinates We just add a z coordinate perpendicular to r and θ. r r û r + z û z v r û r + r θ û θ + z û z û r + r dθ û θ + z û z a r rθ û r + r θ + r θ û θ + z û z d r r dθ ûr + r d θ + dθ ûθ + d z û z

8 Phys 170 Lecture 0 8 Cylinical Coordinates Summary r r û r + z û z v r û r + r θ û θ + z û z û r + r dθ û θ + z û z a r rθ û r + r θ + r θ û θ + z û z d r r dθ ûr + r d θ + Blue is v /r in disguise. Red is the Coriolis Effect dθ ûθ + d z û z

9 Phys 170 Lecture 0 9 Tangent-Normal Coordinates Summary Velocity defines tangent direction Center of curvature defines normal direction and magnitude of curvature ρ. û n û t ρ ρ Position along path defined by s. Velocity is v ds û t s û t Acceleration is a d s û t + Blue terms are just v /r. ( ds ) ρ û n s û t + s ρ ûn

10 Phys 170 Lecture 0 10 Angle Between Tangent & R Direction Sometimes the motion is described r(θ), polar coordinate r as a function of angle θ, and we need to find the local tangent direction (to use tangent-normal coordinates for example. In a time, the particle moves by in the local radial direction, and by r dθ in the local θ direction. The angle ψ between the tangent direction and the local radial direction is given by tanψ Δs θ Δs r r dθ r dθ r dθ r( θ) ψ ψ

11 Phys 170 Lecture 0 11 Angle Between Tangent and θ Direction The angle δ between the tangent direction and the θ direction is tanδ r dθ 1 r dθ r r r( θ) δ û θ

12 Themotion motionofofball ballppisisconstrained constrained by by the the curved slot in OB and th The curved slot in OB and PROBLEM (page 78, 1 edition) theslotted slottedarm armoa. OA. bybythe The motion of ball P is constrained by the curved slot in OB and 3/ by the slotted with arm OA. OA rotates counterclockwise angular speed 3t OA rotates counterclockwise with angular speed 3t3/ rad/s rad/s OA rotates counterclockwise with angular speed 3t3/ rad/s where t is in seconds and when t 0.! 0 where t is in seconds and! 0 when t 0. where t is in seconds and! 0 when t 0. Determine Determine the time when!thetime.! 30 o. 30oowhen Determine the time when! 30. Determine the radial and transverse components of the ball s Determine the radial and transverse components of the ball s o and accelerationcomponents when! 30. of the ball s Determine the radialvelocity and transverse velocity and acceleration when! 30oo. velocity and acceleration when! 30. Phys 170 Lecture 0 1

13 OA rotates counterclockwise with angular speed 3t 3/rad/s OA rotates counterclockwise with angular speed 3t th rad/s (page 78, 1 edition) where t is in seconds and! PROBLEM when t 0. 0 where t is in secondsthe and! 0 when t 0. motion of ball P is constrained by the curved slot in OB and o the slotted OA.o. Determine the timebywhen! arm 30 Determine the timeoawhen.! 30 rotates counterclockwise with angular speed 3t3/ rad/s d θdetermine 6 5 of the ball s 5 components 3 3 and transverse the radial 3t θ 3t 3 t + C t + C where t is in seconds and when t 0.! 0 Determine the radial and transverse of the ball s 5 5 components o velocity and acceleration when! 30 Determine the time when! o. 30 o. 5 velocity and acceleration when! the radial and t transverse components θ 0 when t 0 CDetermine 0 Invert θ of the ball s velocity and acceleration when! 30 o θ π π θ rad 360 π t Phys 170 Lecture sec 13

14 where t is in seconds and! 0 when t 0. OA rotates counterclockwise with angular speed 3t3/ th rad/s PROBLEM (page 78, 1 edition) o Determine the time when! 30. where t is in secondsthe and t 0. by the curved slot in OB and! of0ballwhen motion P is constrained Determine the radial and transverse by the slotted arm OA. components of the ball s o Determine the time when.! 30 OA rotates counterclockwiseo with angular speed 3t3/ rad/s velocity and acceleration when! 30. where t is in seconds and! 0 when t 0. Determine the radial and transverse components of the ball s To find velocity & acceleration components, Determine the time when! o 30. velocity and acceleration when! 30. d r dθdetermine d θ the radial and transverse components of the ball s we need,,, velocity and acceleration when! 30. dθ 3 We have 3t and are given r 4 cos θ dθ t t o o Phys 170 Lecture 0 14

15 ( ) d PROBLEM (page 78, 1th edition) r 4 cos θ The motion of ball P is constrained by the curved slot in OB and d d by the slotted arm OA. r ( 4 cos θ ) OA rotates counterclockwise with angular speed 3t3/ rad/s where t isdinθseconds and! 0 when t 0. r 4 ( sin θ ) Determine the time when! 30. transverse components 4 sin θ dθ Determine 4 sin theθ radial3and sin θ 3 of the ball s velocity and acceleration 3t when 1! 30. t r r r ( ) o o Phys 170 Lecture 0 15

16 d d sin θ 3 PROBLEM (page 78, 1th edition) t Right side gives 3 terms 1 r motion of ball P is constrained by the curved slot in OB and The 3arm OA. dθ by the slotted t 1 3 3/ cos θ + t sin θ OA rotates counterclockwise with angular speed 3t rad/s r r d r where t is in seconds and! 0 when t Determine sin theθ time when! t Determine r the radial and transverse components of the ball s ( ) o velocity and acceleration when! 30 o. Phys 170 Lecture 0 16

17 Phys 170 Lecture 0 17 dθ 3t d θ 9 t r 4 cosθ 4 cos d r 1 sinθ r t 3 1 dθ 1 cosθ t 3 r t1 sinθ r 3.86 sin r t 3 sinθ ( )

18 Phys 170 Lecture 0 18 d ( r ) d r r d ( 4 cosθ ) dθ 4 ( sin θ ) 4sin θ dθ r d 4 sin θ dθ + r d r 4 cosθ Solve for d r 1 r Do another symbolic derivative ( ) dθ 4 cosθ ( ) dθ dθ + sin θ d θ dθ + sin θ d θ

19 Phys 170 Lecture 0 19 Given r( θ) and dθ dθ dθ r dθ d r d r d dθ dθ d dθ d r dθ dθ and d θ, find d dθ dθ + dθ dθ dθ dθ + dθ d θ and d r d + dθ d θ r θ + r θ dθ

20 Phys 170 Lecture 0 0 v a û r + r dθ û r û θ û θ û r û θ ( ) m/s d r r dθ ûr + r d θ û r dθ ûθ ( )1.839 ûθ ( 37.56û r û ) θ m/s

21 Block A is moving down at 4 ft/s. Block C is moving up at ft/s. Determine the speed of block B. Block A is moving down at 6 ft/s. Block C is moving down at 18 ft/s. Determine the speed of block B. If A is fixed, then if C goes down 1 cm, B will go up 0.5 cm. If C is fixed, then if A goes down 1 cm, B will go up 0.5 cm. The motion of B if both A and C move is the sum of the individual motions. A down 4 ft/s B up ft/s C up ft/s B down 1 ft/s Net: B up 1 ft/s Phys 170 Lecture 0 1

22 Block A is moving down at 4 ft/s. Block C is moving up at ft/s. Determine the speed of block B. Block A is moving down at 6 ft/s. Block C is moving down at 18 ft/s. Determine the speed of block B. A down 6 ft/s B up 3 ft/s C down 18 ft/s B up 9 ft/s Net: B up 1 ft/s Phys 170 Lecture 0

23 Phys 170 Lecture 0 3 The end of the cable at A is pulled down with speed m/s. Determine the speed at which block E rises. A down B up 1 B up C up 1/ C up 1/ D up 1/4 Net: E up 0.50 m/s

24 Phys 170 Lecture 0 4 For Next Time Read the rest of Chapter 1