11 Gravity and the Solar System Name Worksheet AP Physics 1

Transcription

1 11 Gravity and the Solar System Name Worksheet AP Physics 1 1. Use Newton s irst Law of Motion to describe how a planet would move if the inward force of ravity from the sun were to suddenly disappear. answer in complete sentences The equation for the law of universal ravitation is: m d 1 where is the force of attraction between masses m 1 and m separated by distance d, and G is Nm /k. show all work with proper equations, units, and sinificant fiures. Calculate the weiht of a 50.0 k mass on the moon usin Newton's Law of Universal Gravitation. (The mass of the moon is approximately k; the radius is m.). Now calculate the weiht of the 50.0 k mass on the moon usin the special case of Newton's Second Law you derived in Unit 6 ( =m). The acceleration due to ravity on the moon is about 1/6 that of earth. express your answer with sinificant fiures Your answers to questions and should be very similar, as you simply used different techniques to calculate the same quantity. Unit 11: Gravity and the Solar System 016 by G. Meador Pae 1 of 6

2 MASS AND DISTANC CHANGS AND THIR GRAVITATIONAL CTS Chanin the mass of an object or its distance from another object will affect the ravitational force that attracts the two objects. By substitutin chanes in any of the variables into the equation for the Law of Universal Gravitation, we can predict how the others chane. Suppose the distance of separation is reduced to one-third of its former value. Then substitutin 1/ d for d in the equation ives: new new And we see the force is increased nine-fold. m m 9 m ( ) 1 1 ( d) d 1 d 9 new G( mm ) m 1 1 ( ) d d m m 1 m ( ) ( d) 4d 4 d old 9 Suppose that the distance did not chane, but one of the masses somehow is doubled. Then substitutin m 1 for m 1 in the equation ives: So we see the force doubles. inally suppose that the distance of separation is doubled. Then substitutin d for d in the equation ives: And we see the force is only 1/4 as much. Use this method to solve the followin problems. Write the equation and make the appropriate substitutions. 4. If both masses are tripled, what happens to the force? 1 4 old old 5. If the masses are not chaned, but the distance of separation is reduced to / the oriinal distance, what happens to the force? 6. If the masses are not chaned, but the distance of separation is tripled, what happens to the force? 7. If both masses are doubled, and the distance of separation is tripled, show what happens to the force. 8. If one of the masses is doubled, the other remains unchaned, and the distance of separation is quadrupled, show what happens to the force. Unit 11: Gravity and the Solar System 016 by G. Meador Pae of 6

3 CALCULATING TH WIGHT O AN OBJCT IN ORBIT One can use the Law of Universal Gravitation to calculate the weiht of an orbitin object. The masses used in the calculation are those of the orbitin object and the planet. The distance used is the distance from the center of the circlin object to the center of the planet. This means that you must add the radius of the planet to the altitude of the orbitin body. It is also important that you use meters for the distance, not kilometers or any other such unit. XAMPL: A 15.0 k object orbits the planet Mars at an altitude of 00 km. What is the weiht of the object if Mars has a radius of,40 km and a mass of k? MTHOD: irst, chane all distances into meters. One kilometer is equal to 1,000 meters, so: 1000, m 1000, m the altitude is 00 km 00, 000 m and the radius is 40, km, 40, 000 m 1 km 1 km Now you must add the altitude to the radius of the planet to et the total distance from the center of the planet to the center of the orbitin object:, 40, 000 m 00, 000 m, 60, 000 m inally, plu in all of the various values into the Law of Universal Gravitation and solve: 11 Nm ( )( k)( 15k) 14 m 1 d k Nm 481. N 1 (, 60000, m) m Show all equations and units on the followin problems, and express answers with sinificant fiures. 9. At closest approach, the 7 k Voyaer probe flew by Neptune at an altitude of 9,40 km. A) What was the probe s weiht at that moment if Neptune has a radius of 4,900 km and a mass of k? B) Use the information from part A to calculate the acceleration due to ravity at that altitude above Neptune. Unit 11: Gravity and the Solar System 016 by G. Meador Pae of 6

4 10. The center of the moon and the center of the earth are km apart. The mass of the moon is approximately k, while earth's mass is about k. A) Calculate the earth's pull on the moon. B) What is the size of the moon's pull on the earth? explain or show work GRAVITATIONAL ILD STRNGTH We can adapt Newton s Law of Universal Gravitation to predict the ravitational field strenth (also called the acceleration due to ravity). or arth: m object r m object object m r and we find that 11 Nm 4 ( )( k) k m m ( m) s s 11. The space station typically orbits 400 km above the earth's surface. The earth has a mass of k and a radius of 6,80 km. A) How much would a 000 k caro pod for the space station weih when it has been lifted to that orbit? B) Use your result from part A (or alternatively you can use the above derived equation for ) to determine the acceleration due to ravity at that altitude. C) Use your knowlede of circular motion to determine the orbital linear speed of the caro pod. DNSITY CONCPTS Unit 11: Gravity and the Solar System 016 by G. Meador Pae 4 of 6

5 You also need to be comfortable with usin the concept of density, often symbolized with the lowercase Greek letter rho (ρ), as the ratio of mass and volume: m V Density can be expressed in k/m and can also be expressed as a fraction of the density of arth, ρ. Knowin that the volume of a sphere is iven by V 4 r, we can make various predictions. or example, if a newly discovered plant has a density / that of arth, or, and radius half that of arth, or, then we could predict its surface ravitational field in terms of that of arth s : 4 irst we need to express the planet s mass in terms of the arth s mass m. Since m=ρv then m ( r ) and thus the new m ( r ) ( r ) m m 4 1 planet s mass is. Next we calculate P 1 r 1 1 G( m ) P 1 1. rp r r 1 4 r 4 1 (. 98 m/s ) 7 m/s 1 ( ) Calculate the ravitational field strenth, to sinificant fiures, of a planet that is twice as dense as arth and which has a radius ½ that of the arth. 1. Calculate the density of a planet, in terms of the arth s density ρ, if it is the same size as arth but has a ravitational field strenth that is 1.0 times that of arth. Unit 11: Gravity and the Solar System 016 by G. Meador Pae 5 of 6

6 GRAVITATIONAL VS. INRTIAL MASS The ravitational mass of an object one uses in =m happens to be the same as the inertial mass in G=ma. These different types of mass are measured in different ways: the ravitational mass is a static measurement taken usin a scale or balance when the object is at rest or in equilibrium, while the inertial mass is a dynamic measurement taken when the object is bein accelerated. Classify each method of measurement by writin I for inertial mass or G for ravitational mass in the blank beside each description: 14. Usin a sprin scale to suspend the object and recordin the readin. 15. Attachin the object to a sprin of known sprin constant, allowin it to oscillate freely horizontally, and measurin the period. 16. Placin the object on one side of a double-pan balance and addin known masses to the other pan until it balances. 17. Attachin the object to a force sensor, pullin the object across a very smooth surface, and measurin acceleration. 18. Thus far in the course we have inored that on the arth s surface an object must have a slihtly unbalanced force to rotate with the arth, pretendin that the normal force precisely balances the object s weiht. A student with a mass of 65.0 k stands at the equator. The radius of the arth is m, and of course it rotates once per day. A) What is the manitude of the centripetal force (in newtons) required to keep the student on the arth s surface? (Don t do a free body diaram and force analysis yet; just use the centripetal force and other circular motion equations.) B) Since the centripetal force is unbalanced, what is the true manitude of the normal force on the student? (The student weihs 67 N, and with a free body diaram and Σ=ma you should find the normal force is actually less than 67 N.) Unit 11: Gravity and the Solar System 016 by G. Meador Pae 6 of 6