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Circular Thinking Ken McGregor Why? It’s a challenge. Can see & feel what’s going on. Instinctive gut reaction that might be right or wrong, but probably needs explanation. Allows for creative & critical thinking – perhaps not so comfortable. Examples A hollow world Frames of reference Rolling, sliding and spinning balls Snooker Collisions Trebuchet Circus see-saw A bola Half pipe Rolling Coin Why does a boomerang come back? Building Blocks Speed, acceleration, force & inertia Having cycled on a bike or on a skateboard Spinning a bucket of water. Hollow Planet Sun Hollow planet Earth ‘Forces’ on the surface The world inside Letting a ball drop & Launching a ‘satellite’ Coriolis Effect Ground frame of reference Carousel frame of reference y y’ x’ x carpet ice Ground frame of reference y’ Carousel frame of reference y y’ x’ G wt x G x’ Frame of Reference Moving frame of reference If stationary, only involves a centrifugal force. stars movement Coriolis force Initial Velocity v’ centrifugal force stars Inertial frame of reference Coriolis Force Transit time 1 hr 20mins Same time no matter what arc is chosen. Bug moving at steady speed along radial spoke of a wheel rotating at constant angular speed w(r+v’Dt/2) v’ wDt w(r-v’Dt/2) v’ Can show the radial acceleration is w r And the transverse acceleration is 2w v ' 2 Rolling, sliding & Spinning Balls Pure rolling v w O Sliding Spinning friction friction Slowing down while Rolling Rolling Rolling v v Friction Friction Rolling v Normal force Friction Skidding to Rolling Rolling v v Sliding to Rolling v Yoyo Rolling v Questions What would the rolling curve be like if you were inside a light rubber tyre that was rolling on the flat? And if you were at the edge of a Yo-yo that was falling under gravity? What would it feel like in both cases? Collisions Snooker Collisions Usually no spin given to the target ball Sliding only 1 v2 v2 2 U w0 2 y x 1 v1 With top spin Rolling only With under spin Height of Rebounding Edge Circus See Saw – Teeterboard M Vi rM rm m The angular acceleration Fm 1 FM 1 on both sides must the m rm M rM same: The work done on m Fm rm FM rM must equal the work expended by M : Combining these we get Fm m rm rM mr 2 Mr 2 m M FM M rM rm So the moments on inertia must the same. r And not - mrm MrM F M Fm rm M 2 1 3 A Bola 2 1 3 V F v L O O r v R r No gain or loss of energy, until the balls strike But angular momentum of the balls decreases as it’s imparted to the earth. Balls strike post radially. Trebuchet R1 R2 m1 m2 Line of centre of mass M m vm vM M m m vm M vM Trebuchet frame moves in this direction If both arms have the same length: Without wheels, we get With wheels, we get mvm MvM Half Pipe – pumping for speed R2 v2 R1 v1 Rolling Coin Balance in the vertical gives N cos mg 2 mv ‘Balance’ in the horizontal N sin r This gives v2 tan gr If the basin is the bottom part of a shell: then r R sin v R sin g tan If the basin has a log shape i.e. Then dz z0 dr r We then get And so r z z0 ln r0 tan v gz0 dz r tan z0 dr r z N i.e. a constant mg r Boomerang V wR V wR V V V wR V wR Ft Fb Bottom end Concepts Stimulates 3D visualisation, through not essential Doesn’t need to be mathematical – simple algebra is enough Don’t need to specify moment of inertia Experience with a bucket of water Don’t need to specify torque Just the example of a spanner and nut THANK YOU FOR YOUR ATTENTION Questions, comments, other experiences? [email protected]