# Traveling on a Rotating Sphere

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1 Traveling on a Rotating Sphere Table of Contents Page Click the titles below to jump through the lesson 2 Spin-offs of a Rotating Sphere 3 What Do You Know? 3 Heated Fluid Circulation 4 Where Do The Trade Winds Blow? 5 Traveling on a Rotating Sphere 7 Sizing-up Inertial Circles 8 Take a Spin With The Coriolis Model 1

5 Traveling On A Rotating Sphere How Did Coriolis Use Math To Understand The Movement Of Objects on a Rotating Sphere? Gaspard-Gustave de Coriolis ( ), pictured on the right, a French mathematician, mechanical engineer, and scientist, worked out the general formulas for motion of objects measured from rotating systems of coordinates. Coriolis was able to determine the following simple rules for the direction of moving objects on the surface of a rotating sphere, now known as the Coriolis effect: The apparent force (or Coriolis force) on moving objects on a rotating sphere is perpendicular to the velocity of the object and the rotation axis. A balance of forces cause objects traveling in the Northern Hemisphere to curve to the right. A balance of forces cause objects traveling in the Southern Hemisphere to curve to the left. 5. Consider the speed at which Earth rotates at different locations. Click to see an animation of the Rotating Earth during the course of one day. Locate the following sites: a marked site on the Equator and London, England. Which site travels the greatest distance during one revolution (24 hours)? Which site has the greatest speed? 6. Imagine an object launched at a high-speed southward from London towards the equator. The object will have two components or parts to its velocity: a high-speed southward (meridional) launch velocity component and an eastward (zonal) velocity component due to the rotation of Earth. Comparing London with a location on the equator, which location has the higher eastward velocity component due to the Earth s rotation? As the object travels southward from London, it will pass over regions of Earth that are moving faster eastward due to the rotation of Earth and the object will appear to fall behind the rotating surface below. This means that the object will appear to curve westward as shown in the figure (curving to the right as viewed by someone facing the direction of the object s motion). 7. Imagine the same object launched northward from the Equator. As it travels northward, it will pass over a surface that moves slower in an eastward direction. Will the object appear to follow a straight line? Curve to the left? Or curve to the right? You have studied the motion of an object launched in the Northern Hemisphere and have seen that the object appears to curve to the right as measured by observers rotating with Earth. You will create additional examples of motion in the Northern Hemisphere in the following investigations and will learn that each one follows the same rule: In the Northern Hemisphere, moving objects appear to curve to the right on the rotating Earth. 5

6 The case of objects launched directly eastward or westward requires special attention. Examine the following figures to verify that in both cases the object s trajectory follows the same rule. North Pole View/Eastward Launch North Pole View/Westward Launch Objects launched eastward from London will appear to curve outward as you rotate eastward with Earth. Objects launched westward from London will appear to curve downward as you rotate eastward with Earth. Westward View/Eastward Launch Westward View/Westward Launch In this view, the launched object (in the small red box) is moving eastward (towards you, out of the paper). It appears to curve outward (red arrow) as you rotate with the Earth after the launch. The object s apparent velocity in the plane of the paper (red arrow) has two components: an upward (blue) component and a southward (green) component. This southward component will cause the eastwardmoving object s path bend southward. It follows the curve right rule. In this view, the launched object (in the small red box) is moving westward (away from you, into the paper). It appears to curve towards the Earth s axis (red arrow) as you rotate with the Earth after the launch. The object s apparent velocity in the plane of the paper (red arrow) has two components: a downward (blue) component and a northward (green) component. This northward component will cause the westward-moving object s path bend northward. It follows the curve right rule. 8. Imagine the same object traveling southward from the Equator. As it moves southward, it will pass over a surface that moves slower in an eastward direction. Will an object appear to follow a straight line? Curve to the left? Or curve to the right? 6

7 The general rules for moving objects on a rotating sphere are: In the Northern Hemisphere, a balance of forces on the rotating Earth influence moving objects to curve right. A balance of forces cause objects traveling in the Southern Hemisphere to curve to the left. Sizing-up Inertial Circles (Oscillations) The equations governing the magnitude (strength) of the Coriolis force, F C, are: F C = m f v f 2 ω sin( φ) Where m is the mass of the object, v is the horizontal component of its velocity, ω is the rotational speed of the Earth (7.27x10-5 radians/second), f is the frequency of the motion and φ is the latitude of the moving object. Under the influence of the Coriolis force, objects follow curved, near-circular paths, called inertial circles. Their motions repeating patterns or inertial oscillations that repeat with a frequency, f, during a time period T=1/f. The radius of the Coriolis inertial circle is: R = v f The Coriolis force is evident in swirling vortex weather patterns (like hurricanes), leading to a counterclockwise rotation in the Northern Hemisphere and a clockwise rotation on the Southern Hemisphere. An example, right, is the beautifully formed low-pressure system swirling off the southwestern coast of Iceland. Because this low-pressure system occurred in the Northern Hemisphere, the winds spun in toward the center of the low-pressure system in a counterclockwise direction. The Aqua MODIS instrument took the image on September 4,

8 9. Use the formulas in the table below, to estimate the frequency of the inertial oscillations and the size of the inertial circles at several latitudes. Do this for both air currents (typical wind speed 10 m/s) and ocean currents (typical ocean current 0.2 m/s). Once you have found the frequency for the latitude, use that number (f in the equation) to find the radious of the inertial circle. Compute the missing values in the following table. Put your answers in the empty boxes below. Latitude (φ, degrees) Frequency, f f=2(7.27x10-5)sin(φ) Current Fluid 20 Air 10 Speed (m/s) Inertial Circle Radius R=v/f (m) 20 Water.2 50 Air Water.2 80 Air Water.2 Take A Spin With The Coriolis Model To help you better understand the Coriolis forces on a rotating sphere, a Coriolis Model has been provided to simulate the motion of an object sliding without friction on a sphere with the same size and rotational speed as Earth. The object is allowed to slide freely for 7 days and you are allowed to set the object s starting velocity (speed and direction) and position. Click on the hypertext word Coriolis Model to open a new window containing the model. Complete the following four trials to determine if the trajectory follows the two Coriolis Rules illustrated below. For each trial: First, Select the object s starting direction and starting speed from the drop down menus. Second, click on the map at a site in the hemisphere indicated in the table below. A pop-up window will appear showing the trajectory of the object tracked over a 1-week time period. 8

9 10. Next, use the following table to make your model settings. Click the map over the ocean at the correct starting latitude and decide if the trajectory follows our curvature rules for the Northern and Southern Hemispheres. Put your answers in the table below. Coriolis Model Initial Settings Your Analysis of the Trajectory Trial Speed (m/sec) Direction 1 50 North 2 50 North 3 50 East 4 50 West Latitude +30 o Northern Hemisphere -30 o Southern Hemisphere +30 o Northern Hemisphere -30 o Southern Hemisphere Trajectory Follows Rules Direction Trajectory Curves Trial 1 Trial 2 Trial 3 Trial The Equator is the dividing line for the two rules that apply to moving objects. What might happen if an object is launched in either hemisphere but crosses over the equator during its trajectory? 12. To test your understanding, make a prediction of what will happen to an object when it is launched in the manner specified in each row of the following table. Check your predictions using the Coriolis Accelerated Motion visualizer, and then put your answers in the table below. Trial Speed (m/sec) Direction Location 5 50 South 15 o North Predicted Trajectory if object crosses Equator Does your prediction agree or disagree with visualizer? 9

10 Trial Predicted Trajectory Does your Speed Direction Location if object crosses Equator prediction agree (m/sec) or disagree with visualizer? 6 50 North 15 o South 7 50 East 15 o North 8 50 West 15 o South Trial 5 Trial 6 Trial 7 Trial 8 As discussed previously in this lesson, the trade winds are driven by heated, light air at the Equator rising up and drawing in cooler surface air slightly north and south of the Equator. It should be clear from the trajectories in trial 5 and trial 6 that air rushing towards the Equator will curve towards the west whether the air comes from the north or south. This creates a pattern of easterly winds (winds blowing from the east) at the Equator. Note that the air masses from the north and south will collide at the Equator and that interaction will strengthen the equatorial wind pattern. The computer model you have been using models a sliding object freely moving over a smooth Earth-sized sphere with nothing blocking its path as it slides above or below the Equator. This is not the case for the air in the atmosphere. The air rushing to the Equator will be driven further in the westward direction by the converging air masses and will not significantly cross the Equator. Tinka Sloss, NewMedia Studio, Inc. 10

11 13. Next, use the Coriolis Model to do a study of how the Coriolis force varies with latitude. You will use the computer model to launch objects at various latitudes. By observing the curvature of the trajectory, you can estimate the relative strength of the deflecting force: strongly deflecting forces result in small, tight circular paths, weakly deflecting forces result on large circular paths. As we have seen, these near-circular paths are called inertial circles and the frequency of rotation, f, and the speed, v, determine the radius of the circle: speed Radius = = frequency v f On a rotating planet, the frequency increases with latitude, φ: f = 2 (7.292x10 5 ) sin( φ) These equations predict that, for constant speed, the radius of a moving object trajectory should decrease with latitude. 14. Fill in the following table and draw a conclusion about how the Coriolis force varies with latitude. Indicate your estimates of radius and force, use the subjective relative scale: small/weak, medium, or large/strong. Note: The speed of the object is kept constant. In this case, the radius and acceleration are inversely proportional (large R gives a small A; small R gives a large A): Position Speed Direction (m/sec) 85 N 50 East 45 N 50 East 15 N 50 East 15 S 50 East 45 S 50 East 85 S 50 East Radius of Curvature of Trajectory (small, medium, large) Coriolis Force (small, medium, large) 15. What do you conclude about the effect of latitude on the Coriolis acceleration? Additional Investigations for the Coriolis Force and Travel How does the Coriolis force impact travel on a rotating sphere? Do airline pilots need to consider the Coriolis force when making their flight plan? Do ship captions need to consider the Coriolis force when charting their course? 11

12 Matrix for Grading Lesson 2 4 Expert 3 Proficient 2 Emergent 1 Novice Responses show an in-depth understanding of models and explorations used to explain scientific concepts and processes used in the lesson. Proficient manipulation of computer models. Data collection and analysis of data are complete and accurate. Predictions and follow through with accuracy of predictions are explained and fully supported with relevant data and examples. Responses show a solid understanding of models and explorations used to explain scientific concepts and processes used in the lesson. Mostly proficient manipulation of computer models. Data collection and analysis of data are mostly complete and accurate. Predictions and follow through with accuracy of predictions are explained and mostly supported with relevant data and examples. Responses show a partial understanding of models and explorations used to explain scientific concepts and processes used in the lesson. Some proficiency in manipulation of computer models. Data collection and analysis of data are partially complete and sometimes accurate. Predictions and follow through with accuracy of predictions are sometimes explained and supported with relevant data and examples. Responses show a very limited understanding of models used to explain scientific concepts and processes used in the lesson. Little or no ability shown to manipulate computer models. Data collection and analysis of data are partially complete and sometimes accurate. Predictions and follow through with accuracy of predictions are not well explained and are not supported with relevant data and examples. 12

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