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Suppose three students performed the Mini Lab from Section 11 several times, starting with springs of the same length With two washers on the spring, student 1 made repeated measurements, which ranged from 144 cm to 148 cm The average of student 1 s measurements was 146 cm, as shown in Figure 1-10 This result was reported as (146 02) cm Student 2 reported finding the spring s length to be (148 03) cm Student 3 reported a length of (140 01) cm Could you conclude that the three measurements are in agreement Is student 1 s result reproducible The results of students 1 and 2 overlap; that is, they have the lengths 145 cm to 148 cm in common However, there is no overlap and, therefore, no agreement, between their results and the result of student 3

Figure 1-10 Three students took multiple measurements Are the measurements in agreement Is student 1 s result reproducible

1 F 2

FIGURE 1-1

1 r2

Figure 7-5 A planet with mass mp and orbital radius r orbits the Sun with mass mS (Illustration not to scale)

Newton stated his law of universal gravitation in terms that applied to the motion of planets about the Sun This agreed with Kepler s third law and confirmed that Newton s law fit the best observations of the day Consider a planet orbiting the Sun, as shown in Figure 7-5 Newton s second law of motion, Fnet ma, can be written as Fnet mpac, where F is the gravitational force, mp is the mass of the planet, and ac is the centripetal acceleration of the planet For simplicity, assume circular orbits Recall from your study of uniform circular motion in 6 that, for a circular orbit, ac 4 2r/T2 This means that Fnet mpac may now be written as 2r/T2 In this equation, T is the time required for the planet to Fnet mp4 make one complete revolution about the Sun If you set the right side of this equation equal to the right side of the law of universal gravitation, you arrive at the following result: G

2

The period of a planet orbiting the Sun can be expressed as follows Period of a Planet Orbiting the Sun T 2

double-click the file If you are running Windows Vista or Windows 7, and a User Account Control box such as that displayed in Figure 1-2, click Yes to continue If you have already installed Zend Server CE and are now installing a newer version, another confirmation box will appear from the installer itself (see Figure 1-3) In this case, click Yes to continue

The period of a planet orbiting the Sun is equal to 2 times the square root of the orbital radius cubed, divided by the product of the universal gravitational constant and the mass of the Sun

Squaring both sides makes it apparent that this equation is Kepler s third law of planetary motion: the square of the period is proportional to the cube of the distance that separates the masses The factor 4 2/GmS depends on the mass of the Sun and the universal gravitational constant Newton found that this derivation applied to elliptical orbits as well

Astronomers have detected three planets that orbit the star Upsilon Andromedae Planet B has an average orbital radius of 0059 AU and a period of 46170 days Planet C has an average orbital radius of 0829 AU and a period of 2415 days Planet D has an average orbital radius of 253 AU and a period of 1284 days (Distances are given in astronomical units (AU) Earth s average distance from the Sun The distance from Earth to the Sun is 100 AU) 1 Do these planets obey Kepler s third law 2 Find the mass of the star Upsilon Andromedae in units of the