Chapter 6: Uniform Circular Motion and Gravitation PDF

Chapter 6 | Uniform Circular Motion and Gravitation 199 6 UNIFORM CIRCULAR MOTION AND GRAVITATION Figure 6.1 This Australian Grand Prix Formula 1 race car moves in a circular path as it makes the...

Chapter 6 | Uniform Circular Motion and Gravitation 199 6 UNIFORM CIRCULAR MOTION AND GRAVITATION Figure 6.1 This Australian Grand Prix Formula 1 race car moves in a circular path as it makes the turn. Its wheels also spin rapidly—the latter completing many revolutions, the former only part of one (a circular arc). The same physical principles are involved in each. (credit: Richard Munckton) Chapter Outline 6.1. Rotation Angle and Angular Velocity Define arc length, rotation angle, radius of curvature and angular velocity. Calculate the angular velocity of a car wheel spin. 6.2. Centripetal Acceleration Establish the expression for centripetal acceleration. Explain the centrifuge. 6.3. Centripetal Force Calculate coefficient of friction on a car tire. Calculate ideal speed and angle of a car on a turn. 6.4. Fictitious Forces and Non-inertial Frames: The Coriolis Force Discuss the inertial frame of reference. Discuss the non-inertial frame of reference. Describe the effects of the Coriolis force. 6.5. Newton’s Universal Law of Gravitation Explain Earth’s gravitational force. Describe the gravitational effect of the Moon on Earth. Discuss weightlessness in space. Examine the Cavendish experiment 6.6. Satellites and Kepler’s Laws: An Argument for Simplicity State Kepler’s laws of planetary motion. Derive the third Kepler’s law for circular orbits. 200 Chapter 6 | Uniform Circular Motion and Gravitation Discuss the Ptolemaic model of the universe. Introduction to Uniform Circular Motion and Gravitation Many motions, such as the arc of a bird’s flight or Earth’s path around the Sun, are curved. Recall that Newton’s first law tells us that motion is along a straight line at constant speed unless there is a net external force. We will therefore study not only motion along curves, but also the forces that cause it, including gravitational forces. In some ways, this chapter is a continuation of Dynamics: Newton's Laws of Motion as we study more applications of Newton’s laws of motion. This chapter deals with the simplest form of curved motion, uniform circular motion, motion in a circular path at constant speed. Studying this topic illustrates most concepts associated with rotational motion and leads to the study of many new topics we group under the name rotation. Pure rotational motion occurs when points in an object move in circular paths centered on one point. Pure translational motion is motion with no rotation. Some motion combines both types, such as a rotating hockey puck moving along ice. 6.1 Rotation Angle and Angular Velocity In Kinematics, we studied motion along a straight line and introduced such concepts as displacement, velocity, and acceleration. Two-Dimensional Kinematics dealt with motion in two dimensions. Projectile motion is a special case of two-dimensional kinematics in which the object is projected into the air, while being subject to the gravitational force, and lands a distance away. In this chapter, we consider situations where the object does not land but moves in a curve. We begin the study of uniform circular motion by defining two angular quantities needed to describe rotational motion. Rotation Angle When objects rotate about some axis—for example, when the CD (compact disc) in Figure 6.2 rotates about its center—each point in the object follows a circular arc. Consider a line from the center of the CD to its edge. Each pit used to record sound along this line moves through the same angle in the same amount of time. The rotation angle is the amount of rotation and is analogous to linear distance. We define the rotation angle Δθ to be the ratio of the arc length to the radius of curvature: (6.1) Δθ = Δs r. Figure 6.2 All points on a CD travel in circular arcs. The pits along a line from the center to the edge all move through the same angle Δθ in a time Δt. This OpenStax book is available for free at http://cnx.org/content/col11406/1.9 Chapter 6 | Uniform Circular Motion and Gravitation 201 Figure 6.3 The radius of a circle is rotated through an angle Δθ. The arc length Δs is described on the circumference. The arc length Δs is the distance traveled along a circular path as shown in Figure 6.3 Note that r is the radius of curvature of the circular path. We know that for one complete revolution, the arc length is the circumference of a circle of radius r. The circumference of a circle is 2πr. Thus for one complete revolution the rotation angle is (6.2) Δθ = 2πr r = 2π. This result is the basis for defining the units used to measure rotation angles, Δθ to be radians (rad), defined so that 2π rad = 1 revolution. (6.3) A comparison of some useful angles expressed in both degrees and radians is shown in Table 6.1. Table 6.1 Comparison of Angular Units Degree Measures Radian Measure π 30º 6 π 60º 3 π 90º 2 120º 2π 3 135º 3π 4 180º π 202 Chapter 6 | Uniform Circular Motion and Gravitation Figure 6.4 Points 1 and 2 rotate through the same angle ( Δθ ), but point 2 moves through a greater arc length (Δs) because it is at a greater distance from the center of rotation (r). If Δθ = 2π rad, then the CD has made one complete revolution, and every point on the CD is back at its original position. Because there are 360º in a circle or one revolution, the relationship between radians and degrees is thus 2π rad = 360º (6.4) so that (6.5) 1 rad = 360º ≈ 57.3º. 2π Angular Velocity How fast is an object rotating? We define angular velocity ω as the rate of change of an angle. In symbols, this is (6.6) ω = Δθ , Δt where an angular rotation Δθ takes place in a time Δt. The greater the rotation angle in a given amount of time, the greater the angular velocity. The units for angular velocity are radians per second (rad/s). Angular velocity ω is analogous to linear velocity v. To get the precise relationship between angular and linear velocity, we again consider a pit on the rotating CD. This pit moves an arc length Δs in a time Δt , and so it has a linear velocity (6.7) v = Δs. Δt From Δθ = Δs r we see that Δs = rΔθ. Substituting this into the expression for v gives v = rΔθ = rω. (6.8) Δt We write this relationship in two different ways and gain two different insights: v = rω or ω = vr. (6.9) The first relationship in v = rω or ω = vr states that the linear velocity v is proportional to the distance from the center of rotation, thus, it is largest for a point on the rim (largest r ), as you might expect. We can also call this linear speed v of a point on the rim the tangential speed. The second relationship in v = rω or ω = v r can be illustrated by considering the tire of a moving car. Note that the speed of a point on the rim of the tire is the same as the speed v of the car. See Figure 6.5. So the faster the car moves, the faster the tire spins—large v means a large ω , because v = rω. Similarly, a larger-radius tire rotating at the same angular velocity ( ω ) will produce a greater linear speed ( v ) for the car. This OpenStax book is available for free at http://cnx.org/content/col11406/1.9 Chapter 6 | Uniform Circular Motion and Gravitation 203 Figure 6.5 A car moving at a velocity v to the right has a tire rotating with an angular velocity ω.The speed of the tread of the tire relative to the axle is v , the same as if the car were jacked up. Thus the car moves forward at linear velocity v = rω , where r is the tire radius. A larger angular velocity for the tire means a greater velocity for the car. Example 6.1 How Fast Does a Car Tire Spin? Calculate the angular velocity of a 0.300 m radius car tire when the car travels at 15.0 m/s (about 54 km/h ). See Figure 6.5. Strategy v = 15.0 m/s. The radius of the tire Because the linear speed of the tire rim is the same as the speed of the car, we have is given to be r = 0.300 m. Knowing v and r , we can use the second relationship in v = rω, ω = v r to calculate the angular velocity. Solution To calculate the angular velocity, we will use the following relationship: ω = vr. (6.10) Substituting the knowns, (6.11) ω = 15.0 m/s = 50.0 rad/s. 0.300 m Discussion When we cancel units in the above calculation, we get 50.0/s. But the angular velocity must have units of rad/s. Because radians are actually unitless (radians are defined as a ratio of distance), we can simply insert them into the answer for the angular velocity. Also note that if an earth mover with much larger tires, say 1.20 m in radius, were moving at the same speed of 15.0 m/s, its tires would rotate more slowly. They would have an angular velocity ω = (15.0 m/s) / (1.20 m) = 12.5 rad/s. (6.12) Both ω and v have directions (hence they are angular and linear velocities, respectively). Angular velocity has only two directions with respect to the axis of rotation—it is either clockwise or counterclockwise. Linear velocity is tangent to the path, as illustrated in Figure 6.6. Take-Home Experiment Tie an object to the end of a string and swing it around in a horizontal circle above your head (swing at your wrist). Maintain uniform speed as the object swings and measure the angular velocity of the motion. What is the approximate speed of the object? Identify a point close to your hand and take appropriate measurements to calculate the linear speed at this point. Identify other circular motions and measure their angular velocities. 204 Chapter 6 | Uniform Circular Motion and Gravitation Figure 6.6 As an object moves in a circle, here a fly on the edge of an old-fashioned vinyl record, its instantaneous velocity is always tangent to the circle. The direction of the angular velocity is clockwise in this case. PhET Explorations: Ladybug Revolution Figure 6.7 Ladybug Revolution (http://cnx.org/content/m42083/1.7/rotation_en.jar) Join the ladybug in an exploration of rotational motion. Rotate the merry-go-round to change its angle, or choose a constant angular velocity or angular acceleration. Explore how circular motion relates to the bug's x,y position, velocity, and acceleration using vectors or graphs. 6.2 Centripetal Acceleration We know from kinematics that acceleration is a change in velocity, either in its magnitude or in its direction, or both. In uniform circular motion, the direction of the velocity changes constantly, so there is always an associated acceleration, even though the magnitude of the velocity might be constant. You experience this acceleration yourself when you turn a corner in your car. (If you hold the wheel steady during a turn and move at constant speed, you are in uniform circular motion.) What you notice is a sideways acceleration because you and the car are changing direction. The sharper the curve and the greater your speed, the more noticeable this acceleration will become. In this section we examine the direction and magnitude of that acceleration. Figure 6.8 shows an object moving in a circular path at constant speed. The direction of the instantaneous velocity is shown at two points along the path. Acceleration is in the direction of the change in velocity, which points directly toward the center of rotation (the center of the circular path). This pointing is shown with the vector diagram in the figure. We call the acceleration of an object moving in uniform circular motion (resulting from a net external force) the centripetal acceleration( a c ); centripetal means “toward the center” or “center seeking.” This OpenStax book is available for free at http://cnx.org/content/col11406/1.9 Chapter 6 | Uniform Circular Motion and Gravitation 205 Figure 6.8 The directions of the velocity of an object at two different points are shown, and the change in velocityΔv is seen to point directly toward the center of curvature. (See small inset.) Because a c = Δv / Δt , the acceleration is also toward the center; a c is called centripetal acceleration. (Because Δθ is very small, the arc length Δs is equal to the chord length Δr for small time differences.) The direction of centripetal acceleration is toward the center of curvature, but what is its magnitude? Note that the triangle formed by the velocity vectors and the one formed by the radii r and Δs are similar. Both the triangles ABC and PQR are isosceles triangles (two equal sides). The two equal sides of the velocity vector triangle are the speeds v 1 = v 2 = v. Using the properties of two similar triangles, we obtain Δv = Δs. (6.13) v r Acceleration is Δv / Δt , and so we first solve this expression for Δv : Δv = vr Δs. (6.14) Then we divide this by Δt , yielding Δv = v × Δs. (6.15) Δt r Δt Finally, noting that Δv / Δt = a c and that Δs / Δt = v , the linear or tangential speed, we see that the magnitude of the centripetal acceleration is (6.16) a c = vr , 2 which is the acceleration of an object in a circle of radius r at a speed v. So, centripetal acceleration is greater at high speeds and in sharp curves (smaller radius), as you have noticed when driving a car. But it is a bit surprising that a c is proportional to speed squared, implying, for example, that it is four times as hard to take a curve at 100 km/h than at 50 km/h. A sharp corner has a small radius, so that a c is greater for tighter turns, as you have probably noticed. It is also useful to express a c in terms of angular velocity. Substituting v = rω into the above expression, we find a c = (rω) / r = rω. We can express the magnitude of centripetal acceleration using either of two equations: 2 2 (6.17) a c = vr ; a c = rω 2. 2 Recall that the direction of a c is toward the center. You may use whichever expression is more convenient, as illustrated in examples below. A centrifuge (see Figure 6.9b) is a rotating device used to separate specimens of different densities. High centripetal acceleration significantly decreases the time it takes for separation to occur, and makes separation possible with small samples. Centrifuges are used in a variety of applications in science and medicine, including the separation of single cell suspensions such as bacteria, viruses, and blood cells from a liquid medium and the separation of macromolecules, such as DNA and protein, 206 Chapter 6 | Uniform Circular Motion and Gravitation from a solution. Centrifuges are often rated in terms of their centripetal acceleration relative to acceleration due to gravity (g) ; maximum centripetal acceleration of several hundred thousand g is possible in a vacuum. Human centrifuges, extremely large centrifuges, have been used to test the tolerance of astronauts to the effects of accelerations larger than that of Earth’s gravity. Example 6.2 How Does the Centripetal Acceleration of a Car Around a Curve Compare with That Due to Gravity? What is the magnitude of the centripetal acceleration of a car following a curve of radius 500 m at a speed of 25.0 m/s (about 90 km/h)? Compare the acceleration with that due to gravity for this fairly gentle curve taken at highway speed. See Figure 6.9(a). Strategy v and r are given, the first expression in a c = vr ; a c = rω 2 is the most convenient to use. 2 Because Solution Entering the given values of v = 25.0 m/s and r = 500 m into the first expression for a c gives (25.0 m/s) 2 (6.18) a c = vr = 2 = 1.25 m/s 2. 500 m Discussion To compare this with the acceleration due to gravity (g = 9.80 m/s 2) , we take the ratio of a c / g = ⎛⎝1.25 m/s 2⎞⎠ / ⎛⎝9.80 m/s 2⎞⎠ = 0.128. Thus, a c = 0.128 g and is noticeable especially if you were not wearing a seat belt. This OpenStax book is available for free at http://cnx.org/content/col11406/1.9 Chapter 6 | Uniform Circular Motion and Gravitation 207 Figure 6.9 (a) The car following a circular path at constant speed is accelerated perpendicular to its velocity, as shown. The magnitude of this centripetal acceleration is found in Example 6.2. (b) A particle of mass in a centrifuge is rotating at constant angular velocity. It must be accelerated perpendicular to its velocity or it would continue in a straight line. The magnitude of the necessary acceleration is found in Example 6.3. Example 6.3 How Big Is the Centripetal Acceleration in an Ultracentrifuge? Calculate the centripetal acceleration of a point 7.50 cm from the axis of an ultracentrifuge spinning at 7.5 × 10 4 rev/min. Determine the ratio of this acceleration to that due to gravity. See Figure 6.9(b). Strategy The term rev/min stands for revolutions per minute. By converting this to radians per second, we obtain the angular velocity ω. Because r is given, we can use the second expression in the equation a c = vr ; a c = rω 2 to calculate the 2 centripetal acceleration. Solution To convert 7.50×10 4 rev / min to radians per second, we use the facts that one revolution is 2πrad and one minute is 60.0 s. Thus, (6.19) ω = 7.50×10 4 rev × 2π rad × 1 min = 7854 rad/s. min 1 rev 60.0 s a c = vr ; a c = rω 2 as 2 Now the centripetal acceleration is given by the second expression in a c = rω 2. (6.20) Converting 7.50 cm to meters and substituting known values gives a c = (0.0750 m)(7854 rad/s) 2 = 4.63×10 6 m/s 2. (6.21) 208 Chapter 6 | Uniform Circular Motion and Gravitation Note that the unitless radians are discarded in order to get the correct units for centripetal acceleration. Taking the ratio of a c to g yields a c 4.63×10 6 5 (6.22) g = 9.80 = 4.72×10. Discussion This last result means that the centripetal acceleration is 472,000 times as strong as g. It is no wonder that such high ω centrifuges are called ultracentrifuges. The extremely large accelerations involved greatly decrease the time needed to cause the sedimentation of blood cells or other materials. Of course, a net external force is needed to cause any acceleration, just as Newton proposed in his second law of motion. So a net external force is needed to cause a centripetal acceleration. In Centripetal Force, we will consider the forces involved in circular motion. PhET Explorations: Ladybug Motion 2D Learn about position, velocity and acceleration vectors. Move the ladybug by setting the position, velocity or acceleration, and see how the vectors change. Choose linear, circular or elliptical motion, and record and playback the motion to analyze the behavior. Figure 6.10 Ladybug Motion 2D (http://cnx.org/content/m42084/1.8/ladybug-motion-2d_en.jar) 6.3 Centripetal Force Any force or combination of forces can cause a centripetal or radial acceleration. Just a few examples are the tension in the rope on a tether ball, the force of Earth’s gravity on the Moon, friction between roller skates and a rink floor, a banked roadway’s force on a car, and forces on the tube of a spinning centrifuge. Any net force causing uniform circular motion is called a centripetal force. The direction of a centripetal force is toward the center of curvature, the same as the direction of centripetal acceleration. According to Newton’s second law of motion, net force is mass times acceleration: net F = ma. For uniform circular motion, the acceleration is the centripetal acceleration— a = a c. Thus, the magnitude of centripetal force F c is F c = ma c. (6.23) a c from a c = vr ; a c = rω 2 , we get two expressions for the centripetal 2 By using the expressions for centripetal acceleration force F c in terms of mass, velocity, angular velocity, and radius of curvature: (6.24) F c = m vr ; F c = mrω 2. 2 You may use whichever expression for centripetal force is more convenient. Centripetal force F c is always perpendicular to the path and pointing to the center of curvature, because a c is perpendicular to the velocity and pointing to the center of curvature. Note that if you solve the first expression for r , you get (6.25) r = mv. 2 Fc This implies that for a given mass and velocity, a large centripetal force causes a small radius of curvature—that is, a tight curve. This OpenStax book is available for free at http://cnx.org/content/col11406/1.9 Chapter 6 | Uniform Circular Motion and Gravitation 209 Figure 6.11 The frictional force supplies the centripetal force and is numerically equal to it. Centripetal force is perpendicular to velocity and causes uniform circular motion. The larger the F c , the smaller the radius of curvature r and the sharper the curve. The second curve has the same v , but a larger Fc produces a smaller r′. Example 6.4 What Coefficient of Friction Do Car Tires Need on a Flat Curve? (a) Calculate the centripetal force exerted on a 900 kg car that negotiates a 500 m radius curve at 25.0 m/s. (b) Assuming an unbanked curve, find the minimum static coefficient of friction, between the tires and the road, static friction being the reason that keeps the car from slipping (see Figure 6.12). Strategy and Solution for (a) F c = mv 2 We know that r. Thus, (900 kg)(25.0 m/s) 2 (6.26) F c = mv 2 r = (500 m) = 1125 N. Strategy for (b) Figure 6.12 shows the forces acting on the car on an unbanked (level ground) curve. Friction is to the left, keeping the car from slipping, and because it is the only horizontal force acting on the car, the friction is the centripetal force in this case. We know that the maximum static friction (at which the tires roll but do not slip) is µ s N , where µ s is the static coefficient of friction and N is the normal force. The normal force equals the car’s weight on level ground, so that N = mg. Thus the centripetal force in this situation is F c = f = µ sN = µ smg. (6.27) Now we have a relationship between centripetal force and the coefficient of friction. Using the first expression for F c from the equation 2 ⎫ (6.28) F c = m vr ⎬, F c = mrω 2⎭ (6.29) m vr = µ smg. 2 We solve this for µ s , noting that mass cancels, and obtain 210 Chapter 6 | Uniform Circular Motion and Gravitation (6.30) µ s = vrg. 2 Solution for (b) Substituting the knowns, (25.0 m/s) 2 (6.31) µs = = 0.13. (500 m)(9.80 m/s 2) (Because coefficients of friction are approximate, the answer is given to only two digits.) Discussion ⎫ F c = m vr ⎬, because m, v, and r are given. The coefficient 2 We could also solve part (a) using the first expression in F c = mrω 2⎭ of friction found in part (b) is much smaller than is typically found between tires and roads. The car will still negotiate the curve if the coefficient is greater than 0.13, because static friction is a responsive force, being able to assume a value less than but no more than µ s N. A higher coefficient would also allow the car to negotiate the curve at a higher speed, but if the coefficient of friction is less, the safe speed would be less than 25 m/s. Note that mass cancels, implying that in this example, it does not matter how heavily loaded the car is to negotiate the turn. Mass cancels because friction is assumed proportional to the normal force, which in turn is proportional to mass. If the surface of the road were banked, the normal force would be less as will be discussed below. Figure 6.12 This car on level ground is moving away and turning to the left. The centripetal force causing the car to turn in a circular path is due to friction between the tires and the road. A minimum coefficient of friction is needed, or the car will move in a larger-radius curve and leave the roadway. Let us now consider banked curves, where the slope of the road helps you negotiate the curve. See Figure 6.13. The greater the angle θ , the faster you can take the curve. Race tracks for bikes as well as cars, for example, often have steeply banked curves. In an “ideally banked curve,” the angle θ is such that you can negotiate the curve at a certain speed without the aid of friction between the tires and the road. We will derive an expression for θ for an ideally banked curve and consider an example related to it. For ideal banking, the net external force equals the horizontal centripetal force in the absence of friction. The components of the normal force N in the horizontal and vertical directions must equal the centripetal force and the weight of the car, respectively. In cases in which forces are not parallel, it is most convenient to consider components along perpendicular axes—in this case, the vertical and horizontal directions. Figure 6.13 shows a free body diagram for a car on a frictionless banked curve. If the angle θ is ideal for the speed and radius, then the net external force will equal the necessary centripetal force. The only two external forces acting on the car are its weight w and the normal force of the road N. (A frictionless surface can only exert a force perpendicular to the surface—that is, a normal force.) These two forces must add to give a net external force that is horizontal toward the center of curvature and has This OpenStax book is available for free at http://cnx.org/content/col11406/1.9 Chapter 6 | Uniform Circular Motion and Gravitation 211 magnitude mv 2 /r. Because this is the crucial force and it is horizontal, we use a coordinate system with vertical and horizontal axes. Only the normal force has a horizontal component, and so this must equal the centripetal force—that is, (6.32) N sin θ = mv 2 r. Because the car does not leave the surface of the road, the net vertical force must be zero, meaning that the vertical components of the two external forces must be equal in magnitude and opposite in direction. From the figure, we see that the vertical component of the normal force is N cos θ , and the only other vertical force is the car’s weight. These must be equal in magnitude; thus, N cos θ = mg. (6.33) Now we can combine the last two equations to eliminate N and get an expression for θ , as desired. Solving the second equation for N = mg / (cos θ) , and substituting this into the first yields (6.34) mg sin θ = mv 2 cos θ r (6.35) mg tan(θ) = mv 2 r tan θ = v2 rg. Taking the inverse tangent gives ⎛ 2⎞ (6.36) θ = tan −1⎝vrg ⎠ (ideally banked curve, no friction). This expression can be understood by considering how θ depends on v and r. A large θ will be obtained for a large v and a small r. That is, roads must be steeply banked for high speeds and sharp curves. Friction helps, because it allows you to take the curve at greater or lower speed than if the curve is frictionless. Note that θ does not depend on the mass of the vehicle. Figure 6.13 The car on this banked curve is moving away and turning to the left. Example 6.5 What Is the Ideal Speed to Take a Steeply Banked Tight Curve? Curves on some test tracks and race courses, such as the Daytona International Speedway in Florida, are very steeply banked. This banking, with the aid of tire friction and very stable car configurations, allows the curves to be taken at very high speed. To illustrate, calculate the speed at which a 100 m radius curve banked at 65.0° should be driven if the road is frictionless. Strategy We first note that all terms in the expression for the ideal angle of a banked curve except for speed are known; thus, we need only rearrange it so that speed appears on the left-hand side and then substitute known quantities. Solution Starting with (6.37) tan θ = vrg 2 212 Chapter 6 | Uniform Circular Motion and Gravitation we get v = (rg tan θ) 1 / 2. (6.38) Noting that tan 65.0º = 2.14, we obtain ⎡ m)(9.80 m/s 2)(2.14)⎤⎦ 1/2 (6.39) v = ⎣(100 = 45.8 m/s. Discussion This is just about 165 km/h, consistent with a very steeply banked and rather sharp curve. Tire friction enables a vehicle to take the curve at significantly higher speeds. Calculations similar to those in the preceding examples can be performed for a host of interesting situations in which centripetal force is involved—a number of these are presented in this chapter’s Problems and Exercises. Take-Home Experiment Ask a friend or relative to swing a golf club or a tennis racquet. Take appropriate measurements to estimate the centripetal acceleration of the end of the club or racquet. You may choose to do this in slow motion. PhET Explorations: Gravity and Orbits Move the sun, earth, moon and space station to see how it affects their gravitational forces and orbital paths. Visualize the sizes and distances between different heavenly bodies, and turn off gravity to see what would happen without it! Figure 6.14 Gravity and Orbits (http://cnx.org/content/m42086/1.9/gravity-and-orbits_en.jar) 6.4 Fictitious Forces and Non-inertial Frames: The Coriolis Force What do taking off in a jet airplane, turning a corner in a car, riding a merry-go-round, and the circular motion of a tropical cyclone have in common? Each exhibits fictitious forces—unreal forces that arise from motion and may seem real, because the observer’s frame of reference is accelerating or rotating. When taking off in a jet, most people would agree it feels as if you are being pushed back into the seat as the airplane accelerates down the runway. Yet a physicist would say that you tend to remain stationary while the seat pushes forward on you, and there is no real force backward on you. An even more common experience occurs when you make a tight curve in your car—say, to the right. You feel as if you are thrown (that is, forced) toward the left relative to the car. Again, a physicist would say that you are going in a straight line but the car moves to the right, and there is no real force on you to the left. Recall Newton’s first law. This OpenStax book is available for free at http://cnx.org/content/col11406/1.9 Chapter 6 | Uniform Circular Motion and Gravitation 213 Figure 6.15 (a) The car driver feels herself forced to the left relative to the car when she makes a right turn. This is a fictitious force arising from the use of the car as a frame of reference. (b) In the Earth’s frame of reference, the driver moves in a straight line, obeying Newton’s first law, and the car moves to the right. There is no real force to the left on the driver relative to Earth. There is a real force to the right on the car to make it turn. We can reconcile these points of view by examining the frames of reference used. Let us concentrate on people in a car. Passengers instinctively use the car as a frame of reference, while a physicist uses Earth. The physicist chooses Earth because it is very nearly an inertial frame of reference—one in which all forces are real (that is, in which all forces have an identifiable physical origin). In such a frame of reference, Newton’s laws of motion take the form given in Dynamics: Newton's Laws of Motion The car is a non-inertial frame of reference because it is accelerated to the side. The force to the left sensed by car passengers is a fictitious force having no physical origin. There is nothing real pushing them left—the car, as well as the driver, is actually accelerating to the right. Let us now take a mental ride on a merry-go-round—specifically, a rapidly rotating playground merry-go-round. You take the merry-go-round to be your frame of reference because you rotate together. In that non-inertial frame, you feel a fictitious force, named centrifugal force (not to be confused with centripetal force), trying to throw you off. You must hang on tightly to counteract the centrifugal force. In Earth’s frame of reference, there is no force trying to throw you off. Rather you must hang on to make yourself go in a circle because otherwise you would go in a straight line, right off the merry-go-round. Figure 6.16 (a) A rider on a merry-go-round feels as if he is being thrown off. This fictitious force is called the centrifugal force—it explains the rider’s motion in the rotating frame of reference. (b) In an inertial frame of reference and according to Newton’s laws, it is his inertia that carries him off and not a real force (the unshaded rider has F net = 0 and heads in a straight line). A real force, F centripetal , is needed to cause a circular path. This inertial effect, carrying you away from the center of rotation if there is no centripetal force to cause circular motion, is put to good use in centrifuges (see Figure 6.17). A centrifuge spins a sample very rapidly, as mentioned earlier in this chapter. Viewed from the rotating frame of reference, the fictitious centrifugal force throws particles outward, hastening their sedimentation. The greater the angular velocity, the greater the centrifugal force. But what really happens is that the inertia of the particles carries them along a line tangent to the circle while the test tube is forced in a circular path by a centripetal force. 214 Chapter 6 | Uniform Circular Motion and Gravitation Figure 6.17 Centrifuges use inertia to perform their task. Particles in the fluid sediment come out because their inertia carries them away from the center of rotation. The large angular velocity of the centrifuge quickens the sedimentation. Ultimately, the particles will come into contact with the test tube walls, which will then supply the centripetal force needed to make them move in a circle of constant radius. Let us now consider what happens if something moves in a frame of reference that rotates. For example, what if you slide a ball directly away from the center of the merry-go-round, as shown in Figure 6.18? The ball follows a straight path relative to Earth (assuming negligible friction) and a path curved to the right on the merry-go-round’s surface. A person standing next to the merry-go-round sees the ball moving straight and the merry-go-round rotating underneath it. In the merry-go-round’s frame of reference, we explain the apparent curve to the right by using a fictitious force, called the Coriolis force, that causes the ball to curve to the right. The fictitious Coriolis force can be used by anyone in that frame of reference to explain why objects follow curved paths and allows us to apply Newton’s Laws in non-inertial frames of reference. Figure 6.18 Looking down on the counterclockwise rotation of a merry-go-round, we see that a ball slid straight toward the edge follows a path curved to the right. The person slides the ball toward point B, starting at point A. Both points rotate to the shaded positions (A’ and B’) shown in the time that the ball follows the curved path in the rotating frame and a straight path in Earth’s frame. Up until now, we have considered Earth to be an inertial frame of reference with little or no worry about effects due to its rotation. Yet such effects do exist—in the rotation of weather systems, for example. Most consequences of Earth’s rotation can be qualitatively understood by analogy with the merry-go-round. Viewed from above the North Pole, Earth rotates counterclockwise, as does the merry-go-round in Figure 6.18. As on the merry-go-round, any motion in Earth’s northern hemisphere experiences a This OpenStax book is available for free at http://cnx.org/content/col11406/1.9 Chapter 6 | Uniform Circular Motion and Gravitation 215 Coriolis force to the right. Just the opposite occurs in the southern hemisphere; there, the force is to the left. Because Earth’s angular velocity is small, the Coriolis force is usually negligible, but for large-scale motions, such as wind patterns, it has substantial effects. The Coriolis force causes hurricanes in the northern hemisphere to rotate in the counterclockwise direction, while the tropical cyclones (what hurricanes are called below the equator) in the southern hemisphere rotate in the clockwise direction. The terms hurricane, typhoon, and tropical storm are regionally-specific names for tropical cyclones, storm systems characterized by low pressure centers, strong winds, and heavy rains. Figure 6.19 helps show how these rotations take place. Air flows toward any region of low pressure, and tropical cyclones contain particularly low pressures. Thus winds flow toward the center of a tropical cyclone or a low-pressure weather system at the surface. In the northern hemisphere, these inward winds are deflected to the right, as shown in the figure, producing a counterclockwise circulation at the surface for low-pressure zones of any type. Low pressure at the surface is associated with rising air, which also produces cooling and cloud formation, making low-pressure patterns quite visible from space. Conversely, wind circulation around high-pressure zones is clockwise in the northern hemisphere but is less visible because high pressure is associated with sinking air, producing clear skies. The rotation of tropical cyclones and the path of a ball on a merry-go-round can just as well be explained by inertia and the rotation of the system underneath. When non-inertial frames are used, fictitious forces, such as the Coriolis force, must be invented to explain the curved path. There is no identifiable physical source for these fictitious forces. In an inertial frame, inertia explains the path, and no force is found to be without an identifiable source. Either view allows us to describe nature, but a view in an inertial frame is the simplest and truest, in the sense that all forces have real origins and explanations. Figure 6.19 (a) The counterclockwise rotation of this northern hemisphere hurricane is a major consequence of the Coriolis force. (credit: NASA) (b) Without the Coriolis force, air would flow straight into a low-pressure zone, such as that found in tropical cyclones. (c) The Coriolis force deflects the winds to the right, producing a counterclockwise rotation. (d) Wind flowing away from a high-pressure zone is also deflected to the right, producing a clockwise rotation. (e) The opposite direction of rotation is produced by the Coriolis force in the southern hemisphere, leading to tropical cyclones. (credit: NASA) 6.5 Newton’s Universal Law of Gravitation What do aching feet, a falling apple, and the orbit of the Moon have in common? Each is caused by the gravitational force. Our feet are strained by supporting our weight—the force of Earth’s gravity on us. An apple falls from a tree because of the same force acting a few meters above Earth’s surface. And the Moon orbits Earth because gravity is able to supply the necessary centripetal force at a distance of hundreds of millions of meters. In fact, the same force causes planets to orbit the Sun, stars to orbit the center of the galaxy, and galaxies to cluster together. Gravity is another example of underlying simplicity in nature. It is the weakest of the four basic forces found in nature, and in some ways the least understood. It is a force that acts at a distance, without physical contact, and is expressed by a formula that is valid everywhere in the universe, for masses and distances that vary from the tiny to the immense. Sir Isaac Newton was the first scientist to precisely define the gravitational force, and to show that it could explain both falling bodies and astronomical motions. See Figure 6.20. But Newton was not the first to suspect that the same force caused both our weight and the motion of planets. His forerunner Galileo Galilei had contended that falling bodies and planetary motions had the same cause. Some of Newton’s contemporaries, such as Robert Hooke, Christopher Wren, and Edmund Halley, had also made some progress toward understanding gravitation. But Newton was the first to propose an exact mathematical form and to use that form to show that the motion of heavenly bodies should be conic sections—circles, ellipses, parabolas, and hyperbolas. This theoretical prediction was a major triumph—it had been known for some time that moons, planets, and comets follow such paths, but no one had been able to propose a mechanism that caused them to follow these paths and not others. 216 Chapter 6 | Uniform Circular Motion and Gravitation Figure 6.20 According to early accounts, Newton was inspired to make the connection between falling bodies and astronomical motions when he saw an apple fall from a tree and realized that if the gravitational force could extend above the ground to a tree, it might also reach the Sun. The inspiration of Newton’s apple is a part of worldwide folklore and may even be based in fact. Great importance is attached to it because Newton’s universal law of gravitation and his laws of motion answered very old questions about nature and gave tremendous support to the notion of underlying simplicity and unity in nature. Scientists still expect underlying simplicity to emerge from their ongoing inquiries into nature. The gravitational force is relatively simple. It is always attractive, and it depends only on the masses involved and the distance between them. Stated in modern language, Newton’s universal law of gravitation states that every particle in the universe attracts every other particle with a force along a line joining them. The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. Figure 6.21 Gravitational attraction is along a line joining the centers of mass of these two bodies. The magnitude of the force is the same on each, consistent with Newton’s third law. Misconception Alert The magnitude of the force on each object (one has larger mass than the other) is the same, consistent with Newton’s third law. The bodies we are dealing with tend to be large. To simplify the situation we assume that the body acts as if its entire mass is concentrated at one specific point called the center of mass (CM), which will be further explored in Linear Momentum and This OpenStax book is available for free at http://cnx.org/content/col11406/1.9 Chapter 6 | Uniform Circular Motion and Gravitation 217 Collisions. For two bodies having masses m and M with a distance r between their centers of mass, the equation for Newton’s universal law of gravitation is F = G mM , (6.40) r2 where F is the magnitude of the gravitational force and G is a proportionality factor called the gravitational constant. G is a universal gravitational constant—that is, it is thought to be the same everywhere in the universe. It has been measured experimentally to be 2 (6.41) G = 6.674×10 −11 N ⋅ m kg 2 in SI units. Note that the units of G are such that a force in newtons is obtained from F = G mM , when considering masses in r2 kilograms and distance in meters. For example, two 1.000 kg masses separated by 1.000 m will experience a gravitational attraction of 6.674×10 −11 N. This is an extraordinarily small force. The small magnitude of the gravitational force is consistent with everyday experience. We are unaware that even large objects like mountains exert gravitational forces on us. In fact, our body weight is the force of attraction of the entire Earth on us with a mass of 6×10 24 kg. Recall that the acceleration due to gravity g is about 9.80 m/s 2 on Earth. We can now determine why this is so. The weight of an object mg is the gravitational force between it and Earth. Substituting mg for F in Newton’s universal law of gravitation gives mg = G mM , (6.42) r2 where m is the mass of the object, M is the mass of Earth, and r is the distance to the center of Earth (the distance between the centers of mass of the object and Earth). See Figure 6.22. The mass m of the object cancels, leaving an equation for g : g = G M2. (6.43) r Substituting known values for Earth’s mass and radius (to three significant figures), ⎛ 2 ⎞ 5.98×10 24 kg (6.44) g = ⎜6.67×10 −11 N ⋅ m ⎟× , ⎝ kg 2 ⎠ (6.38×10 6 m) 2 and we obtain a value for the acceleration of a falling body: g = 9.80 m/s 2. (6.45) Figure 6.22 The distance between the centers of mass of Earth and an object on its surface is very nearly the same as the radius of Earth, because Earth is so much larger than the object. This is the expected value and is independent of the body’s mass. Newton’s law of gravitation takes Galileo’s observation that all masses fall with the same acceleration a step further, explaining the observation in terms of a force that causes objects to fall—in fact, in terms of a universally existing force of attraction between masses. Take-Home Experiment Take a marble, a ball, and a spoon and drop them from the same height. Do they hit the floor at the same time? If you drop a piece of paper as well, does it behave like the other objects? Explain your observations. 218 Chapter 6 | Uniform Circular Motion and Gravitation Making Connections Attempts are still being made to understand the gravitational force. As we shall see in Particle Physics, modern physics is exploring the connections of gravity to other forces, space, and time. General relativity alters our view of gravitation, leading us to think of gravitation as bending space and time. In the following example, we make a comparison similar to one made by Newton himself. He noted that if the gravitational force caused the Moon to orbit Earth, then the acceleration due to gravity should equal the centripetal acceleration of the Moon in its orbit. Newton found that the two accelerations agreed “pretty nearly.” Example 6.6 Earth’s Gravitational Force Is the Centripetal Force Making the Moon Move in a Curved Path (a) Find the acceleration due to Earth’s gravity at the distance of the Moon. (b) Calculate the centripetal acceleration needed to keep the Moon in its orbit (assuming a circular orbit about a fixed Earth), and compare it with the value of the acceleration due to Earth’s gravity that you have just found. Strategy for (a) This calculation is the same as the one finding the acceleration due to gravity at Earth’s surface, except that r is the distance from the center of Earth to the center of the Moon. The radius of the Moon’s nearly circular orbit is 3.84×10 8 m. Solution for (a) Substituting known values into the expression for g found above, remembering that M is the mass of Earth not the Moon, yields ⎛ 2 ⎞ 5.98×10 24 kg (6.46) g = G M2 = ⎜6.67×10 −11 N ⋅ m ⎟× r ⎝ kg 2 ⎠ (3.84×10 8 m) 2 = 2.70×10 −3 m/s. 2 Strategy for (b) Centripetal acceleration can be calculated using either form of 2⎫ (6.47) a c = vr ⎬. a c = rω 2⎭ We choose to use the second form: a c = rω 2, (6.48) where ω is the angular velocity of the Moon about Earth. Solution for (b) Given that the period (the time it takes to make one complete rotation) of the Moon’s orbit is 27.3 days, (d) and using (6.49) 1 d×24 hr ×60 min ×60 s = 86,400 s d hr min we see that 2π rad (6.50) ω = Δθ = = 2.66×10 −6 rad s. Δt (27.3 d)(86,400 s/d) The centripetal acceleration is a c = rω 2 = (3.84×10 8 m)(2.66×10 −6 rad/s) 2 (6.51) = 2.72×10 −3 m/s. 2 The direction of the acceleration is toward the center of the Earth. Discussion The centripetal acceleration of the Moon found in (b) differs by less than 1% from the acceleration due to Earth’s gravity found in (a). This agreement is approximate because the Moon’s orbit is slightly elliptical, and Earth is not stationary (rather the Earth-Moon system rotates about its center of mass, which is located some 1700 km below Earth’s surface). The clear implication is that Earth’s gravitational force causes the Moon to orbit Earth. This OpenStax book is available for free at http://cnx.org/content/col11406/1.9 Chapter 6 | Uniform Circular Motion and Gravitation 219 Why does Earth not remain stationary as the Moon orbits it? This is because, as expected from Newton’s third law, if Earth exerts a force on the Moon, then the Moon should exert an equal and opposite force on Earth (see Figure 6.23). We do not sense the Moon’s effect on Earth’s motion, because the Moon’s gravity moves our bodies right along with Earth but there are other signs on Earth that clearly show the effect of the Moon’s gravitational force as discussed in Satellites and Kepler's Laws: An Argument for Simplicity. Figure 6.23 (a) Earth and the Moon rotate approximately once a month around their common center of mass. (b) Their center of mass orbits the Sun in an elliptical orbit, but Earth’s path around the Sun has “wiggles” in it. Similar wiggles in the paths of stars have been observed and are considered direct evidence of planets orbiting those stars. This is important because the planets’ reflected light is often too dim to be observed. Tides Ocean tides are one very observable result of the Moon’s gravity acting on Earth. Figure 6.24 is a simplified drawing of the Moon’s position relative to the tides. Because water easily flows on Earth’s surface, a high tide is created on the side of Earth nearest to the Moon, where the Moon’s gravitational pull is strongest. Why is there also a high tide on the opposite side of Earth? The answer is that Earth is pulled toward the Moon more than the water on the far side, because Earth is closer to the Moon. So the water on the side of Earth closest to the Moon is pulled away from Earth, and Earth is pulled away from water on the far side. As Earth rotates, the tidal bulge (an effect of the tidal forces between an orbiting natural satellite and the primary planet that it orbits) keeps its orientation with the Moon. Thus there are two tides per day (the actual tidal period is about 12 hours and 25.2 minutes), because the Moon moves in its orbit each day as well). Figure 6.24 The Moon causes ocean tides by attracting the water on the near side more than Earth, and by attracting Earth more than the water on the far side. The distances and sizes are not to scale. For this simplified representation of the Earth-Moon system, there are two high and two low tides per day at any location, because Earth rotates under the tidal bulge. The Sun also affects tides, although it has about half the effect of the Moon. However, the largest tides, called spring tides, occur when Earth, the Moon, and the Sun are aligned. The smallest tides, called neap tides, occur when the Sun is at a 90º angle to the Earth-Moon alignment. 220 Chapter 6 | Uniform Circular Motion and Gravitation Figure 6.25 (a, b) Spring tides: The highest tides occur when Earth, the Moon, and the Sun are aligned. (c) Neap tide: The lowest tides occur when the Sun lies at 90º to the Earth-Moon alignment. Note that this figure is not drawn to scale. Tides are not unique to Earth but occur in many astronomical systems. The most extreme tides occur where the gravitational force is the strongest and varies most rapidly, such as near black holes (see Figure 6.26). A few likely candidates for black holes have been observed in our galaxy. These have masses greater than the Sun but have diameters only a few kilometers across. The tidal forces near them are so great that they can actually tear matter from a companion star. Figure 6.26 A black hole is an object with such strong gravity that not even light can escape it. This black hole was created by the supernova of one star in a two-star system. The tidal forces created by the black hole are so great that it tears matter from the companion star. This matter is compressed and heated as it is sucked into the black hole, creating light and X-rays observable from Earth. ”Weightlessness” and Microgravity In contrast to the tremendous gravitational force near black holes is the apparent gravitational field experienced by astronauts orbiting Earth. What is the effect of “weightlessness” upon an astronaut who is in orbit for months? Or what about the effect of weightlessness upon plant growth? Weightlessness doesn’t mean that an astronaut is not being acted upon by the gravitational force. There is no “zero gravity” in an astronaut’s orbit. The term just means that the astronaut is in free-fall, accelerating with the acceleration due to gravity. If an elevator cable breaks, the passengers inside will be in free fall and will experience weightlessness. You can experience short periods of weightlessness in some rides in amusement parks. This OpenStax book is available for free at http://cnx.org/content/col11406/1.9 Chapter 6 | Uniform Circular Motion and Gravitation 221 Figure 6.27 Astronauts experiencing weightlessness on board the International Space Station. (credit: NASA) Microgravity refers to an environment in which the apparent net acceleration of a body is small compared with that produced by Earth at its surface. Many interesting biology and physics topics have been studied over the past three decades in the presence of microgravity. Of immediate concern is the effect on astronauts of extended times in outer space, such as at the International Space Station. Researchers have observed that muscles will atrophy (waste away) in this environment. There is also a corresponding loss of bone mass. Study continues on cardiovascular adaptation to space flight. On Earth, blood pressure is usually higher in the feet than in the head, because the higher column of blood exerts a downward force on it, due to gravity. When standing, 70% of your blood is below the level of the heart, while in a horizontal position, just the opposite occurs. What difference does the absence of this pressure differential have upon the heart? Some findings in human physiology in space can be clinically important to the management of diseases back on Earth. On a somewhat negative note, spaceflight is known to affect the human immune system, possibly making the crew members more vulnerable to infectious diseases. Experiments flown in space also have shown that some bacteria grow faster in microgravity than they do on Earth. However, on a positive note, studies indicate that microbial antibiotic production can increase by a factor of two in space-grown cultures. One hopes to be able to understand these mechanisms so that similar successes can be achieved on the ground. In another area of physics space research, inorganic crystals and protein crystals have been grown in outer space that have much higher quality than any grown on Earth, so crystallography studies on their structure can yield much better results. Plants have evolved with the stimulus of gravity and with gravity sensors. Roots grow downward and shoots grow upward. Plants might be able to provide a life support system for long duration space missions by regenerating the atmosphere, purifying water, and producing food. Some studies have indicated that plant growth and development are not affected by gravity, but there is still uncertainty about structural changes in plants grown in a microgravity environment. The Cavendish Experiment: Then and Now As previously noted, the universal gravitational constant G is determined experimentally. This definition was first done accurately by Henry Cavendish (1731–1810), an English scientist, in 1798, more than 100 years after Newton published his universal law of gravitation. The measurement of G is very basic and important because it determines the strength of one of the four forces in nature. Cavendish’s experiment was very difficult because he measured the tiny gravitational attraction between two ordinary-sized masses (tens of kilograms at most), using apparatus like that in Figure 6.28. Remarkably, his value for G differs by less than 1% from the best modern value. One important consequence of knowing G was that an accurate value for Earth’s mass could finally be obtained. This was done by measuring the acceleration due to gravity as accurately as possible and then calculating the mass of Earth M from the relationship Newton’s universal law of gravitation gives mg = G mM , (6.52) r2 where m is the mass of the object, M is the mass of Earth, and r is the distance to the center of Earth (the distance between the centers of mass of the object and Earth). See Figure 6.21. The mass m of the object cancels, leaving an equation for g : g = G M2. (6.53) r Rearranging to solve for M yields gr 2 (6.54) M=. G 222 Chapter 6 | Uniform Circular Motion and Gravitation So M can be calculated because all quantities on the right, including the radius of Earth r , are known from direct measurements. We shall see in Satellites and Kepler's Laws: An Argument for Simplicity that knowing G also allows for the determination of astronomical masses. Interestingly, of all the fundamental constants in physics, G is by far the least well determined. The Cavendish experiment is also used to explore other aspects of gravity. One of the most interesting questions is whether the gravitational force depends on substance as well as mass—for example, whether one kilogram of lead exerts the same gravitational pull as one kilogram of water. A Hungarian scientist named Roland von Eötvös pioneered this inquiry early in the 20th century. He found, with an accuracy of five parts per billion, that the gravitational force does not depend on the substance. Such experiments continue today, and have improved upon Eötvös’ measurements. Cavendish-type experiments such as those of Eric Adelberger and others at the University of Washington, have also put severe limits on the possibility of a fifth force and have verified a major prediction of general relativity—that gravitational energy contributes to rest mass. Ongoing measurements there use a torsion balance and a parallel plate (not spheres, as Cavendish used) to examine how Newton’s law of gravitation works over sub-millimeter distances. On this small-scale, do gravitational effects depart from the inverse square law? So far, no deviation has been observed. Figure 6.28 Cavendish used an apparatus like this to measure the gravitational attraction between the two suspended spheres ( m ) and the two on the stand ( M ) by observing the amount of torsion (twisting) created in the fiber. Distance between the masses can be varied to check the dependence of the force on distance. Modern experiments of this type continue to explore gravity. 6.6 Satellites and Kepler’s Laws: An Argument for Simplicity Examples of gravitational orbits abound. Hundreds of artificial satellites orbit Earth together with thousands of pieces of debris. The Moon’s orbit about Earth has intrigued humans from time immemorial. The orbits of planets, asteroids, meteors, and comets about the Sun are no less interesting. If we look further, we see almost unimaginable numbers of stars, galaxies, and other celestial objects orbiting one another and interacting through gravity. All these motions are governed by gravitational force, and it is possible to describe them to various degrees of precision. Precise descriptions of complex systems must be made with large computers. However, we can describe an important class of orbits without the use of computers, and we shall find it instructive to study them. These orbits have the following characteristics: 1. A small mass m orbits a much larger mass M. This allows us to view the motion as if M were stationary—in fact, as if from an inertial frame of reference placed onM —without significant error. Mass m is the satellite of M , if the orbit is gravitationally bound. 2. The system is isolated from other masses. This allows us to neglect any small effects due to outside masses. The conditions are satisfied, to good approximation, by Earth’s satellites (including the Moon), by objects orbiting the Sun, and by the satellites of other planets. Historically, planets were studied first, and there is a classical set of three laws, called Kepler’s laws of planetary motion, that describe the orbits of all bodies satisfying the two previous conditions (not just planets in our solar system). These descriptive laws are named for the German astronomer Johannes Kepler (1571–1630), who devised them after careful study (over some 20 years) of a large amount of meticulously recorded observations of planetary motion done by Tycho Brahe (1546–1601). Such careful collection and detailed recording of methods and data are hallmarks of good science. Data constitute the evidence from which new interpretations and meanings can be constructed. Kepler’s Laws of Planetary Motion Kepler’s First Law The orbit of each planet about the Sun is an ellipse with the Sun at one focus. This OpenStax book is available for free at http://cnx.org/content/col11406/1.9 Chapter 6 | Uniform Circular Motion and Gravitation 223 Figure 6.29 (a) An ellipse is a closed curve such that the sum of the distances from a point on the curve to the two foci ( f1 and f 2 ) is a constant. You can draw an ellipse as shown by putting a pin at each focus, and then placing a string around a pencil and the pins and tracing a line on paper. A circle is a special case of an ellipse in which the two foci coincide (thus any point on the circle is the same distance from the center). (b) For any closed gravitational orbit, m follows an elliptical path with M at one focus. Kepler’s first law states this fact for planets orbiting the Sun. Kepler’s Second Law Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal times (see Figure 6.30). Kepler’s Third Law The ratio of the squares of the periods of any two planets about the Sun is equal to the ratio of the cubes of their average distances from the Sun. In equation form, this is T 12 r 13 (6.55) = , T 22 r23 where T is the period (time for one orbit) and r is the average radius. This equation is valid only for comparing two small masses orbiting the same large one. Most importantly, this is a descriptive equation only, giving no information as to the cause of the equality. 224 Chapter 6 | Uniform Circular Motion and Gravitation Figure 6.30 The shaded regions have equal areas. It takes equal times for m to go from A to B, from C to D, and from E to F. The mass m moves fastest when it is closest to M. Kepler’s second law was originally devised for planets orbiting the Sun, but it has broader validity. Note again that while, for historical reasons, Kepler’s laws are stated for planets orbiting the Sun, they are actually valid for all bodies satisfying the two previously stated conditions. Example 6.7 Find the Time for One Orbit of an Earth Satellite 8 Given that the Moon orbits Earth each 27.3 d and that it is an average distance of 3.84×10 m from the center of Earth, calculate the period of an artificial satellite orbiting at an average altitude of 1500 km above Earth’s surface. Strategy The period, or time for one orbit, is related to the radius of the orbit by Kepler’s third law, given in mathematical form in T 12 r 13 =. Let us use the subscript 1 for the Moon and the subscript 2 for the satellite. We are asked to find T 2. The T 22 r 23 given information tells us that the orbital radius of the Moon is r 1 = 3.84×10 m , and that the period of the Moon is 8 T 1 = 27.3 d. The height of the artificial satellite above Earth’s surface is given, and so we must add the radius of Earth (6380 km) to get r 2 = (1500 + 6380) km = 7880 km. Now all quantities are known, and so T 2 can be found. Solution Kepler’s third law is T 12 r 13 (6.56) =. T 22 r 23 To solve for T 2 , we cross-multiply and take the square root, yielding ⎛r ⎞ 3 (6.57) T 22 = T 12 ⎝r 2 ⎠ 1 ⎛r ⎞ 3/2 (6.58) T 2 = T 1 ⎝r 2 ⎠. 1 Substituting known values yields ⎛ 7880 km ⎞ 3/2 (6.59) T 2 = 27.3 d× 24.0 h × d ⎝3.84×10 5 km ⎠ = 1.93 h. This OpenStax book is available for free at http://cnx.org/content/col11406/1.9 Chapter 6 | Uniform Circular Motion and Gravitation 225 Discussion This is a reasonable period for a satellite in a fairly low orbit. It is interesting that any satellite at this altitude will orbit in the same amount of time. This fact is related to the condition that the satellite’s mass is small compared with that of Earth. People immediately search for deeper meaning when broadly applicable laws, like Kepler’s, are discovered. It was Newton who took the next giant step when he proposed the law of universal gravitation. While Kepler was able to discover what was happening, Newton discovered that gravitational force was the cause. Derivation of Kepler’s Third Law for Circular Orbits We shall derive Kepler’s third law, starting with Newton’s laws of motion and his universal law of gravitation. The point is to demonstrate that the force of gravity is the cause for Kepler’s laws (although we will only derive the third one). Let us consider a circular orbit of a small mass m around a large mass M , satisfying the two conditions stated at the beginning of this section. Gravity supplies the centripetal force to mass m. Starting with Newton’s second law applied to circular motion, (6.60) F net = ma c = m vr. 2 The net external force on mass m is gravity, and so we substitute the force of gravity for F net : (6.61) G mM = m vr. 2 r2 The mass m cancels, yielding GM r =v. 2 (6.62) The fact that m cancels out is another aspect of the oft-noted fact that at a given location all masses fall with the same acceleration. Here we see that at a given orbital radius r , all masses orbit at the same speed. (This was implied by the result of the preceding worked example.) Now, to get at Kepler’s third law, we must get the period T into the equation. By definition, period T is the time for one complete orbit. Now the average speed v is the circumference divided by the period—that is, (6.63) v = 2πr. T Substituting this into the previous equation gives 4π 2 r 2. (6.64) GM r = T2 Solving for T 2 yields 2 (6.65) T 2 = 4π r 3. GM Using subscripts 1 and 2 to denote two different satellites, and taking the ratio of the last equation for satellite 1 to satellite 2 yields T 12 r 13 (6.66) =. T 22 r23 This is Kepler’s third law. Note that Kepler’s third law is valid only for comparing satellites of the same parent body, because only then does the mass of the parent body M cancel. 2 Now consider what we get if we solve T 2 = 4π r 3 for the ratio r 3 / T 2. We obtain a relationship that can be used to GM determine the mass M of a parent body from the orbits of its satellites: r 3 = G M. (6.67) T 2 4π 2 If r and T are known for a satellite, then the mass M of the parent can be calculated. This principle has been used extensively to find the masses of heavenly bodies that have satellites. Furthermore, the ratio r 3 / T 2 should be a constant for all satellites of the same parent body (because r / T 2 = GM / 4π 2 ). (See Table 6.2). 3 226 Chapter 6 | Uniform Circular Motion and Gravitation It is clear from Table 6.2 that the ratio ofr 3 / T 2 is constant, at least to the third digit, for all listed satellites of the Sun, and for those of Jupiter. Small variations in that ratio have two causes—uncertainties in the r and T data, and perturbations of the orbits due to other bodies. Interestingly, those perturbations can be—and have been—used to predict the location of new planets and moons. This is another verification of Newton’s universal law of gravitation. Making Connections Newton’s universal law of gravitation is modified by Einstein’s general theory of relativity, as we shall see in Particle Physics. Newton’s gravity is not seriously in error—it was and still is an extremely good approximation for most situations. Einstein’s modification is most noticeable in extremely large gravitational fields, such as near black holes. However, general relativity also explains such phenomena as small but long-known deviations of the orbit of the planet Mercury from classical predictions. The Case for Simplicity The development of the universal law of gravitation by Newton played a pivotal role in the history of ideas. While it is beyond the scope of this text to cover that history in any detail, we note some important points. The definition of planet set in 2006 by the International Astronomical Union (IAU) states that in the solar system, a planet is a celestial body that: 1. is in orbit around the Sun, 2. has sufficient mass to assume hydrostatic equilibrium and 3. has cleared the neighborhood around its orbit. A non-satellite body fulfilling only the first two of the above criteria is classified as “dwarf planet.” In 2006, Pluto was demoted to a ‘dwarf planet’ after scientists revised their definition of what constitutes a “true” planet. Table 6.2 Orbital Data and Kepler’s Third Law Parent Satellite Average orbital radius r(km) Period T(y) r3 / T2 (km3 / y2) Earth Moon 3.84×10 5 0.07481 1.01×10 19 Sun Mercury 5.79×10 7 0.2409 3.34×10 24 Venus 1.082×10 8 0.6150 3.35×10 24 Earth 1.496×10 8 1.000 3.35×10 24 Mars 2.279×10 8 1.881 3.35×10 24 Jupiter 7.783×10 8 11.86 3.35×10 24 Saturn 1.427×10 9 29.46 3.35×10 24 Neptune 4.497×10 9 164.8 3.35×10 24 Pluto 5.90×10 9 248.3 3.33×10 24 Jupiter Io 4.22×10 5 0.00485 (1.77 d) 3.19×10 21 Europa 6.71×10 5 0.00972 (3.55 d) 3.20×10 21 Ganymede 1.07×10 6 0.0196 (7.16 d) 3.19×10 21 Callisto 1.88×10 6 0.0457 (16.19 d) 3.20×10 21 The universal law of gravitation is a good example of a physical principle that is very broadly applicable. That single equation for the gravitational force describes all situations in which gravity acts. It gives a cause for a vast number of effects, such as the orbits of the planets and moons in the solar system. It epitomizes the underlying unity and simplicity of physics. Before the discoveries of Kepler, Copernicus, Galileo, Newton, and others, the solar system was thought to revolve around Earth as shown in Figure 6.31(a). This is called the Ptolemaic view, for the Greek philosopher who lived in the second century AD. This model is characterized by a list of facts for the motions of planets with no cause and effect explanation. There tended to be a different rule for each heavenly body and a general lack of simplicity. This OpenStax book is available for free at http://cnx.org/content/col11406/1.9 Chapter 6 | Uniform Circular Motion and Gravitation 227 Figure 6.31(b) represents the modern or Copernican model. In this model, a small set of rules and a single underlying force explain not only all motions in the solar system, but all other situations involving gravity. The breadth and simplicity of the laws of physics are compelling. As our knowledge of nature has grown, the basic simplicity of its laws has become ever more evident. Figure 6.31 (a) The Ptolemaic model of the universe has Earth at the center with the Moon, the planets, the Sun, and the stars revolving about it in complex superpositions of circular paths. This geocentric model, which can be made progressively more accurate by adding more circles, is purely descriptive, containing no hints as to what are the causes of these motions. (b) The Copernican model has the Sun at the center of the solar system. It is fully explained by a small number of laws of physics, including Newton’s universal law of gravitation. Glossary angular velocity: ω , the rate of change of the angle with which an object moves on a circular path arc length: Δs , the distance traveled by an object along a circular path banked curve: the curve in a road that is sloping in a manner that helps a vehicle negotiate the curve center of mass: the point where the entire mass of an object can be thought to be concentrated centrifugal force: a fictitious force that tends to throw an object off when the object is rotating in a non-inertial frame of reference centripetal acceleration: the acceleration of an object moving in a circle, directed toward the center centripetal force: any net force causing uniform circular motion Coriolis force: the fictitious force causing the apparent deflection of moving objects when viewed in a rotating frame of reference fictitious force: a force having no physical origin gravitational constant, G: a proportionality factor used in the equation for Newton’s universal law of gravitation; it is a universal constant—that is, it is thought to be the same everywhere in the universe ideal angle: the angle at which a car can turn safely on a steep curve, which is in proportion to the ideal speed ideal banking: the sloping of a curve in a road, where the angle of the slope allows the vehicle to negotiate the curve at a certain speed without the aid of friction between the tires and the road; the net external force on the vehicle equals the horizontal centripetal force in the absence of friction ideal speed: the maximum safe speed at which a vehicle can turn on a curve without the aid of friction between the tire and the road microgravity: an environment in which the apparent net acceleration of a body is small compared with that produced by Earth at its surface Newton’s universal law of gravitation: every particle in the universe attracts every other particle with a force along a line joining them; the force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them non-inertial frame of reference: an accelerated frame of reference 228 Chapter 6 | Uniform Circular Motion and Gravitation pit: a tiny indentation on the spiral track moulded into the top of the polycarbonate layer of CD radians: a unit of angle measurement radius of curvature: radius of a circular path rotation angle: the ratio of the arc length to the radius of curvature on a circular path: Δθ = Δs r ultracentrifuge: a centrifuge optimized for spinning a rotor at very high speeds uniform circular motion: the motion of an object in a circular path at constant speed Section Summary 6.1 Rotation Angle and Angular Velocity Uniform circular motion is motion in a circle at constant speed. The rotation angle Δθ is defined as the ratio of the arc length to the radius of curvature: Δθ = Δs r , where arc length Δs is distance traveled along a circular path and r is the radius of curvature of the circular path. The quantity Δθ is measured in units of radians (rad), for which 2π rad = 360º= 1 revolution. The conversion between radians and degrees is 1 rad = 57.3º. Angular velocity ω is the rate of change of an angle, ω = Δθ , Δt where a rotation Δθ takes place in a time Δt. The units of angular velocity are radians per second (rad/s). Linear velocity v and angular velocity ω are related by v = rω or ω = vr. 6.2 Centripetal Acceleration Centripetal acceleration a c is the acceleration experienced while in uniform circular motion. It always points toward the center of rotation. It is perpendicular to the linear velocity v and has the magnitude a c = vr ;