Is velocity constant? A = πr 2

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1 Physics R Date: Circular Motion & Gravity Uniform Circular Motion What does uniform mean? Equations: (on reference table) Uniform circular motion means circular motion with C = 2πr = Is velocity constant? A = πr 2 1. A bicycle is traveling around a circle with radius 30 m. It completes one circle in 40 seconds. a. What was the bike s average speed? b. What was the bike s average velocity? 2. A jogger is at a constant speed of 4 m/s around a circle of diameter 50 m. How long does it take him to complete one circle? 3. A car traveling at 20 m/s completes a circle in 120 second. What is the diameter of the circle? Motion/Acceleration in 1 dimension Motion/Acceleration in 2 dimensions 1

2 Velocity Centripetal Acceleration Acceleration Centripetal Acceleration (on reference table) Centripetal means: Centripetal Force Centripetal force is not a new type of force. It is any center seeking force (or sum of forces) that causes circular/curved motion Tennis ball on a string Car driving in a circle Satellite orbiting Earth Center-seeking Center-seeking Center-seeking Force: Force: Force: 4. A car is traveling in a circle with radius 16 m at 12 m/s. What is the centripetal accelereation of the car? 2

3 5. A 4 kilogram object is swing in a horizontal circle from the end of a 0.5m rope at a speed of 5 meters per second. a. What is the centripetal acceleration acting on the object? b. What is the centripetal force acting on the object? 6. A ball on a string is traveling in a circle of radius 1 meter. It experience centripetal acceleration of 4 m/s 2. What is the speed of the ball? 7. A ball with mass of 0.2 kg is traveling in a circle with a speed of 5 m/s. If it has centripetal acceleration of 6 m/s 2, what is the radius of the circle? 8. What is the centripetal force on a 2000 kg airplane making a turn with a radius of 1000m if it is moving at 33 m/s? 9. A bicycle wheel has a radius of 0.5 m. When it spins, it completes one full turn in 1.6 seconds. A pebble wedged in the tread has a mass of 10 grams. a. Calculate the speed of the pebble b. Calculate the centripetal force on the pebble. 10. The diagram shows the top view of a 65-kilogram student at point A on an amusement park ride. The ride spins the student in a horizontal circle of radius 2.5 meters, at a constant speed of 8.6 meters per second. The floor is lowered and the student remains against the wall without falling to the floor. Use the following vectors for a through d a. Which vector best represents the direction of the student s velocity? b. Of the centripetal acceleration? c. Of the centripetal force? d. Of the student s momentum? e. Calculate the magnitude of the centripetal force acting on the student. 3

4 11. A bicycle is traveling in a circle of radius 30 m. It is traveling at 9 m/s. a. What is the magnitude of the centripetal acceleration? b. What would happen to the centripetal acceleration if the radius doubled? c. What if the radius was halved? d. What if the velocity doubled? e. What if the velocity was halved? 12. A ball attached to a string is moved at constant speed in a horizontal circular path. A target is located near the path of the ball as shown in the diagram. At which point along the ball s path should the string be released, if the ball is to hit the target? 13. A child is riding on a merry-go-round. As the speed of the merry-go-round is doubled, the magnitude of the centripetal force acting on the child 1. remains the same 2. is doubled 3. is halved 4. is quadrupled 14. A baby and a stroller have a total mass of 20 kilograms. A force of 36 newtons keeps the stroller moving in a circular path with a radius of 5.0 meters. Calculate the speed at which the stroller moves around the curve. 15. A 1750-kilogram car travels at a constant speed of 15 meters per second around a horizontal circular track with a radius of 45 meters. The magnitude of the centripetal force acting on the car is 16. A point on a car tire rotating at a constant rate. Which graph best represents the magnitude of the centripetal acceleration of point S as a function of time? 17. A car round a horizontal curve of constant radius at a constant speed. Which diagram best represents the directions of both the car s velocity, v, and acceleration, a? 4

5 18. The moon orbits the earth every 27.3 days (2.359 x 10 6 s). The Earth-moon radius is approximately 3.8 x 10 8 m. What is the speed of the moon in m/s? What is the centripetal acceleration of the moon? What force causes this acceleration? Isaac Newton: The Apple & The Moon How far is an apple from the center of the Earth? What is the acceleration due to gravity for an apple that falls from a tree, on the surface of the Earth? If apples fall to the Earth, why doesn t the moon? Could it be that the same law applies? Imagine a projectile launched horizontally from 4.9 meters above the ground. It will hit the ground after 1 second. If it is launched at 30 m/s, how far will it travel before it hits the ground? But what if the projectile is launched so fast that the curvature of the Earth becomes a concern? What if it is launched so fast, that after one second, it travels far enough that the Earth curves away a distance of 4.9 m. What will its height be after one second? Then after another second, the Earth will have curved away another 4.9 m. What will its height be? This is called An object in orbit is in a perpetual state of free fall. As it falls each meter, the Earth curves away a meter, so it never reaches the Earth Newton knew that the moon was approximately 60 Earth radii away. He hypothesized that gravity followed a 1/r 2 relationship. If this is true, what should acceleration due to gravity be for the moon? 5

6 Was he correct? How did he know? Force of attraction between Universal Gravitation Yes, F g = (on reference table) 19. What is the force of attraction between two 60 kg students separated by 1 meter? Why is it that we don t feel this force in everyday life? 20. What is the force of attraction between a 3 kg mass and a 9 kg mass separated by 0.05 m? 21. The force of attraction between two 10 kg masses is 2 x 10-6 N. What is the distance between the two masses? 22. The force of attraction between the Earth and a 2000 kg satellite is 5000 N. What is the distance between the center of the Earth and the satellite? Why is this important? This was the first time in history that the laws of physics governing the world around us were proven to be the same as the laws of physics governing the heavens. The same force that causes an apple to fall from the tree, causes the moon, sun, planets, and stars to behave the way they do. For the first time in 6

7 history, these could be considered real physical objects with physical properties that behave according to physical laws that is why this is called Newton s law of Universal Gravitation it applies to all objects in the universe. Calculator Practice Use the capital E or EE button. Using this, you will be less prone to mistakes. The EE button replaces x10^ 8 x 10 3 = 8E3 7E-14 = 7 x Practice 23. Using the reference table, calculate the force of gravity between the Earth and the Moon. 24. Calculate the force of gravity between two 5x10-4 kg objects separated by 12 micrometers. Gravitational Fields Force of gravity for an object near the surface of the Earth F g = We say that near the surface of the Earth, there is a gravitational field strength of Gravitational field is like an around any object that has mass. It causes other objects to experience a force and acceleration. 25. What is the gravitational field strength on the moon, where a dropped object falls 0.8 meters in 1 second? 7

8 26. What is the gravitational field strength on Mars, where acceleration due to gravity is 3.6 m/s 2? Henry Cavendish The man who measured the mass of the Earth Newton knew that gravitational field strength on Earth, (G) x (M Earth ) / (radius of Earth) 2 = He knew the radius of the Earth, but how could he know G or M Earth? Henry Cavendish used very massive objects and a very well balanced machine to measure the very small force of attraction between two objects on Earth. Once he successfully measured G, he could calculate the mass of the Earth! 27. If Cavendish used two 600 kg spherical masses separated by 1 meter, what would the force of gravity be between them? Draw the direction of force for both masses. What acceleration would each mass experience? 28. The space station orbits the Earth from an altitude of 400 km above Earth s surface. a. What radial distance should you use? b. Draw the direction of force on both the space station and Earth. c. What is the force of attraction between the 4 x 10 6 kg space station and the Earth? d. What is the acceleration of the space station due to this force? e. If the space station travels in uniform circular motion, calculate the speed it is traveling. 8

9 Ratios To find a ratio for F g (or any quantity, this applies to more than just force or F g ) First take the equation: Fg = Then put a 1 in for anything that stays the same. Put a 2 in for anything that doubles, 3 for triples, 0.5 for halved, etc. Your answer will then be the multiple of the original F g! 29. What happens to the force of gravity between two objects if the distance between both objects doubles? 30. What happens to centripetal acceleration when speed is halved? 31. A car going around a circle undergoes a centripetal acceleration of 2 m/s 2. What would the acceleration be if the speed doubled? Circular Motion Quick Review Universal Gravitation 9

10 Draw vectors to represent centripetal force; centripetal acceleration; and/or velocity in circular motion systems. Calculate speed, force, and acceleration in circular motion systems. 1. A merry go round of radius 6 meters completes one rotation every 15 seconds. Calculate the speed of a rider on the outside of the merry-go-round. 2. Sketch velocity, force, and acceleration vectors for each: a. b. Review Checklist Determine the effect of changing variable(s) on an equation particularly F g 10. A gravitational force F attracts two objects of mass M toward each other when R meters apart. a. What happens to the force if both masses are doubled? b. the distance is changed to 3R? c. both masses are doubled and the distance is half? d. one mass is 3M, the other is 2M, and the distance is R/2? 11. An object undergoes centripetal acceleration of 5 m/s 2. a. What would the acceleration be if the speed doubles? b. What if the radius doubles? c. What if both double? 3. What is the centripetal acceleration of a toy ball on the end of a 1.44 meter long string if it is moving at 12 meters per second? 4. What is the centripetal force acting on a 2000 kilogram airplane if it turns with a radius of 1000 meters while moving at 300 meters per second? 5. Calculate the speed experienced by a rider on a carnival Graviton ride if the ride provides an acceleration of 14 m/s 2 at a distance of 4 meters from the center? 6. A 3.5 kilogram object is swung in a circular path on the end of a 0.4 meter long string. The object makes one trip around the circle every 0.2 seconds. a. Calculate the speed of the object (hint: v = d/t) b. Determine the centripetal acceleration of the object. c. Calculate the centripetal force acting on the object. Answers m/s 2. a. velocity right, acceleration/force up b. velocity up, acceleration/force right m/s ,000 N m/s 6. a m/s b m/s 2 c N x N N x 10-3 m 10. a. 4 X as much b. 1/9 X c. 16 X d. 24 X 11. a. 20 m/s 2 b. 2.5 m/s 2 c. 10 m/s 2 Solve for an unknown variable using the universal gravitation formula 7. Determine the gravitational force of attraction experienced by two 5.0 kilogram masses separated by a distance of 2.5 meters. 8. Determine the force of gravity between the Earth and a 1200kg satellite orbiting with a distance of 2x10 7 m from the center of the Earth. 9. The force of gravity between two 10 kg objects is 1 x 10-3 N. Determine the distance between both objects. 10