1. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu 1 PHYS 1441 – Section 001 Lecture #10 Thursday, June 19, 2014 Dr. Jaehoon Yu Uniform Circular Motion Centripetal Acceleration Unbanked and Banked highways Newton’s Law of Universal Gravitation Weightlessness Work done by a constant force Today’s homework is homework #6, due 11pm, Tuesday, June 24!!

2. Announcements Quiz 3 –Beginning of the class Monday, June 23 –Covers CH 4.7 to what we finish today –Bring your calculator but DO NOT input formula into it! Your phones or portable computers are NOT allowed as a replacement! –You can prepare a one 8.5x11.5 sheet (front and back) of handwritten formulae and values of constants for the exam  no solutions, derivations or definitions! No additional formulae or values of constants will be provided! Mid-term result –Class average: 67/97 Equivalent to 69.1/100 Previous exam: 61.8/100 –Top score: 89/97 Mid-term grade discussion bottom half of the class Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu 2

3. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu Reminder: Special Project #3 Using the fact that g=9.80m/s 2 on the Earth’s surface, find the average density of the Earth. –Use the following information only but without computing the volume explicitly The gravitational constant The radius of the Earth 20 point extra credit Due: Monday, June 23 You must show your OWN, detailed work to obtain any credit!! Much more than in this lecture note! 3

4. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu Uniform circular motion is the motion of an object traveling at a constant speed on a circular path. Definition of the Uniform Circular Motion 4

5. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu Let T be the period of this motion, the time it takes for the object to travel once around the complete circle whose radius is r. Speed of a uniform circular motion? 5

6. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu The wheel of a car has a radius of 0.29m and is being rotated at 830 revolutions per minute on a tire-balancing machine. Determine the speed at which the outer edge of the wheel is moving. Ex. : A Tire-Balancing Machine 6

7. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu Newton’s Second Law & Centripetal Force The centripetal * acceleration is always perpendicular to the velocity vector, v, and points to the center of the axis (radial direction) in a uniform circular motion. The force that causes the centripetal acceleration acts toward the center of the circular path and causes the change in the direction of the velocity vector. This force is called the centripetal force. Are there forces in this motion? If so, what do they do? What do you think will happen to the ball if the string that holds the ball breaks? The external force no longer exist. Therefore, based on Newton’s 1st law, the ball will continue its motion without changing its velocity and will fly away following the tangential direction to the circle. *Mirriam Webster: Proceeding or acting in the direction toward the center or axis 7

8. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu Ex. Effect of Radius on Centripetal Acceleration The bobsled track at the 1994 Olympics in Lillehammer, Norway, contain turns with radii of 33m and 23m. Find the centripetal acceleration at each turn for a speed of 34m/s, a speed that was achieved in the two–man event. Express answers as multiples of g=9.8m/s 2. Centripetal acceleration: R=33m R=24m 8

9. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu Example 5.1: Uniform Circular Motion A ball of mass 0.500kg is attached to the end of a 1.50m long cord. The ball is moving in a horizontal circle. If the string can withstand maximum tension of 50.0 N, what is the maximum speed the ball can attain before the cord breaks? Centripetal acceleration: When does the string break? when the required centripetal force is greater than the sustainable tension. Calculate the tension of the cord when speed of the ball is 5.00m/s. 9

10. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu On an unbanked curve, the static frictional force provides the centripetal force. Unbanked Curve and Centripetal Force 10

11. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu On a frictionless banked curve, the centripetal force is the horizontal component of the normal force. The vertical component of the normal force balances the car’s weight. Banked Curves 11

12. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu Ex. The Daytona 500 The Daytona 500 is the major event of the NASCAR season. It is held at the Daytona International Speedway in Daytona, Florida. The turns in this oval track have a maximum radius (at the top) of r=316m and are banked steeply, with θ=31 o. Suppose these maximum radius turns were frictionless. At what speed would the cars have to travel around them? x comp. x y y 12

13. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu Newton’s Law of Universal Gravitation People have been very curious about the stars in the sky, making observations for a long~ time. The data people collected, however, have not been explained until Newton has discovered the law of gravitation. Every object in the Universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. How would you write this law mathematically? G is the universal gravitational constant, and its value is This constant is not given by the theory but must be measured by experiments. With G Unit? This form of forces is known as the inverse-square law, because the magnitude of the force is inversely proportional to the square of the distances between the objects. 13

14. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu Ex. Gravitational Attraction What is the magnitude of the gravitational force that acts on each particle in the figure, assuming m 1 =12kg, m 2 =25kg, and r=1.2m? 14

15. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu Why does the Moon orbit the Earth? 15

16. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu Gravitational Force and Weight Since weight depends on the magnitude of gravitational acceleration, g, it varies depending on geographical location. The attractive force exerted on an object by the Earth Gravitational Force, FgFg Weight of an object with mass M is By measuring the forces one can determine masses. This is why you can measure mass using the spring scale. What is the SI unit of weight? N 16

17. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu Gravitational Acceleration What is the SI unit of g? m/s 2 Gravitational acceleration at distance r from the center of the earth! 17

18. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu Gravitational force on the surface of the earth: Magnitude of the gravitational acceleration on the surface of the Earth 18

19. Thursday, June 19, 2014 Example for Universal Gravitation Using the fact that g=9.80m/s 2 on the Earth’s surface, find the average density of the Earth. Since the gravitational acceleration is Therefore the density of the Earth is Solving for MEME Solving for g PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu 19

20. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu There is only one speed that a satellite can have if the satellite is to remain in an orbit with a fixed radius. Satellite in Circular Orbits What acts as the centripetal force? The gravitational force of the earth pulling the satellite! 20

21. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu Determine the speed of the Hubble Space Telescope orbiting at a height of 598 km above the earth’s surface. Ex. Orbital Speed of the Hubble Space Telescope 21

22. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu Period of a Satellite in an Orbit Speed of a satellite Period of a satellite Square either side and solve for T2 This is applicable to any satellite or even for planets and moons. Kepler’s 3 rd Law 22

23. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu Geo-synchronous Satellites What period should these satellites have? Satellite TV Global Positioning System (GPS) The same as the earth!! 24 hours 23

24. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu In each case, what is the weight recorded by the scale? Ex. Apparent Weightlessness and Free Fall 0 24 0

25. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu At what speed must the surface of the space station move so that the astronaut experiences a push on his feet equal to his weight on earth? The radius is 1700 m. Ex. Artificial Gravity 25

26. Thursday, June 19, 2014 Motion in Resistive Forces Medium can exert resistive forces on an object moving through it due to viscosity or other types frictional properties of the medium. These forces are exerted on moving objects in opposite direction of the movement. Some examples? These forces are proportional to such factors as speed. They almost always increase with increasing speed. Two different cases of proportionality: 1.Forces linearly proportional to speed: Slowly moving or very small objects 2.Forces proportional to square of speed: Large objects w/ reasonable speed Air resistance, viscous force of liquid, etc PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu 26

27. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu 27 x y Work Done by a Constant Force A meaningful work in physics is done only when the net forces exerted on an object changes the energy of the object. M F  Free Body Diagram M d  Which force did the work?Force How much work did it do? What does this mean? Physically meaningful work is done only by the component of the force along the movement of the object. Unit? Work is an energy transfer!! Why? What kind?Scalar

28. Let’s think about the meaning of work! A person is holding a grocery bag and walking at a constant velocity. Are his hands doing any work ON the bag? –No –Why not? –Because the force hands exert on the bag, Fp,Fp, is perpendicular to the displacement!! –This means that hands are not adding any energy to the bag. So what does this mean? –In order for a force to perform any meaningful work, the energy of the object the force exerts on must change due to that force!! What happened to the person? –He spends his energy just to keep the bag up but did not perform any work on the bag. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu 28

29. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu 29 Work done by a constant force s

30. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu 30 Scalar Product of Two Vectors Product of magnitude of the two vectors and the cosine of the angle between them Operation is commutative Operation follows the distribution law of multiplication How does scalar product look in terms of components? Scalar products of Unit Vectors =0

31. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu 31 Example of Work by Scalar Product A particle moving on the xy plane undergoes a displacement d =(2.0 i +3.0 j )m as a constant force F =(5.0 i +2.0 j ) N acts on the particle. a) Calculate the magnitude of the displacement and that of the force. b) Calculate the work done by the force F. Y X d F Can you do this using the magnitudes and the angle between d and F?F?

32. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu 32 Ex. Pulling A Suitcase-on-Wheel Find the work done by a 45.0N force in pulling the suitcase in the figure at an angle 50.0 o for a distance s=75.0m. Does work depend on mass of the object being worked on?Yes Why don’t I see the mass term in the work at all then? It is reflected in the force. If an object has smaller mass, it would take less force to move it at the same acceleration than a heavier object. So it would take less work. Which makes perfect sense, doesn’t it?

33. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu 33 Ex. 6.1 Work done on a crate A person pulls a 50kg crate 40m along a horizontal floor by a constant force F p =100N, which acts at a 37 o angle as shown in the figure. The floor is rough and exerts a friction force F fr =50N. Determine (a) the work done by each force and (b) the net work done on the crate. What are the forces exerting on the crate? F G =-mg So the net work on the crate Work done on the crate by FGFG FpFp F fr Which force performs the work on the crate? FpFp F fr Work done on the crate by Fp:Fp: Work done on the crate by F fr : This is the same as F N =+mg Work done on the crate by FNFN

34. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu 34 Ex. Bench Pressing and The Concept of Negative Work A weight lifter is bench-pressing a barbell whose weight is 710N a distance of 0.65m above his chest. Then he lowers it the same distance. The weight is raised and lowered at a constant velocity. Determine the work in the two cases. What is the angle between the force and the displacement? What does the negative work mean? The gravitational force does the work on the weight lifter!

35. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu 35 The truck is accelerating at a rate of +1.50 m/s 2. The mass of the crate is 120-kg and it does not slip. The magnitude of the displacement is 65 m. What is the total work done on the crate by all of the forces acting on it? Ex. Accelerating a Crate What are the forces acting in this motion? Gravitational force on the crate, weight, W or F g Normal force force on the crate, F N Static frictional force on the crate, f s

36. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu 36 Ex. Continued… Lets figure out what the work done by each of the forces in this motion is. Work done by the gravitational force on the crate, W or F g Work done by Normal force force on the crate, F N Work done by the static frictional force on the crate, f s Which force did the work? Static frictional force on the crate, f s How? By holding on to the crate so that it moves with the truck!

37. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu 37 Kinetic Energy and Work-Kinetic Energy Theorem Some problems are hard to solve using Newton’s second law –If forces exerting on an object during the motion are complicated –Relate the work done on the object by the net force to the change of the speed of the object M ΣF M s vivi vfvf Suppose net force ΣF ΣF was exerted on an object for displacement d to increase its speed from vi vi to vf.vf. The work on the object by the net force ΣF ΣF is Using the kinematic equation of motion Work Kinetic Energy Work done by the net force causes change in the object’s kinetic energy. Work-Kinetic Energy Theorem

38. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu 38 When a net external force by the jet engine does work on an object, the kinetic energy of the object changes according to Work-Kinetic Energy Theorem

39. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu 39 The mass of the space probe is 474-kg and its initial velocity is 275 m/s. If a 56.0-mN force acts on the probe parallel through a displacement of 2.42×10 9 m, what is its final speed? Ex. Deep Space 1 Solve for v f

40. Thursday, June 19, 2014PHYS 1441-001, Summer 2014 Dr. Jaehoon Yu 40 A satellite is moving about the earth in a circular orbit and an elliptical orbit. For these two orbits, determine whether the kinetic energy of the satellite changes during the motion. Ex. Satellite Motion and Work By the Gravity For a circular orbit For an elliptical orbit No change!Why not? Gravitational force is the only external force but it is perpendicular to the displacement. So no work. Changes!Why? Gravitational force is the only external force but its angle with respect to the displacement varies. So it performs work.