"Everything should be made as simple as possible, but not simpler." - Albert Einstein

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1 Physics Schedule Chapter Five Centripetal Motion & Gravity October 20 th November 3 rd, 2015 Date Activity/Topic Assignment due next day Tuesday, Universal Law of Gravity Worksheet 1 October 20 Block, Gravity & Worksheet 2 Worksheet 3 October 21,22 Centripetal Acceleration Friday, October 23 Centripetal Forces Worksheet 4 Monday, October 26 Tuesday, October 27 Block, October 28,29 Friday, October 30 Monday, November 2 Applications of Centripetal Forces Review Day Centripetal Motion Lab No School P/T Conferences Test Worksheet 5 Review Finish Lab TBA "All science is either physics or stamp collecting." - Ernest Rutherford "Physicists define stress as force per unit area. The rest of humanity defines stress as physics." - Unknown "Everything should be made as simple as possible, but not simpler." - Albert Einstein

2 V. Unit 5: Circular Motion & Gravitation A. Gravitation 1. Students should understand the relationship between gravitational force, the masses of objects and their separation. 2. Students should be able to make calculations using Newton s Law of Universal Gravitation. 3. Students should be able to calculate the factor by which the gravitational force between two objects changes based upon changes to mass and separation data. (1) EX: If two objects are separated by a distance d, what would happen to the gravitational force between them if they were moved to a separation of 3d? 4. Students should understand the nature of weightlessness for people and objects in orbit. B. Uniform Circular Motion (UCM) 1. Students should be able to identify the requirements for UCM. They are (1) Constant speed (2) Circular path of constant radius 2. Students should be able to use the equation for centripetal acceleration. 3. Students should be able to identify the direction of the following for an object in UCM at any given moment in its path; velocity, acceleration, net force and direction of motion if the centripetal force stops. 4. Students should be able to calculate the velocity of an object in UCM based upon the circumference of its path and its period. 5. Students should be able to use the centripetal acceleration in conjunction with Newton s 2 nd law to determine the net force acting on an object in UCM. 6. Students should be able to identify the force or forces causing centripetal motion. 7. Students should be able to find the velocity for an object to: (1) do a loop-the-loop safely. (2) turn a car around a level curve safely.

3 Physics - Chapter 5: Worksheet 1 (A1,A2,A4) Universal Gravity Part 1: Gravity in general 1. a. What is the formula for the force of gravity? b. Why are there two m s in the equation? c. What is G in the gravity equation and what is its value? d. What are the units for mass, force and distance in the gravity equation? Technically, this simple gravitational formula works only if the two objects are relatively far away compared to their size. However, we will simplify this for the time being. 2. Solve the gravitational equation for m Solve the gravitational equation for r. 4. What is the gravitational force between two 1.0 kg objects separated by 1.0 meter? 5. a. What is the gravitational force between a 1.0 kg object and a 2.0 kg object separated by 1.0 meter? b. How does this answer compare to your answer to the last question? 6. a. What is the gravitational force between two 1.0 kg objects separated by 2.0 meters? b. How does this answer compare to the answer to question 4? 7. By what factor will the gravitational force between two objects increase if the mass of each object is doubled and the distance between them is doubled? 8. By what factor will the gravitational force between two objects increase if the mass of one object is doubled and one is tripled and the distance between them is halved? 9. What is the range of gravity (how far can it work)? Why don t I attract things that are right next to me with gravity? Interlude: Metric review Convert each of these quantities into the unit shown. 1.3 x 10 5 km = m 3.4 Mm = m Continued on Next Page

4 Important information: Mass of Earth = 5.97 x kg Mass of the Moon = 7.35 x kg Radius of Earth = 6,380,000 m Radius of the Moon = 1,740,000 m Mass of the Sun = 1.99 x kg Radius of the Sun = 6.96 x 10 8 m Earth Sun distance (mean) = x 10 9 m Earth Moon distance (mean) = 384,000,000 m 1. What is the weight of a 100 kg man? (Do this the old fashioned way, use g = 9.8m/s 2.) 2. Use the gravitational force equation to find the gravitational force from the Earth on a 100 kg man sitting on the Earth s surface. Show the equation and the numbers that you use for each variable. (Your answer should not be a surprise.) 3. Let s say our 100 kg man was taken up to the International Space Station. The station orbits at an altitude of about 200,000 m over the Earth s surface. What would the gravitational force be on him now? 4. What is the gravitational force of the Earth on the Moon? (The Earth-Moon distance given above is the distance between centers.) 5. What is the gravitational force of the Moon on the Earth? 6. Given the answers to the last two questions, why does the Moon orbit the Earth and not the other way around? 7. What is the force of gravity exerted on the Earth by the Sun? 8. Given that this is by far the largest force exerted on the Earth (and so essentially the net force), what acceleration does the Earth have as a result?

5 Physics - Chapter 5: Worksheet 2 (A2,A3) Some More Work with Gravity 1. How far from each other are two 1 kg masses if they are exerting a gravitational force of 2.3 x N on each other? 2. Once a month, the Sun and Moon are both on the same side of the Earth. Because the sunlit side of the Moon faces away from us it is difficult to see. This is referred to as a new Moon. Using the calculations you made on the previous worksheet, find the net force exerted on the Earth by the Moon and the Sun 3. What is the net force on the Earth from the Sun and Moon during a full Moon when the Sun and the Moon are on opposite sides of the Earth? (Include both magnitude and direction.) 4. What happens to the force of gravity between two objects if a. the mass of both objects is doubled? b. the distance between the objects is halved? c. the distance between the objects is tripled? d. the distance between the objects is halved and the mass of one of them is tripled. e. the mass of one object is halved, the other is quadrupled and the distance between them is decreased by a factor of It is possible to determine the value of g on the surface of a planet if one knows the mass of the planet and the radius. Planet X has a mass of 300,000,000 kg and a radius of 1000 m. The weight of an apple on this planet s surface can be determined according to the following formulas: Weight = m apple g Weight = (G) (m apple ) ( m planet ) R 2 a. Determine g for this planet by equating these two weight formulas, substituting in given values and solving for g for planet X. b. If the Earth were to shrink to half of its present radius, but retained its present mass, what would the value of g be? How many times heavier would you be on this Earth?

6 Physics - Chapter 5: Worksheet 3 (B1,B2,B3,B4) Uniform Circular Motion 1. Which of the following quantities change during uniform circular motion? For the ones that do, in what way are they changing? a. velocity b. speed c. direction d. radius e. acceleration 2. Acceleration is defined as the change in velocity per unit time. If objects don t speed up or slow down during uniform circular motion, how are they accelerating? 3. The black dot to right is traveling on the path shown by the circle. If the dot is traveling clockwise, show the direction of the velocity and acceleration at this time. 4. Write the formula for centripetal acceleration (a c ) and describe each of the variables. 5. What is the centripetal acceleration of a car moving at 15.0 m/s around a curve of radius 40.0 meters? 6. A pitcher is going through his windup. As he prepares to release the ball, it is traveling at 36 m/s. If we consider his shoulder the center of rotation and his arm is 95 centimeters long, what is the centripetal acceleration of the ball? Warning: Math refresher section!!! 7. What is the circumference of each of the following circles (in meters)? a. radius = 24 meters b. radius = 23 cm c. radius = 6580 km 8. What is the radius of a quarter-mile circular track? 9. Mr. Ballew jogs around a circular track at a constant 2.5 m/s for 20 minutes and does 7.5 laps. a. What is the circumference of the track? (Hint: How far does he go total?) b. What is the radius of the track? c. What is his centripetal acceleration? 10. A 250-gram ball at the end of a string is revolving uniformly in a horizontal circle of radius m. The ball makes exactly 3.00 revolutions in a second. a. What is the time and distance for 1 revolution? b. What is the speed of the ball? c. What is the centripetal acceleration?

7 Physics - Chapter 5: Worksheet 4 (B5,B6) Centripetal Acceleration & Force 1. What is the equation for Newton s 2 nd law? (Never Forget!) 2. Substitute our equation for the acceleration of an object in UCM into Newton s 2 nd law. This gives you an equation for the net force on an object in UCM. What equation do you get? 3. In what direction does a centripetal force act? 4. A 1000 kg car goes around a flat level curve of radius 50 meters at a constant speed of 14 m/s. a. What is the acceleration of the car? b. What is the force on the car? c. What type of force turns a car as it goes around a level curve? (I don t mean centripetal force. That is a general term for any force or combination of forces that cause an object to move in a circle. I want to know the specific kind of force that allows the car to turn. If you aren t getting an idea here, think about what would make it difficult to turn.) 5. A 25-kg child is on a Ferris Wheel moving at a speed of 1.35 m/s at a distance of 12.0 meters from the center. a. What is the acceleration of the child? b. What is the magnitude of the net force on the child? What direction is the force? c. When an object moves in a circle we say that it is experiencing a centripetal force. This centripetal force is the net force on the object and may be from a single force or it may be the net result of several forces in combination. At the lowest point in the ride there are two forces acting on the child. What are they and which is greater? d. When the child is at the highest point in the ride what forces are acting on the child and which is greater? e. How much force does the seat of the ride have to put on the seat of the child at the bottom of the ride? 6. Consider an object that requires 0.20 seconds to make one complete circle. Assume that this object is undergoing uniform circular motion. a. What is the period of this object b. How many revolutions will the object make in one second? c. What is the frequency of revolution of this object in revolutions per second. 7. A horizontal net force of 200 N is exerted on a 2.0 kg discus to keep it revolving uniformly in a horizontal circle or radius 1.00 m. a. What is the centripetal acceleration of the object? b. What is the speed of the object? c. If the mass of the object were doubled, what force would be required to keep it turning at this speed?

8 Physics - Chapter 5: Worksheet 5 (B7) Applications of Centripetal Force CAR AROUND A LEVEL CURVE PROBLEMS 1. When a car goes around a level curve, friction supplies the force needed to keep moving in a circle. What kind of friction is at work here? 2. What is the formula for centripetal force? 3. An object of mass m is sitting on level ground. What is the normal force on the object? 4. How do we find the maximum possible friction force on an object of mass m on level ground? (Don t use F N. Use your answer for #3.) 5. A car of mass m is going around a level curve of radius r at velocity v. If the car speeds up any more, friction will not be able to keep them turning. a. Set the formula for maximum possible friction equal to the centripetal force equation. b. Solve the equation you got in 5a for v (speed in this case). c. Solve the equation you got in 5a for μ s. d. Solve the equation you got in 5a for r. 6. A car is going around a level curve of radius 25 meters. The tires have a coefficient of static friction of What is the maximum speed the car can travel around the curve? 7. Same as the last question, but the radius is 50 meters. 8. A car is going around a level curve of radius 40 meters. If the car begins to slip at a speed of 9.3 m/s, what is the coefficient of friction? 11. If bald tires are expected to have a μ s of only 0.19 on wet roads, what radius should the onramp to a highway have if cars are to move at 35 miles per hour? 12. What would be the safest possible speed for a car going around a frictionless curve? LOOP-THE-LOOP PROBLEMS 13. When a car does a loop-the-loop at the minimum possible speed, the weight of the car is all that provides the centripetal force at the top of the loop. (At higher speeds the track pushes down with an extra normal force if needed.) a. Set the weight of a car of mass m equal to the formula for centripetal force. b. Solve the equation from 13a for speed. 14. How fast does a car need to go at the top of a loop-the-loop of radius 10 meters to make it? 15. How fast does a car need to go at the top of a loop-the-loop of radius 20 meters to make it?

9 Physics - Chapter 5: Review Worksheet 1. What supplies the force required for a car to move around a level curve? 2. When the tires of a car are moving, the friction between the tires and the road is considered static friction rather than kinetic. Why is that? 3. When braking, it is considered much worse to have your tires lock up and stop rotating than if they keep rolling. In the first case, the tires slide rather than roll across the pavement. From a friction point of view, why is that worse? (Think about the last question.) 4. It would be impossible to move around a frictionless level curve in a car. Why would a frictionless turn around a banked curve be possible? Should the curve be banked toward or away from the center of curvature? 5. What is the fastest speed a car of mass 400 kg can move around a curve of radius 20 meters if the coefficient of friction between the tires and the road is 0.96? 6. What is the formula for the acceleration of an object moving in a circle? a. What is the direction of centripetal acceleration? b. What is the direction of the velocity of an object moving with uniform circular motion? c. What is the direction of a centripetal force? 7. What minimum speed does a roller skater have to be moving to do a loop-the-loop in a 2-meter radius vertical circle? (Krusty the Clown once did this trick with Homer on his back.) 8. How fast does a car have to move to do a loop-the-loop of radius 14 meters? 9. A 30-kg child and a 60-kg adult are riding on a Ferris wheel. At the bottom of the ride they are affected by two forces, the normal force and gravity. Which of these is greater at this point in the ride? How do you know? a. Which of the two riders has the greatest acceleration? b. Which of the two riders has the greater centripetal force acting on them? 10. Make a list of the material objects you own not affected by gravity.

10 11. What is the universal law of gravitation? What is the universal gravitational constant? What is implied by the word universal? 12. What happens to the force of gravity between two objects if one of the masses is doubled? If both masses are doubled? 13. If I am standing on the surface of the Earth, isn t the distance between us zero? If that is true then how can I do that gravity equation thing? 14. If the gravitational force depends on the mass of both objects, then why don t massive things fall more quickly than less massive things? Do they experience more force? 15. Two 1 kg objects are separated by 1 meter. What is the gravitational force of each on the other? 16. A 1000 kg object (object A) and a 2000 kg object (object B) are separated by 2 meters. a. What is the magnitude of the gravitational force between the masses? b. How does the force on A by B compare to the force on B by A? c. What would happen to the gravitational force if only the mass of A were doubled? d. What would happen to the gravitational force if the distance were halved? e. What would happen to the gravitational force if the mass of A was doubled, the mass of B was doubled and the separation was doubled? 17. Two objects have equal masses and are separated by a distance, d. The gravitational force they exert on each other is F. What will the gravitational force be if the distance is increased to 3d and both masses are doubled. a. 4F b. F/9 c. 4F/9 d. 4F/3 18. Calculate the force of gravity on the Enterprise in orbit 12,760 km above the Earth s surface if the mass of the ship is 5 x 10 8 kg. a. What is the acceleration of the spacecraft? In what direction is the acceleration? Challenge: For an object of mass m moving in orbit around a planet of mass M, the gravitational force provides the centripetal force that causes the object to move in a circle. Set the centripetal and gravitational formulas equal and solve for v to find the velocity of an object in orbit.