Circular Orbits. Slide Pearson Education, Inc.

Size: px

Start display at page:

Download "Circular Orbits. Slide Pearson Education, Inc."

Clyde Fowler
6 years ago
Views:

1 Circular Orbits The figure shows a perfectly smooth, spherical, airless planet with one tower of height h. A projectile is launched parallel to the ground with speed v 0. If v 0 is very small, as in trajectory A, it simply falls to the ground along a parabolic trajectory. This is the flat-earth approximation. Slide 8-58

2 Circular Orbits As the initial speed v 0 is increased, the range of the projectile increases as the ground curves away from it. Trajectories B and C are of this type. If v 0 is sufficiently large, there comes a point where the trajectory and the curve of the earth are parallel. In this case, the projectile falls but it never gets any closer to the ground! This is trajectory D, called an orbit. Slide 8-59

3 Circular Orbits In the flat-earth approximation, shown in figure (a), the gravitational force on an object of mass m is: Since actual planets are spherical, the real force of gravity is toward the center of the planet, as shown in figure (b). Slide 8-60

4 Circular Orbits An object in a low circular orbit has acceleration: If the object moves in a circle of radius r at speed v orbit the centripetal acceleration is: The required speed for a circular orbit near a planet s surface, neglecting air resistance, is: Slide 8-61

5 Example 1 - Communications satellites are placed in circular orbits where they stay directly over a fixed point on the equator as the earth rotates*. What is the required altitude* of a geostationary satellite? What is the satellite s speed? Slide 8-48

6 Fictitious Forces If you are riding in a car that makes a sudden stop, you seem to be hurled forward. You can describe your experience in terms of fictitious forces. Fictitious forces are not real because no agent is exerting them. Fictitious forces describe your motion relative to a noninertial reference frame. Slide 8-64

7 Centrifugal Force? The figure shows a bird seye view of you riding in a car as it makes a left turn. From the perspective of an inertial reference frame, the normal force from the door points inward, keeping you on the road with the car. Relative to the noninertial reference frame of the car, you feel pushed toward the outside of the curve. The fictitious force which seems to push an object to the outside of a circle is called the centrifugal force. There really is no such force in an inertial reference frame. Slide 8-65

8 QuickCheck 8.6 A coin sits on a turntable as the table steadily rotates ccw. The free-body diagrams below show the coin from behind, moving away from you. Which is the correct diagram? Slide 8-66

9 QuickCheck 8.7 A coin sits on a turntable as the table steadily rotates ccw. What force or forces act in the plane of the turntable? Slide 8-68

QuickCheck 8.8 Two coins are on a turntable that steadily speeds up, starting from rest, with a ccw rotation. Which coin flies off the turntable first?

10 QuickCheck 8.8 Two coins are on a turntable that steadily speeds up, starting from rest, with a ccw rotation. Which coin flies off the turntable first? A. Coin 1 flies off first. B. Coin 2 flies off first. C. Both coins fly off at the same time. D. We can t say without knowing their masses. Slide 8-70

11 Example 2 - A 5.0 g coin is placed 15 cm from the center of a turnable. The coin has static and kinetic friction with the turntable surface of 0.80 and 0.50, respectively. The turntable very slowly* speeds up to 60 rpm. Does the coin slide off? Slide 8-48

12 Loop-the-Loop The figure shows a rollercoaster going around a vertical loop-the-loop of radius r. Note this is not uniform circular motion; the car slows down going up one side, and speeds up going down the other. At the very top and very bottom points, only the car s direction is changing, so the acceleration is purely centripetal. Because the car is moving in a circle, there must be a net force toward the center of the circle. Slide 8-73

13 Loop-the-Loop The figure shows the roller-coaster free-body diagram at the bottom of the loop. Since the net force is toward the center (upward at this point), n > F G. This is why you feel heavy at the bottom of the valley on a roller coaster. The normal force at the bottom is larger than mg. Slide 8-74

14 Loop-the-Loop The figure shows the roller-coaster free-body diagram at the top of the loop. The track can only push on the wheels of the car, it cannot pull, therefore presses downward. The car is still moving in a circle, so the net force is also downward: The normal force at the at the top can exceed mg if v top is large enough. Slide 8-75

15 Loop-the-Loop At the top of the roller coaster, the normal force of the track on the car is: As v top decreases, there comes a point when n reaches zero. The speed at which n = 0 is called the critical speed: This is the slowest speed at which the car can complete the circle without falling off the track near the top. Slide 8-76

16 Loop-the-Loop A roller-coaster car at the top of the loop. Slide 8-77

17 QuickCheck 8.9 A physics textbook swings back and forth as a pendulum. Which is the correct free-body diagram when the book is at the bottom and moving to the right? Slide 8-78

18 QuickCheck 8.10 A car that s out of gas coasts over the top of a hill at a steady 20 m/s. Assume air resistance is negligible. Which free-body diagram describes the car at this instant? Slide 8-80

19 QuickCheck 8.11 A roller coaster car does a loopthe-loop. Which of the free-body diagrams shows the forces on the car at the top of the loop? Rolling friction can be neglected. Slide 8-82

20 Example 3 - A 500 g ball moves in a vertical circle on a 102- cm-long string. If the speed at the top is 4.0 m/s, then the speed at the bottom will be 7.5 m/s. Find the tension in the string at the top and the bottom. Also, find the minimum speed the ball could travel and maintain circular motion.

21 Gravity on a Rotating Earth The figure shows an object being weighed by a spring scale on the earth s equator. The observer is hovering in an inertial reference frame above the north pole. If we pretend the spring-scale reading is F sp = F G = mg, this has the effect of weakening gravity. Slide 8-72

22 Gravity on a Rotating Earth Example 4 - A 75 kg man weighs himself at the north pole and at the equator. Which scale reading is higher and by how much? Slide 8-72

23 Drag The air exerts a resistive (drag) force on objects as they move through the air. Faster objects experience a greater drag force than slower objects. For example, The drag force on a high-speed motorcyclist is significant. The drag force direction is opposite the object s velocity. Slide 6-93

24 Drag For normal-sized objects on earth traveling at a speed v which is less than a few hundred meters per second, air resistance can be modeled as: A is the cross-section area of the object. ρ is the density of the air, which is about 1.3 kg/m 3. C is the drag coefficient, which is a dimensionless number that depends on the shape of the object. Slide 6-94

25 Drag Cross-section areas for objects of different shape. Effects due to drag are mass dependent. For an object falling: D mg = -ma y so a y = (g D/m) Slide 6-95

26 Terminal Speed The drag force from the air increases as an object gains speed. If an object falls far enough, it will eventually reach a speed at which D = F G. At this speed, the net force is zero, so the object falls at a constant speed, called the terminal speed v term. Slide 6-100

27 Terminal Speed The figure shows the velocity-versus-time graph of a falling object with and without drag. Without drag, the velocity graph is a straight line with a y = g. When drag is included, the vertical component of the velocity asymptotically approaches v term. Slide 6-101

28 Example 5 - The terminal velocity of a person falling in air depends upon the weight and the area of the person facing the fluid. Find the terminal velocity of an 80.0-kg skydiver falling in a pike (headfirst) position with a surface area of 0.140m 2.

PHYSICS. Chapter 8 Lecture FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E RANDALL D. KNIGHT Pearson Education, Inc.

PHYSICS. Chapter 8 Lecture FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E RANDALL D. KNIGHT Pearson Education, Inc. PHYSICS FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E Chapter 8 Lecture RANDALL D. KNIGHT Chapter 8. Dynamics II: Motion in a Plane IN THIS CHAPTER, you will learn to solve problems about motion