Circular Motion and Gravitation. a R = v 2 v. Period and Frequency. T = 1 f. Centripetal Acceleration acceleration towards the center of a circle.

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1 Circular Motion and Gravitation Centripetal Acceleration acceleration towards the center of a circle. a.k.a. Radial Acceleration (a R ) v v Ball rolling in a straight line (inertia) Same ball, hooked to a string a R a R a R = v 2 v r If you are on a carousel at constant speed, are you experiencing acceleration? If you twirl a yo-yo and let go of the string, what way will it fly? Period and Frequency Period (T) Time for one complete (360 o ) revolution seconds Frequency Number of revolutions per second rev/s or Hertz (Hz) T = 1 f 1

2 Formulas A 150-g ball is twirled at the end of a m string. It makes 2.00 revolutions per second. Find the period, velocity, and acceleration. a R = v 2 r v = 2pr T T = 1 f v = 2prf (0.500 s, 7.54 m/s, 94.8 m/s 2 ) The moon has a radius with the earth of about 384,000 km and a period of 27.3 days. a. Calculate the acceleration of the moon toward the earth. b. Convert the answer to g s. Jupiter is about 778 X 10 6 km from the sun. It takes days to orbit the sun. a) Calculate the circumference of Jupiter s orbit. (4.89 X m) b) Calculate Jupiter s period in seconds. (3.74 X 10 8 s) c) Calculate Jupiter s orbital speed. (1.30 X 10 4 m/s) d) Calculate Jupiter s centripetal acceleration towards the sun. (2.18 X 10-4 m/s 2 ) (Ans: 2.72 X 10-3 m/s 2, 2.78 X 10-4 g ) Centripetal Force Centripetal Force center seeking force that pulls an object in a circular path. Yo-yo Planets Merry-go-round Car rounding a curve? 2

3 Centrifugal Force? Doesn t exist apparent outward force When you let the string go, the ball will continue in a straight line path. No new acceleration involved. Water in swinging cup example SF = ma R = mv 2 r Direction water wants to go A kg yo-yo is attached to a m string and twirled at 2 revolutions per second. It is twirled horizontally. a) Calculate the circumference of the circle (3.77 m) b) Calculate the linear speed (7.54 m/s) c) Calculate the centripetal force in the string (14.2 N) Centripetal Force of string 3

4 An electron orbits the nucleus with a radius of 0.5 X m. The electron has a mass of 9 X kg and a speed of 2.3 X 10 6 m/s. a. Calculate the centripetal force on the electron. (9.52 X 10-8 N) b. Calculate the frequency of an electron. (7.32 X Hz) c. Convert the velocity to miles/s. (1429 miles/s) d. What provides the centripetal force? miles = km A rotating space station is shaped like a wheel. It has a radius of m. a) Calculate the speed at which it must rotate to provide 1 g of gravity. (17.15 m/s) b) Calculate the centripetal force on a 126 lb man (1 lb is equivalent to kg) (561 N) c) Calculate the frequency and period of the space station. ( Hz, 11.0 s) d) Where would the floor be on such a space station? Circular Motion: Example 2 Thor s Hammer (mjolnir) has a mass of 10 kg and the handle and loop have a length of 50 cm. If he can swing the hammer at a speed of 3 m/s, what force is exerted on Thor s hands? (Ans: 180 N) Can Thor swing his hammer so that it is perfectly parallel to the ground? F H What angle will the hammer take with the horizontal? Also calculate the force of the handle. q Mass = 10 kg r = 50 cm v = 3 m/s mg 4

5 mv 2 = Tcosq r 0 = Tsinq mg A kg walnut is swung in a horizontal circle of radius of 50.0 cm. The walnut makes 2.50 revolutions per second. a) Calculate the linear speed of the walnut. (7.85 m/s) b) Draw a free-body diagram of the walnut. c) Calculate the centripetal force needed to keep it in a circle. (0.616 N) d) Calculate the force of tension and the angle the string makes with the horizontal. (0.618 N, 4.50 o ) A 0.25 kg yo-yo is swung in a horizontal radius of 65.0 cm. The string makes a 10.0 o angle with the horizontal. a) Draw a free-body diagram of the yo-yo. b) Calculate the tension in the string. (14.1 N) c) Calculate the linear speed of the yo-yo. (6.00 m/s) d) Calculate the period and frequency (0.68 s, 1.47 Hz) Circular Motion: Example 3 A kg ball is swung on a 1.10-m string in a vertical circle. What minimum speed must it have at the top of the circle to keep moving in a circle? mg T What is the tension in the cord at the bottom of the arc if the ball moves at the minimum speed? (v = 3.28 m/s) T mg (ANS: 2.94 N) 5

6 A rollercoaster vertical loop has a radius of 20.0 m. Assume the coaster train has a mass of 3,000 kg. a) Calculate the minimum speed the coaster needs to have to make the loop. (14.0 m/s) b) Calculate the normal force the tracks provide to the train at the bottom of the curve if the train is travelling at 25.0 m/s. (123,150 N) c) Calculate the normal force the tracks provide at the top of the curve if the train is travelling at 25.0 m/s. (64, 350 N) The ferris wheel at Knoebels has a radius of 16.8 m and travels at a speed of 3.52 m/s. a) Calculate the frequency and period (0.033 Hz, 30 s) b) Calculate the normal force that the seat provides to a 56.0 kg rider at the top. (507 N) c) Calculate the normal force that the seat provides to a 56.0 kg rider at the bottom. (590 N) A kg motorcycle is driven around a 12.0 meter tall vertical circular track. a. Calculate the minimum speed needed to make the loop. (7.67 m/s) b. The bike is driven at a constant speed of 11 m/s. Calculate the normal force on the tires at the bottom. (7490 N) A car travels over a round hill (radius = 50.0 m). a) Calculate the maximum speed at which the car can take the hill. (22.0 m/s) b) Calculate the normal force on a kg car if it is travelling over the hill at 10.0 m/s. (7.80 X 10 3 N) Car Rounding a Turn Friction provides centripetal force Use (m s ). Wheels are turning, not sliding, across the surface Wheel lock = kinetic friction takes over. m k is always less than m s, so the car is much more likely to skid. A 1000-kg car rounds a curve (r=50 m). n mg f s = F c a) Calculate the maximum speed the car can take if the road is dry and m s = 0.60 (17 m/s) b) Calculate the maximum speed if the road is icy and m s = 0.25 (11 m/s) c) If the car is travelling at 14.0 m/s, under which conditions will they skid? (0.40) 6

7 A 15,000-kg truck can safely round a 150 m curve at a speed of 20 m/s. a) Calculate the centripetal force needed to keep the truck on the road (40,000 N ) b) Calculate the coefficient of static friction (0.27) c) Calculate the maximum speed a 1000 kg Cube car can take the turn. (20 m/s) The Rotor The Rotor at an amusement park has a radius of 7.0 m and makes 30 rev/min. a) Calculate the speed of the rotor. ( 22.0 m/s) b) Draw a free body diagram of a person in the rotor. What causes the normal force? c) Calculate the coefficient of static friction between the person and the wall. (0.14) The Rotor at an amusement park has a radius of 6.0 m and a m s of a) Draw a free body diagram of a person in the rotor. What causes the normal force? b) Calculate the speed of the rotor. (17 m/s) c) Calculate the period and frequency of the rotor. (2.22 s, 0.45 rev/s) The Rotor at an amusement park has a radius of 4.00 m and a m s of a) Draw a free body diagram of a person in the rotor. What causes the normal force? b) Calculate the speed of the rotor. (12.5 m/s) c) Calculate the period and frequency of the rotor. ( 2.00 s, 0.50 rev/s ) Banked Curves Banked to reduce the reliance on friction Part of the Normal Force now contributes to the centripetal force v 2 = tanq gr F C = f s + F N sinq (ideally, we bank the road so that no friction is required: f s = 0) 7

8 A 1000-kg car rounds a 50 m radius turn at 14 m/s. What angle should the road be banked so that no friction is required? q mg F N q Now we will simply work with the Normal Force to find the component that points to the center of the circle F N cosq q q mg F N F N sinq q First consider the y forces. SF y = F N cosq - mg Since the car does not move up or down: SF y = 0 0 = F N cosq mg F N cosq = mg F N = mg/cosq mv 2 = F N sinq r mv 2 = mgsinq r cosq v 2 = gtanq r v 2 = gtanq r v 2 = tanq gr tan q = (14 m/s) 2 = 0.40 (50 m)(9.8m/s 2 ) q = 22 o A 2,000-kg Nascar car rounds a 300 m radius turn at 200 miles/hr. (1 mile=1609 m) a. Convert the speed to m/s. (89.4 m/s) b. What angle should the road be banked so that no friction is required? (70 o ) c. Suppose a track is only banked at 35.0 o, calculate the maximum speed that a car can take the turn. (45.3 m/s, 101 mph) d. Looking at the formula for banking angle, how could a track designer decrease that angle? 8

9 Weightlessness True weightlessness exists only very far from planets Apparent weightlessness can be achieved on earth Apparent weight is the Normal Force Apparent weightlessness (n = 0) Elevator at Constant Velocity a= 0 SF = n mg 0 = n mg n = mg Suppose Mr. Saba has a mass of 102 kg: n = mg = (102kg)(9.8m/s 2 ) n = 1000 N n mg a is zero Elevator Accelerating Upward Elevator Accelerating Downward a = 4.9 m/s 2 SF = n mg ma = n mg n = ma + mg n = m(a + g) n = (102kg)(4.9m/s m/s 2 ) n = 1500 N n a = 4.9 m/s 2 SF = mg - n ma = mg - n n = mg - ma n = m(g - a) n = (102kg)(9.8m/s m/s 2 ) n = 500 N n mg a is upward mg a is down At what acceleration will he feel weightless? n = 0 SF = mg - n ma = mg -n ma = mg - 0 ma = mg a = 9.8 m/s 2 Apparent weightlessness occurs if a > g n mg Calculate the apparent weight of a 56.0 kg man in an elevator if the elevator is: a) Accelerating upwards at 2.00 m/s 2. (661 N) b) Accelerating downwards at 2.00 m/s 2 (436 N) c) Accelerating downwards at 9.80 m/s 2 (0 N) d) Accelerating sideways at 9.80 m/s 2. (549 N) 9

10 Mr. Saba is riding an elevator that accelerates upward at 3.00 m/s 2. His apparent weight is 768 N. a) Calculate his mass. (60.0 kg) b) Calculate his apparent weight if the elevator accelerates downwards at 3.00 m/s 2. (408 N) c) Calculate the acceleration required to have him weigh 1000 N. Is this up or down? (6.87 m/s 2 ) Mr. Fredericks wishes he were a God of Math like me. Other examples of apparent weightlessness Even when you are running, you fell weightless between strides. Why don t satellites fall back onto the earth? Speed They are falling due to the pull of gravity Can feel weightless (just like in the elevator) An kg car rounds a turn of 60.0 m radius. a. If the m s is 0.47, calculate the maximum speed at which he can take the turn. (16.6 m/s) b. Suppose you wished to bank the curve so that no friction is required. Calculate the banking angle. (25.1 o ) A 2.00 kg book is placed in a spinning drum (rotor) of radius 0.75 m and m s of Draw a free body diagram Calculate the speed at which the drum needs to rotate to suspend the book. (6.06 m/s) Gravitation Is gravity caused by the earth s rotation? Will a man down here fall off if the earth stops rotating? 10

11 Gravitation Newton s Law of Universal Gravitation 1. Every object in the universe is attracted to every other object. (based on mass) 2. The force drops off with the distance squared. (As distance increases, the force of gravity drops very quickly) F= GMm r 2 Gravitation: Formula Cavendish proves the law in 1798 G = 6.67 X N m 2 /kg 2 M = mass of one object m = mass of second object r = distance from center of objects Gravitation: The Solar System Everything in the solar system pulls on everything else. Sun pulls on Earth All the other planets also pull on the Earth Some comets/meteors are actually from outside our solar system and were captured by our sun s gravity. 11

12 What is the force of gravity between two 60.0 kg (132 lbs) people who stand 2.00 m apart? F = 6.00 X 10-8 N What is the force of gravity between a 60 kg person and the earth? Assume the earth has a mass of 5.98 X kg and a radius of 6,378,000 m (~4,000 miles). A 2000-kg satellite orbits the earth at an altitude of 6380 km (the radius of the earth)above the earth s surface. What is the force of gravity on the satellite? F= GMm = (6.67 X N m 2 /kg 2 )(2000kg)(5.98 X kg) r 2 (6,380,000 m + 6,380,000 m) 2 F = 4900 N F = 588 N Calculating the Mass of the Earth Calculate the mass of the earth knowing that it has a radius of 6.38 X 10 6 m. Start using the weight formula. 12

13 g = GM r 2 Calculating g Calculating g : Example 1 Calculate the value of g at the top of Mt. Everest, 8848 m above the earth s surface. g = (6.67 X N-m 2 /kg 2 )(5.98 X kg) (6.38 X 10 6 m) 2 g = 9.80 m/s 2 g = 9.77 m/s 2 g varies with: Altitude Location Earth is not a perfect sphere (roughly 22 km greater in radius at equator, 13.7 miles) Different mineral deposits change density salt domes - low density salt regions near petroleum deposits Objects weigh about 1/6 their weight on Earth on the Moon. Calculate the mass of the moon, knowing that the radius of the moon is 1734 km. An object weighs 200 N on Earth. a) Calculate the acceleration of gravity on Mars (3.71 m/s 2 ) b) Calculate its weight on Mars (75.5 N) R m = 3440 km M m = 0.11M e M E = 5.98 X kg Pluto has a mass of kg, and a radius of 1180 km. a. Calculate the gravity of Pluto at its surface. (0.628 m/s 2 ) b. If a person weighs, 550 N on earth, calculate his weight on Pluto. (35.2 N) c. Pluto is 5.9 X 10 9 km from the sun, and has a year equal to 248 earth years. Calculate its orbital speed. (4740 m/s) 13

14 Gravitation: Example 3 Calculate each force separately: F ME What is the net force on the moon when it is at a right angle with the sun and the earth? Relevant Data: M M = 7.35 X kg M E = 5.98 X kg M S = 1.99 X kg r MS = 1.50 X m r ME = 3.84 X 10 8 m F ME = 1.99 X N F MS = 4.34 X N F R 2 = F ME 2 + F MS 2 F R = 4.77 X N tan q = opp = F ME adj F MS q = 24.6 o F MS q Sun F R Earth Three 5.00 kg bowling balls are placed at the corners of an equilateral triangle whose sides are 1.50 m long. Calculate the magnitude and direction of the gravitational force on the top ball. Four 5.00 kg bowling balls are placed at the corners of a square whose sides are 1.50 m long. Calculate the magnitude and direction of the gravitational force on the lower left ball. (1.42 X 10-9 N, 45 o ) A geosynchronous satellite has a period of one day. The radius of the Earth is 6380 km and the mass of the Earth is 5.98 X kg. a) Convert the period to seconds (8.64 X 10 4 s) b) Calculate the height above the earth that a geosynchronous satellite must orbit. (Hint: use mv 2 /r) (3.59 X 10 7 m, 4.23 X 10 7 m total) c) Calculate the speed of the orbit. (3070 m/s) A satellite orbits with a period of 5.00 hours. The radius of the Earth is 6380 km and the mass of the Earth is 5.98 X kg. a) Calculate the height of the satellite above the earth. (8.47 X 10 6 m) b) Calculate the speed of the orbit. (5.17 X 10 3 m/s) 14

15 Kepler s Laws ( ) 1. The orbit of each planet is an ellipse, with the sun at one focus 2. Each planet sweeps out equal areas in equal time The orbit of each planet is an ellipse, with the sun at one focus 2. Each planet sweeps out equal areas in equal time Deriving the Third Law Suppose the travel time in both cases is three days. Shaded areas are exactly the same area m 2 Jupiter m 1 Gm 1 m J = m 1 v 2 r 2 r Gm J = v 2 r Gm J = 4p 2 r 2 r T 2 T 2 = 4p 2 r 3 Gm J Substitute v=2pr T T 2 = 4p 2 We can do this for two r 3 Gm J different moons T 2 1 = 4p 2 T 2 2 = 4p 2 r 3 1 Gm J r 3 2 Gm J 15

16 Useful Forms r 3 = GM T 2 4p 2 ONE SATELLITE lrejyw TWO SATELLITES The Third Law: Example 1 Mars has a year that is about 1.88 earth years. What is the distance from Mars to the Sun, using the Earth as a reference (r ES = X 10 8 m) r M 3 = T M2 r E 3 T E 2 r 3 M = (1.88y) 2 (1.496 X 10 8 m) 3 (1 y) 2 r M 3 = 1.18 X m 3 r M = 2.28 X 10 8 m Third Law: Example 2 How long is a year on Jupiter if Jupiter is 5.2 times farther from the Sun than the earth? T J 2 = r J 3 T E 2 = (5.2) 3 (1 y) 2 r E 3 (1) 3 T J 2 = 141 y 2 T J = 11.9 y 16

17 Kepler s Third Law: Example 3 How high should a geosynchronous satellite be placed above the earth? Assume the satellite s period is 1 day, and compare it to the moon, whose period is 27 days. The average distance between the earth and the moon is 384,000 km. Third Law: Example 4 What is the mass of the sun, knowing that the earth is X m from the sun. (3.63 X 10 7 m, or 3.63 X 10 4 km) T 2 = 4p 2 r 3 Gm S m S = 4p 2 r 3 GT 2 Calculate the mass of the earth, knowing that the moon has a period of days and an average distance of 384,400 km. m S = 4p 2 (1.496 X m) 3 (6.67 X N-m 2 /kg 2 ) (3.16 X 10 7 s) 2 m S = 2.0 X kg The dwarf planet Eris is 1.01 X km from the sun. The earth is X m from the sun. Calculate the length of a year on Eris. The Gemini 11 spacecraft sent two astronauts to a height of 1374 km above the earth s surface. The radius of the Earth is 6380 km and the mass of the Earth is 5.98 X kg. a) Calculate the period of the satellite (1.89 hour) b) Calculate the speed of the orbit. (7173 m/s) (~555 yrs) 17

18 The period of Neptune s moon Galatea is days, and the radius is 61,953 km from the center of Neptune. a. Calculate the mass of Neptune (1.02 X kg) b. Calculate the speed of the orbit. (10,500 m/s) The Gemini 11 spacecraft had a period of 1.89 hours. The radius of the Earth is 6380 km and the mass of the Earth is 5.98 X kg. a) Calculate the height above the earth that it orbited. (1374 km) b) Calculate the speed of the orbit. (7166 m/s) The mass of Mars is 6.40 X kg. Calculate the period (in hours) of its moon Phobos if Phobos has an orbital radius of 9377 km. Pluto has a radius of 1150 km and a mass of 1.20 X kg. a) Calculate the acceleration of gravity on Pluto (0.605 m/s 2 ) b) Calculate the weight of a 70.0 kg person on Pluto. (42.4 N) c) Calculate the acceleration of gravity on Pluto in terms of g s ( g s) (7.67 h) The radius of the sun is 695,500 km, and its mass is 1.99 X kg. a. Calculate the acceleration due to gravity at the surface of the sun. (274 m/s 2 ) b. Calculate your weight at the surface of the sun. Choose any planet except earth. Look up its mass, radius, distance from the sun, and the length of its day. 1. Calculate the surface gravity on your planet 2. Calculate the weight of a 56.0 kg person on our planet. 3. Calculate the height that a satellite must be placed above the surface of the planet to be in geosynchronous orbit (T = one day of that planet). 4. The earth is 1.5 X m from the sun. Calculate the time it takes (in years) your planet to orbit the sun. (Use Kepler s laws) 18

19 A student is given the following data and asked to calculate the mass of Saturn. The data describes the orbital periods and radii of several of Saturn s moons. Orbital Period, T Orbital Radius, R (seconds) (m) 8.14 X X X X X X X X 10 8 Let s use this equation: T 2 = 4p 2 r 3 Gm S And rearrange it: Gm S = 1 4p 2 r 3 T 2 Once more: 1 = Gm S T 2 4p 2 r 3 Calculate the following values and graph them. 1 G T 2 4p 2 r E E E E E E E E E E E E E E E E-37 Calculate the slope of the graph y = m x + b 1 = m S G T 2 4p 2 r 3 The Four Fundamental Forces 1. Gravity 2. Electromagnetic 3. Strong Nuclear Force 4. Weak Nuclear Force m s = 5.9 X kg 19

20 2. a) 1.52 m/s 2, center b) 38.0 N, center m/s m/s, no effect 8. a) 3.14 N b) 9.02 N m/s, rev/s m/s rev/min 16.F T1 = 4p 2 f 2 (m 1 r 1 + m 2 r 2 ) F T2 = 4p 2 f 2 m 2 r a) 3000 m b) 5500 N c) 3900 N m/s m/s g surface X 10 7 m 34. a) 9.8 m/s 2 b) 4.3 m/s X N away from the Sun X kg m/s 2 upward X 10 3 s (1.41 h), independent of mass 44.a) 58 kg b) 58 kg c) 77 kg d) 39 kg e) 0 44.a) 58 kg (569N) b) 58 kg(569n) c) 77 kg (755 N) d) 39 kg (382N) e) R Icarus = 1.62 X m X kg X kg, 1.7 X Suns 58.R Europa = 6.71 X 10 5 km R Ganymede = 1.07 X 10 6 km R Callisto = 1.88 X 10 6 km rev/day 70. a) 3000 m b) 5500 N c) 3900 N Graphing Centripetal Force A A 1 kg yo-yo was swung in a circle at a constant speed. The force on the string was measured as the string was let out slowly. Radius (m) Force (N) Graphing Centripetal Force B In a second experiment, the speed was changed while the length of the string (r) was kept constant Speed (m/s) Force (N)

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23 In an experiment, a yo-yo is twirled in a horizontal circle. 1. Suppose the speed of the yo-yo gradually increases. Sketch of graph of centripetal acceleration (a R ) versus speed. 2. Suppose the length of the string is gradually increased. Sketch of graph of centripetal acceleration (a R ) versus radius. Considering a Sports Management Degree? What causes the seasons R34o 23