Simple Nature - Light and Matter

Determining G example 15The constant G is not easy to determine, and Newton went to hisgrave without knowing an accurate value for it. If we knew themass of the earth, then we could easily determine G from experimentswith terrestrial gravity, but the only way to determine themass of the earth accurately in units of kilograms is by finding Gand reasoning the other way around! (If you estimate the averagedensity of the earth, you can make at least a rough estimate ofG.) Figures g and h show how G was first measured by HenryCavendish in the nineteenth century.The rotating arm is releasedfrom rest, and the kinetic energy of the two moving balls is measuredwhen they pass position C. Conservation of energy gives−2 GMmr BA− 2 GMmr BD= −2 GMmr CA− 2 GMmr CD+ 2K ,where M is the mass of one of the large balls, m is the mass ofone of the small ones, and the factors of two, which will cancel,occur because every energy is mirrored on the opposite side ofthe apparatus. (As discussed on page 102, it turns out that we getthe right result by measuring all the distances from the center ofone sphere to the center of the other.) This can easily be solvedfor G. The best modern value of G, from later versions of thesame experiment, is 6.67 × 10 −11 J · m/kg 2 .Escape velocity example 16⊲ The Pioneer 10 space probe was launched in 1972, and continuedsending back signals for 30 years. In the year 2001, not longbefore contact with the probe was lost, it was about 1.2 × 10 13 mfrom the sun, and was moving almost directly away from the sunat a velocity of 1.21×10 4 m. The mass of the sun is 1.99×10 30 kg.Will Pioneer 10 escape permanently, or will it fall back into the solarsystem?⊲ We want to know whether there will be a point where the probewill turn around. If so, then it will have zero kinetic energy at theturnaround point:g / Cavendish’s original drawingof the apparatus for hisexperiment, discussed in example15. The room was sealedto exclude air currents, and themotion was observed throughtelescopes sticking through holesin the walls.h / A simplified drawing of theCavendish experiment, viewedfrom above. The rod with the twosmall masses on the ends hangsfrom a thin fiber, and is free torotate.K i + U i = U f12 mv 2 − GMmr i12 v 2 − GMr i= − GMmr f= − GMr f,where M is the mass of the sun, m is the (irrelevant) mass ofthe probe, and r f is the distance from the sun of the hypotheticalturnaround point. Plugging in numbers on the left, we get a positiveresult. There can therefore be no solution, since the right sideis negative. There won’t be any turnaround point, and Pioneer 10is never coming back.i / The Pioneer 10 space probe’strajectory from 1974 to 1992,with circles marking its positionat one-year intervals. After its1974 slingshot maneuver aroundJupiter, the probe’s motion wasdetermined almost exclusively bythe sun’s gravity.Section 2.3 Gravitational Phenomena 101