Ch5. Uniform Circular Motion

1. Ch5. Uniform Circular Motion Uniform circular motion is the motion of an object traveling at a constant (uniform) speed on a circular path. Period T is the time required to travel once around the circle, that is, to make one complete revolution. r

2. Example 1: A Tire-Balancing Machine The wheel of a car has a radius of r = 0.29m and is being rotated at 830 revolutions per minute (rpm) on a tire-balancing machine. Determine the speed (in m/s) at which the outer edge of the wheel is moving. The speed v can be obtained directly from , but first the period T is needed. It must be expressed in seconds.

3. 830 revolutions in one minute T=1.2*10-3 min, which corresponds to 0.072s

4. Uniform circular motion emphasizes that 1. The speed, or the magnitude of the velocity vector, is constant. 2. Direction of the vector is not constant. 3. Change in direction, means acceleration 4. “Centripetal acceleration” , it points toward the center of the circle.

5. Centripetal Acceleration Magnitude ac of the centripetal acceleration depends on the speed v of the object and the radius r of the circular path. ac=v2/r

6. Sector of the circle COP. is very small, the arc length OP is approximately a straight line whose length is the distance v traveled by the object. in velocity divided by the elapsed time or a= /

7. COP is an isosceles triangle. Both triangles have equal apex angles . ac=v2/r The direction is toward the center of the circle.

8. Conceptual Example 2: Which way will the object go? An object on a guideline is in uniform circular motion. The object is symbolized by a dot, and at point O it is release suddenly from its circular path. If the guideline is cut suddenly, will the object move along OA or OP ?

9. Newton’s first law of motion guides our reasoning. An object continues in a state of rest or in a state of motion at a constant speed along a straight line unless compelled to changes that state by a net force. When the object is suddenly released from its circular path, there is no longer a net force being applied to the object. In the case of a model airplane, the guideline cannot apply a force, since it is cut. Gravity certainly acts on the plane, but the wings provide a lift force that balances the weight of the plane.

10. In the absence of a net force, then, the plane or any object would continue to move at a constant speed along a straight line in the direction it had at the time of release. This speed and direction are given in Figure 5.4 by the velocity vector v. As a result, the object would move along the straight line between points O and A, not on the circular arc between points O and P.

11. Example 3: The Effect of Radius on Centripetal Acceleration The bobsled track at the 1994 Olympics in Lillehammer, Norway, contained turns with radii of 33 m and 24 m, as the figure illustrates. Find the centripetal acceleration at each turn for a speed of 34 m/s, a speed that was achieved in the two-man event. Express the answers as multiples of g=9.8m/s2.

12. From ac=v2/r it follows that Radius=33m Radius=24m

13. Conceptual Example 4: Uniform Circular Motion and Equilibrium A car moves at a constant speed, and there are three parts to the motion. It moves along a straight line toward a circular turn, goes around the turn, and then moves away along a straight line. In each of three parts, is the car in equilibrium?

14. An object in equilibrium has no acceleration, according to the definition given in Section 4.11. As the car approaches the turn, both the speed and direction of the motion are constant. Thus, the velocity vector does not change, and there is no acceleration. The same is true as the car moves away from the turn. For these parts of the motion, then, the car is in equilibrium. As the car goes around the turn, however, the direction of travel changes, so the car has a centripetal acceleration that is characteristic of uniform circular motion. Because of this acceleration, the car is not in equilibrium during the turn. In general, an object that is in uniform circular motion can never be in equilibrium.

15. Check your understanding 1 The car in the drawing is moving clockwise around a circular section of road at a constant speed. What are the directions of its velocity and acceleration at (a) position 1 and (b) position 2?

16. (a) The velocity is due south, and the acceleration is due west. (b) The velocity is due west, and the acceleration is due north.

17. Centripetal Force

18. Concepts at a glance: Newton’s second law indicates that whenever an object accelerates, there must be a net force to create the acceleration. Thus, in uniform circular motion there must be a net force to produce the centripetal acceleration. As the Concept-at-a-glance chart, the second law gives this net force as the product of the object’s mass m and its acceleration v2/r. This chart is an expanded version of the chart shown previously in Figure 4.9. The net force causing the centripetal acceleration is called the centripetal force FC and points in the same direction as the acceleration- that is, toward the center of the circle.

19. “centripetal force” does not denote a new and separate force created by nature.

20. Example 5: The Effect of Speed on Centripetal Force The model airplane has a mass of 0.90 kg and moves at a constant speed on a circle that is parallel to the ground. The path of the airplane and its guideline lie in the same horizontal plane, because the weight of the plane is balanced by the lift generated by its wings. Find the tension T in the guideline(length=17m) for speeds of 19 and 38m/s.

21. Speed =38m/s Equation 5.3 gives the tension directly: FC=T=mv2/r Speed =19m/s

22. Conceptual Example 6: A Trapeze Act In a circus, a man hangs upside down from a trapeze, legs bent over the bar and arms downward, holding his partner. Is it harder for the man to hold his partner when the partner hangs straight down and is stationary or when the partner is swinging through the straight-down position?

23. Reasoning and Solution: When the man and his partner are stationary, the man’s arms must support his partner’s weight. When the two are swinging, however, the man’s arms must do an additional job. Then the partner is moving on a circular arc and has a centripetal acceleration. The man’s arms must exert and additional pull so that there will be sufficient centripetal force to produce this acceleration. Because of the additional pull, it is harder for the man to hold his partner while swinging than while stationary.

24. --dry = 0.9 --icy = 0.1 N FS FS mg Example 7: Centripetal Force and Safe Driving Compare the maximum speeds at which a car can safely negotiate an unbanked turn (r= 50.0m)

25. N fS mg The car does not accelerate , FN – mg = 0 FN = mg.

26. Dry road ( =0.900) Icy road ( =0.100) As expected, the dry road allows the greater maximum speed.

27. Upward on the wing surfaces with a net lifting force L, the plane is banked at an angle , a component L sin of the lifting force is directed toward the center of the turn. Greater speeds and/or tighter turns require greater centripetal forces. Banking into a turn also has an application in the construction of high-speed roadways.

28. Check your understanding 2 • A car is traveling in uniform circular motion on a section of road whose radius is r. The road is slippery, and the car is just on the verge of sliding. • If the car’s speed was doubled, what would have to be the smallest radius in order that the car does not slide? Express your answer in terms of r. • What would be your answer to part (a) if the car were replaced by one that weighted twice as much?

29. (a) 4r (b) 4r

30. Banked Curves

31. A car is going around a friction-free banked curve. The radius of the curve is r. FN sin that points toward the center C FN cos and, since the car does not accelerate in the vertical direction, this component must balance the weight mg of the car.

32. At a speed that is too small for a given , a car would slide down a frictionless banked curve: at a speed that is too large, a car would slide off the top.

33. Example 8:The Daytona 500 The Daytona 500 is the major event of the NASCAR (National Association for Stock Car Auto Racing) 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 . Suppose these maximum-radius turns were frictionless. At what speed would the cars have to travel around them?

34. From Equation 5.4, it follows that (96 mph) Drivers actually negotiate the turns at speeds up to 195 mph, however, which requires a greater centripetal force than that implied by Equation 5.4 for frictionless turns.

35. Satellites in Circular Orbits

36. If the satellite is to remain in an orbit of radius r, the speed must have precisely this value. The closer the satellite is to the earth, the smaller is the value for r and the greater the orbital speed must be. Mass m of the satellite does not appear. For a given orbit, a satellite with a large mass has exactly the same orbital speed as a satellite with a small mass.

37. Example 9: Orbital Speed of the Hubble Space Telescope Determine the speed of the Hubble Space Telescope orbiting at a height of 598 km above the earth’s surface. Orbital radius r must be determined relative to the center of the earth. The radius of the earth is approximately 6.38*106m, r=6.98*106m

38. The orbital speed is

39. Global Positioning System(GPS)

40. Example 10: A Super-massive Black Hole The Hubble Telescope has detected the light being emitted from different regions of galaxy M87. The black circle identifies the center of the galaxy. From the characteristics of this light, astronomers have determined an orbiting speed of 7.5*105m/s for matter located at a distance of 5.7*1017m from the center. Find the mass M of the object located at the galactic center.

41. Replacing ME with M =4.8*1039kg

42. The ratio of this incredibly large mass to the mass of our sun is (4.8*1039kg)/(2.0*1030kg)=2.4*109. Matter equivalent to 2.4 billion suns is located at the center of galaxy M87. The volume of space in which this matter is located contains relatively few visible star. There are strong evidences for the existence of a super-massive black hole. “black hole” tremendous mass prevents even light from escaping. The light that forms the image comes not from the black hole itself, but from matter that surrounds it.

43. Period T of a satellite is the time required for one orbital revolution.

44. Period is proportional to the three-halves power of the orbital radius is know as Kepler’s third law. (Johannes Kepler, 1571-1630). Kepler’s third law also holds for elliptical orbits. “synchronous satellites”: orbital period is chosen to be one day, the time it takes for the earth to turn once about its axis. Satellites move around their orbits in a way that is synchronized with the rotation of the earth, appearing in fixed positions in the sky and can serve as “stationary”.

45. Example 11: The Orbital Radius for Synchronous Satellites The period T of a synchronous satellite is one day. Find the distance r from the center of the earth and the height H of the satellite above the earth’s surface. The earth itself has a radius of 6.38*106m. T=8.64*104s

46. r = 4.23*107m H=4.23*107m-0.64*107m=3.59*107m (22300 mi)

47. MMars ---smaller Since v is the same, M small -----r small. Check your understanding 3 Two satellites are placed in orbit, one about Mars and the other about Jupiter, such that the orbital speeds are the same. Mars has the smaller mass. Is the radius of the satellite in orbit about Mars less than, greater than, or equal to the radius of the satellite orbiting Jupiter? Less than.

48. Apparent Weightlessness and Artificial Gravity The idea of life on board an orbiting satellite conjures up visions of astronauts floating around in a state of “weightlessness”

49. Conceptual Example 12: Apparent Weightlessness and Free-Fall Objects in uniform circular motion continually accelerate or “fall” toward the center of the circle, in order to remain on the circular path. The only difference between the satellite and the elevator is that the satellite moves on a circle, so that its “falling” does not bring it closer to the earth. True weight is the gravitational force (F=GmME/r2) that the earth exerts on an object and is not zero.

50. Example 13: Artificial Gravity At what speed must the surface of the space station (r=1700m) move in the figure, so that the astronaut at point P experiences a push on his feet that equals his earth weight?