The Role of Gravity in Orbital Motion

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Marylou Peters
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1 ! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State Univ.) Oveview Gavity is the natual phenomenon by which all objects in the univese ae attacted to one anothe. Gavity allows stas to fom fom clouds of hydogen gas, planets to fom fom molecules of cosmic dust, and is esponsible fo the obits of all celestial bodies. But, what measuable quantities affect the stength of gavity? In this module, students exploe how gavity affects celestial bodies and thei obits. Science Standads (NGSS) MS-ESS1-2 Develop and use a model to descibe the ole of gavity in the motions within galaxies and the sola system. MS-PS2-2 Plan an investigation to povide evidence that the change in an object s motion depends on the sum of the foces on the object and the mass of the object. MS-PS2-4 Constuct and pesent aguments using evidence to suppot the claim that gavitational inteactions ae attactive and depend on the masses of inteacting objects. MS-PS2-5 Conduct an investigation and evaluate the expeimental design to povide evidence that fields exist between objects exeting foces on each othe even though the objects ae not in contact. Focus Question How does the mass of objects and thei distance fom each othe affect the stength of gavitational attaction? Objectives Though this lesson, students will: Constuct and test a hypothesis as a team Detemine the dependence of mass and sepaation on gavitational stength Lean how these same popeties affect escape velocity Undestand gavitational attaction as a field (distoted spacetime)

2 Backgound Gavity acts as an attactive foce that opeates on all objects with mass. The stength of the gavitational attaction is dependent on only two vaiables: the mass of the objects and thei distance of sepaation. As gavity is an attactive foce between all massive objects, the gavitational field pemeates all of space, affecting objects both on and off Eath. This is how Newton came to undestand that gavity was as esponsible fo the apple falling fom the tee as the Moon obiting the Eath. Mateials Fishing line (3 lb. test) 2 wooden batons to hold fishing line (optional: +2 caabines) Mete stick Masking tape Pepaation Tajectoies: With the masking tape, mak an aea fo the Sun as shown in the diagam below. Then make thee small additional makings following the ed line at one half, one, and two metes away fom the Sun. Gavity (batons): Dill holes in the batons fo the fishing line to pass though. Anothe option is to have a caabine attached to the baton this makes it easie to switch out the fishing line duing the activity. Gavity (sting): Cut 7x(2-mete) pieces, 1x(4-mete) piece, and 1x(8-mete) piece of fishing line. Tie each end of fishing line into a loop big enough to fit though the baton/ caabine. It helps to cut the fishing line a little longe than the distance equied to account fo making the loops. Make one copy of the woksheet fo each student and distibute.

3 Pocedue Wam up: Example of Newton s Fist Law of Motion Using the baton attached to a sting, twil the baton aound in a cicle. Ask the students what would happen if you let go of the sting. Ask the students what keeps the baton cicling aound you hand (A: the tension in the sting poduces a foce). Now imagine that the sting was invisible and that this is the same concept as gavity. This is a demonstation of Newton s Fist Law of Motion: An object at est stays at est and an object in motion stays in motion with the same speed and in the same diection unless acted upon by an unbalanced foce. Review the activity with the students and have them make pedictions based on thei intuition and fill in the fist table. Activity: Gavity and Obits In this activity, students will assume the oles of the Sun, gavity, and a neaby planet. Split the class up into goups of 3-5 students. Choose one student to be the Sun, gavity, and the planet. In the case of 4-5 student goups, additional students can act as exta mass fo the Sun in the fist few tials. The student epesenting the Sun should stand on the X. The student epesenting the planet should begin some distance away as indicated by the aows. The student epesenting gavity should stand along the ed line on the opposite side of the planet s tajectoy, facing the Sun. Both the Sun and Gavity will hold one baton, both ends attached to the length of sting. The Planet will pass along seveal tajectoies acoss the path of the Sun at a distance of (one-half, one, and two metes). At the point of closest appoach (eaching the ed line) the planet will expeience the foce of gavity fom the Sun epesented by the fishing line. When the Planet eaches this point they will gab a baton fom Gavity and continue on thei path. Once unde the influence of gavity, the Planet will expeience one of thee possible outcomes: 1. The sting holds. The foce of gavity is stong enough to captue the Planet in obit aound the Sun. 2. The sting beaks. The foce of gavity is not stong enough to captue the Planet, but the staight-line tajectoy is changed. 3. The sting beaks. The foce of gavity is too weak to captue Planet o change it s tajectoy. Assessment Constuct Newton s Law of Univesal Gavitation and discuss the affects that a change in mass and a change in sepaation distance has on the stength of the field. Calculate the stength of the gavitational foce of the othe planets in the Sola System elative to the Eath (mass: Eath mass, distance: AU).

4 The Role of Gavity in Obital Motion Intoduction: Today you will investigate the ole gavity plays in obital motion, like the Moon and Eath, o Eath and the Sun. The foce of gavity acts between these celestial bodies and changes thei motions, depicting Newton s Fist Law of Motion. What is Newton s Fist Law of Motion? Pediction: Tial Sun Mass Distance # of sting Sting beak? Path changed? m m m m m 1 (4 m fold x4) m 1 (8 m fold x16) Expeiment: Planet speed Tial Sun Mass Distance # of sting Sting beak? Path changed? m m m m m 1 (4 m fold x4) m 1 (8 m fold x16)

5 Follow-up Questions Compae you initial pedictions with the esults of you expeiment. Does the dependence on mass and sepaation distance agee with you pedictions? Is gavity diectly o indiectly popotional to the mass of an object? How do the esults of you expeiment suppot this? Knowing how mass elates to the stength of gavity, what would you expect to find if you incease the mass of the planet instead of the Sun? Would gavity s dependence on mass change? Is gavity diectly o indiectly popotional to the distance of sepaation between two objects? How do the esults of you expeiment suppot this? Accoding to the esults of you expeiment, can you pedict the dependance of gavity on Mass M and sepaation distance? (Hint: Fo, look closely at # of stings) F gav /

6 Exta Cedit Add you own exta tials to you Results chat and test what happens with the dependence on the velocity of the Planet. Keep the mass of the Sun and the distance of the Planet appoach constant and vay only the speed at which the Planet follows the initial tajectoy. Descibe you pedictions hee: How does changing the planet s velocity affect the esults? What does this tell you about the ole kinetic enegy plays in detemining the obit?

7 Extensions Fo an object like a planet to escape the gavitational attaction of anothe object, say the Sun, the planet must be moving with enough speed o the attactive foce will be too geat and the planet will be captue into obit. The speed needed to escape is called the escape velocity. If the planet gets too close and is not moving fast enough, it will be captue into obit aound the moe massive Sun. If the planet is moving fast enough then it can escape! We can deive the escape velocity fom consevation of enegy. Fo a bound system (think objects in obit), the kinetic enegy must always be less than the potential enegy. We define kinetic and potential enegy as the following: Consevation of enegy states that: KE = 1 2 mv2 PE = GMm (KE + PE) initial =(KE + PE) final If an object escaped and taveled in infinitely lage distance away, then the final kinetic and potential enegy would be zeo. In that case the above equation becomes: (KE + PE) initial =0 1 2 mv2 = GMm 2GM v escape = Hee we see that given any mass of a lage object M at a given distance, we can calculate the speed needed by the smalle object to beak out of obit. Supplemental escape velocity: Vaiable speed: The students can discove thei escape speed fom the system. Using the 1 mete distance tajectoy, have the students eun the expeiment using diffeent Planet speeds and detemine how fast they must move in ode to beak the sting. The students should find that at slowe speeds, the sting does not beak and they ae captued into obit, but as they incease the speed, eventually the sting will beak and they have eached the escape velocity fo the system.

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses, 3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects