Randall D. Knight — Newton's Theory of Gravity: Problem 56

Figure 13.17 showed a graph of \(\log T\) versus \(\log r\) for the planetary data given in Table 13.2. Such a graph is called a log-log graph. The scales in Figure 13.17 are logarithmic, not linear, meaning that each division along the axis corresponds to a factor of 10 increase in the value. Strictly speaking, the “correct” labels on the y-axis should be 7, 8, 9, and 10 because these are the logarithms of \(10^7, \dots, 10^{10}\). a. Consider two quantities \(u\) and \(v\) that are related by the expression \(v^p = Cu^q\), where \(C\) is a constant. The exponents \(p\) and \(q\) are not necessarily integers. Define \(x = \log u\) and \(y = \log v\). Find an expression for \(y\) in terms of \(x\). b. What shape will a graph of \(y\) versus \(x\) have? Explain. c. What slope will a graph of \(y\) versus \(x\) have? Explain. d. Use the experimentally determined “best-fit” line in Figure 13.17 to find the mass of the sun.