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Chapter 6 • Circular Motion, Orbits, and Gravity Uniform Circular Motion Slide 6-23 For uniform circular motion, the acceleration A. is parallel to the velocity B. is directed toward the center of the circle C. is large for a larger orbital the same speed D. is always due to gravity 100% is la rg e gr av ity 0% .. is al w ay sd ue to ta lt h. or bi fo ra to w ar d la rg er th e ce n ve lo c it th e is di re ct ed ra lle lt o pa is 0% te r.. . y 0% Forces in Circular Motion v = r a= v2 2r r mv2 F net = m a = , tow ard center of circle r Slide 6-24 Time of Circular Motion 1 = The frequency is just the inverse of the period Example of frequency units: 2 = = 2 © 2015 Pearson Education, Inc. rev min What happens when speed decreases and radius decreases? 2 = © 2015 Pearson Education, Inc. Comparing centripetal accelerations Rank the acceleration of the following uniform circular motions from smallest to largest: They’re all going to have 4 2 so we just need to find 2 for each one 1 2 3 2 = 4 © 2015 Pearson Education, Inc. = 2 2 2 = = 4 2 2 5 Comparing centripetal accelerations Rank the acceleration of the following uniform circular motions from smallest to largest 1 12 10cm 1 rev 50 min 2 22 2 20cm 2 rev 50 min 2 3 32 20cm 3 rev 100 min 2 4 42 40cm 4 © 2015 Pearson Education, Inc. rev 100 min = 2 2 = 5 52 2 40cm 5 rev 200 min 2 Driving on a curvy road 2 = =0 2 = © 2015 Pearson Education, Inc. Problem solving uniform circular motion = 2 2 = © 2015 Pearson Education, Inc. Problem solving uniform circular motion A level curve on a country road has a radius of 150 m. What is the maximum speed at which this curve can be safely negotiated on a rainy day when the coefficient of friction between the tires on a car and the road is 0.40? Slide 6-28 Problem solving uniform circular motion A level curve on a country road has a radius of 150 m. What is the maximum speed at which this curve can be safely negotiated on a rainy day when the coefficient of friction between the tires on a car and the road is 0.40? = 2 = = = Slide 6-28 When a car turns a corner on a level road, which force provides the necessary centripetal acceleration? 76% 12% 10% Ai r Gr av ity re sis ta n ce Fo rc e 0% No rm al sio n 2% Te n Friction Tension Normal Force Air resistance Gravity Fr ict io n A. B. C. D. E. Additional Questions A coin sits on a rotating turntable. 1. At the time shown in the figure, which arrow gives the direction of the coin’s velocity? Slide 6-45 Answer A coin sits on a rotating turntable. 1. At the time shown in the figure, which arrow gives the direction of the coin’s velocity? A Slide 6-46 Additional Questions A coin sits on a rotating turntable. 2. At the time shown in the figure, which arrow gives the direction of the frictional force on the coin? Slide 6-47 Answer A coin sits on a rotating turntable. 2. At the time shown in the figure, which arrow gives the direction of the frictional force on the coin? D Slide 6-48 Additional Questions A coin sits on a rotating turntable. 3. At the instant shown, suppose the frictional force disappeared. In what direction would the coin move? Slide 6-49 Answer A coin sits on a rotating turntable. 3. At the instant shown, suppose the frictional force disappeared. In what direction would the coin move? A Slide 6-50 A ball moves CW in circular motion. What happens when it reaches the opening? 95% A B C D 5% D 0% C B 0% A A. B. C. D. Problem solving uniform circular motion In the hammer throw, an athlete spins a heavy mass in a circle at the end of a chain, then lets go of the chain. For male athletes, the “hammer” is a mass of 7.3 kg at the end of a 1.2 m chain. 2 A world-class thrower can get the hammer up to a speed of 29 m/s. If an athlete swings the mass in a horizontal circle centered on the handle he uses to hold the chain, what is the tension in the chain? 2 m 29 s = = 7.3kg 1.2m 2 Slide 6-29 Problem solving uniform circular motion A car of mass 1500 kg goes over a hill at a speed of 20 m/s. The shape of the hill is approximately circular, with a radius of 60 m, as in the figure at right. When the car is at the highest point of the hill, a. What is the force of gravity on the car? b. What is the normal force of the road on the car at this point? = Slide 6-30 Problem solving uniform circular motion A car of mass 1500 kg goes over a hill at a speed of 20 m/s. The shape of the hill is approximately circular, with a radius of 60 m, as in the figure at right. When the car is at the highest point of the hill, a. What is the force of gravity on the car? b. What is the normal force of the road on the car at this point? 2 = + = − 2 = − + Slide 6-30 Maximum walking speed v m ax gr Slide 6-31 Example Problem A roller coaster car goes through a vertical loop at a constant speed. For positions A to E, rank order the: • centripetal acceleration • normal force • apparent weight 2 = Slide 6-32 Example Problem A roller coaster car goes through a vertical loop at a constant speed. For positions A to E, rank order the: • centripetal acceleration • normal force • apparent weight Slide 6-32 Example Problem A roller coaster car goes through a vertical loop at a constant speed. For positions A to E, rank order the: • centripetal acceleration • normal force • apparent weight Slide 6-32 Net Force causes centripetal acceleration = net © 2015 Pearson Education, Inc. Net Force causes centripetal acceleration © 2015 Pearson Education, Inc. Friction force as a Net Force © 2015 Pearson Education, Inc. Problem solving circular motion with Net Force 2 = = © 2015 Pearson Education, Inc. = A car travels around a corner with the cruise control on. Which vector is the car’s centripetal acceleration? A B C D E 94% E 2% D 2% C 2% B 0% A A. B. C. D. E. A car travels around a corner while speeding up. Which vector is the net force on the car? A B C D E 84% 6% E C B 0% 2% D 8% A A. B. C. D. E. A car travels around a corner while speeding up. Which vector is the force of friction from the road on the car? A B C D E 96% D E 2% 0% C 0% B 2% A A. B. C. D. E. Fictitious Forces in circular motion Feels like something is pushing you out of the car © 2015 Pearson Education, Inc. A car travels around a corner at a constant speed. Which vector is the fictitious force on the car? 0% 0% D E 0% B 0% C A B C D E A A. B. C. D. E. 100% Simulating Gravity This one guy you knew from a while back says that in order to avoid loss of bone density on his trip to mars so, the ship must be spinning to simulate gravity. How fast must the craft below spin to simulate gravity? = 2 = = © 2015 Pearson Education, Inc. Tilted roads in favor of circular motion © 2015 Pearson Education, Inc. Example Problem: Over the Top A handful of professional skaters have taken a skateboard through an inverted loop in a full pipe. For a typical pipe with diameter 14 ft, what is the minimum speed the skater must have at the very top of the loop? Slide 6-33 Example Problem: Over the Top A handful of professional skaters have taken a skateboard through an inverted loop in a full pipe. For a typical pipe with diameter 14 ft, what is the minimum speed the skater must have at the very top of the loop? 2 = = 2 − = − = =0 ft help: = 32 2 s Slide 6-33 Keeping riders safe in circular motion © 2015 Pearson Education, Inc. When a ball on the end of a string is swung in a vertical circle, the ball’s acceleration is because 100% ar e. .. cc el re ct io n Th e ba ll is n di an d ee d 0% er at io n 0% is ch an gin g Th e sp di re ct io n Th e Th e sp ee d is ch a ng in g 0% ot a A. The speed is changing B. The direction is changing C. The speed and direction are changing D. The ball is not acceleration The Force of Gravity Slide 6-35 The inverse-square law versus in Newton’s law of gravitation is a value unique to the mass that’s pulling is the same throughout the whole universe! © 2015 Pearson Education, Inc. Example Problem A typical bowling ball is spherical, weighs 16 pounds, and has a diameter of 8.5 in. Suppose two bowling balls are right next to each other in the rack. What is the gravitational force between the two—magnitude and direction? What is the magnitude and direction of the force of gravity on a 60 kg person? Slide 6-36 A 60 kg person stands on each of the following planets. Ranks order her weight on the three bodies, from highest to lowest 19% 4% C> A> B 0% C> B> A B> C >A 0% B> A> C A>B>C B>A>C B>C>A C>B>A C>A>B A> B> C A. B. C. D. E. 77% Which vector pair matches the car’s motion over the hill? 94% 2% C B 4% A A. A B. B C. C © 2015 Pearson Education, Inc. The Force of Gravity = © 2015 Pearson Education, Inc. e ~17,000mph e Example Problem: Orbital Motion Phobos is one of two small moons that orbit Mars. Phobos is a very small moon, and has correspondingly small gravity—it varies, but a typical value is about 6 mm/s2. Phobos isn’t quite round, but it has an average radius of about 11 km. What would be the orbital speed around Phobos, assuming it was round with gravity and radius as noted? = = e 3 .006 −11 24 kg 6.67 ∙ 10 ∙ 5.97 ∙ 10 kg ∙ s2 9.8 6.4 ∙ 106 m Slide 6-34 A satellite orbits the earth. A space shuttle crew is sent to boost the satellite into a higher orbit. Which of these quantities increases? 94% 0% Pe rio et d al ac ce tio le na ra lf t io or n ce of th e ea rth nt rip Gr av ita Ce ed ar s pe ed 4% 2% 0% An gu l Speed Angular speed Period Centripetal acceleration Gravitational force of the earth Sp e A. B. C. D. E. Summary Slide 6-41 Summary Slide 6-42