PHYSICS 231 PH 2 1 Lecture 16 Centripetal acceleration PHY 231 1 Review f i Average angular t f ti t velocity rad s Instantaneous lim t 0 t Angular velocity f i Average angular t f ti t acceleration rad s2 Instantaneous angular lim t 0 t acceleration PHY 231 2 rad 3600 10 2 360 rad 1 rad 360 2 deg 2 And Angular and linear velocity are related v r Angular and linear acceleration are related l t d a r Here a is also along the tangential direction demo tangential velocity PHY 231 3 Constant Angular Acceleration Kinematical equations for constant angular acceleration are obtained in complete analogy to th those for f linear li motion ti with ith constant t t acceleration l ti 0 0t t 1 2 2 0 t 0 t 12 0 2 02 2 0 PHY 231 4 gears If 1 3 rad s how fast Is the bike going b What Wh t if r1 0 1 01m r3 0 7 0 7 m r1 0 3 r2 0 15 b V1 1r1 3 0 3 0 9 m s V1 V2 because of the chain 2 V2 r2 0 9 0 15 6 rad s 3 2 connected V3 3r3 6 0 7 4 2 m s V1 1r1 3 0 1 0 3 m s V1 V2 because of the chain 2 V2 r2 0 3 0 15 2 rad s 3 2 connected V3 3r3 2 0 7 1 4 m s PHY 231 5 question peddles rear gear front gear a b c d e rear wheel equall to t equall to t equal to larger than larger g than larger g than smaller than larger than smaller than equal to PHY 231 A boy is riding his bike gear is rotating g If the front g with linear velocity v1 and angular velocity 1 then the rear gear has a linear velocity v1 and angular velocity 1 The Th f frontt gear iis llarger in radius than the rear gear 6 quiz extra credit An object is moving in a circle with radius R 2 m If the radius di is i d doubled bl d and d th the angular l velocity l it remains i th the same the linear velocity will a b c d e half remain the same double quadruple whatever PHY 231 7 Driving a car through a bend Is there I th af force th thatt pushes h you away from the center of the circle Newton s first law If no net force is acting on an object it will ill continue ti with ith th the same velocity l it i inertia ti of f mass Velocity is a vector points to a direction If no net force is acting g on an object j it will not change g its direction A force is acting on the car steering friction but you tend to go in the same direction as you were going It is not a force that pushes you but the lack of it The side door will keep you from falling out it exerts a f force on you and d you exertt a force f on th the door d F21 F F12 PHY 231 8 Centripetal acceleration v f vi v a t f ti t The change in velocity is not the change g in speed p but in direction Sin 2 s 2 r Sin 2 v 2 v s 2 r v 2 v v s v r t t ac v2 r PHY 231 v s t 9 Centripetal acceleration ac v2 r directed to the center of the circular motion Also v r so ac 2r This acceleration Thi l ti can be b caused d by b various i forces f gravity objects attracted by earth tension object j making g circular motion on a rope p friction car driving through a curve etc This acceleration is NOT caused by a mysterious force PHY 231 10 Forces that can cause centripetal acceleration Object swinging on a rope T F ma T T mac T mv2 r m 2r An object with m 1 kg is swung with a rope of length 3 m around with angular velocity 2 2 rad s rad s What is the tension in the rope T m 2r 1 2 T 1 22 3 12 3 12 N PHY 231 11 Lifting by swinging Swinging mass m1 with velocity v What is the relation between v and r that will keep m2 stationary T r v out of paper T H Hanging mass m 2 Fg m2g demo m1 T m1ac m2 T m2g ac m2g m1 Also ac v2 r v linear tangential velocity of m1 v2 r m2g m 1 If m1 slows down i e v decreases r must go down so m2 sinks PHY 231 12 question Swinging mass m1 with velocity v T r v2 r m2g m1 v out of paper T H Hanging mass m 2 Fg m2g If f the velocity y of f m1 is increased to twice the original value the radius of orbit a decreases to of its original value b decreases to of its original value c remains the same d increases to 2x 2 its originall value l e increases to 4x its original value PHY 231 13 A car going through a bend A car is passing through a bend with radius 100 m The static coefficient of friction of the tires on the road is 0 5 What is the maximum velocity the car can have without flying out of the bend F ma sn mac smg mv2 r 0 5 9 81 v2 100 v 22 22 m s If smg mv2 r r then the car will fly out of the bend bend PHY 231 14 2001 A space odyssey A space ship rotates with a linear velocity of 50 m s What shou should the th distance stanc from the central axis to the crew s cabin s be so that the crew feels like they are on earth the floor of the cabins is the inside of the outer edge d of f the h spaceship h The rotating spaceship has an acceleration directed towards the center of the ship the lack of forces acting on the crew pushes them against the ship F ma mg mv2 r so r v2 9 8 PHY 231 and thus r 255 m 15 Conical motion Tcos T Tsin mg demo What is the centripetal acceleration if the mass is 1 kg g and 20o Vertical direction F ma F ma Tcos mg 0 So T mg cos Horizontal direction F mac Tsin mac mgsin cos mgtan mac ac gtan 9 8 0 36 3 6 m s2 PHY 231 16 A general strategy As usual make a drawing of the problem if not given i Draw all the forces that are acting on the object s under investigation investigation Decompose each of these into directions toward the center of the circular …
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