An Earth satellite used in the global positioning system moves in a circular orbit with period $11 \mathrm{~h} 58 \mathrm{~min}$. (a) Determine the radius of its orbit. (b) Determine its speed. (c) The satellite contains an oscillator producing the principal nonmilitary GPS signal. Its frequency is $1575.42 \mathrm{MHz}$ in the reference frame of the satellite. When it is received on the Earth's surface, what is the fractional change in this frequency due to time dilation, as described by special relativity? (d) The gravitational blue shift of the frequency according to general relativity is a separate effect. The magnitude of that fractional change is given by

$$\frac{\Delta f}{f}=\frac{\Delta U_g}{m c^2}$$

where $\Delta U_{\mathrm{g}}$ is the change in gravitational potential energy of an object-Earth system when the object of mass $m$ is moved between the two points at which the signal is observed. Calculate this fractional change in frequency. (e) What is the overall fractional change in frequency? Superposed on both of these relativistic effects is a Doppler shift that is generally much larger. It can be a red shift or a blue shift, depending on the motion of a particular satellite relative to a GPS receiver

Put each fractional expression into standard form by rationalizing the denominator.

(a) $\frac{1}{\sqrt[3]{x}} \quad$ (b) $\frac{1}{\sqrt[6]{x^5}}\quad$ (c) $\frac{1}{\sqrt[7]{x^3}}$

Each circle below represents a whole, or 1. Determine the unknown fractional part of each circle.

Put each fractional expression into standard form by rationalizing the denominator.

(a) $\frac{1}{\sqrt[3]{x^2}}\qquad$ (b) $\frac{1}{\sqrt[4]{x^3}}\qquad$ (c) $\frac{1}{\sqrt[3]{x^4}}$

A steel wire with radius $r_{\mathrm {steel }}$ has a fractional increase in length of $\left(\Delta l / l_{0}\right)_{\mathrm {steel }}$ when the tension in the wire is increased from zero $T_{\mathrm {steel }}$. An aluminum wire has radius $r_{\mathrm{al}}$ that is twice the radius of the steel wire: $r_{\mathrm{al}}=2 r_{\mathrm{steel}}$. In terms of $T_{\mathrm {steel }}$, what tension in the aluminum wire produces the same fractional change in length as in the steel wire?

The linear fractional transformation $T(z)=\frac{z-z_{1}}{z-z_{3}} \frac{z_{2}-z_{3}}{z_{2}-z_{1}}$ maps the triple $z_{1}, z_{2}$, and $z_{3}$ to ______.

Put each fractional expression into standard form by rationalizing the denominator. $\text { (a) } \frac{1}{\sqrt{6}} \text { (b) } \sqrt{\frac{3}{2}} \text { (c) } \frac{9}{\sqrt[4]{2}}$

(a) Assuming a separation of 0.193 nm, compute the expected electric dipole moment of NaF. (b) The measured dipole moment is $27.2 \times 10^{-30} \mathrm{C} \cdot \mathrm{m}$. What is the fractional ionic character of NaF?

Construct a linear fractional transformation that maps the given triple $z_{1}, z_{2}$, and $z_{3}$ to the triple $w_{1}, w_{2}$, and $w_{3}$. $-1,0,1$ to $i, \infty, 0$

Construct a linear fractional transformation that maps the given triple $z_{1}, z_{2}$, and $z_{3}$ to the triple $w_{1}, w_{2}$, and $w_{3}$. $0,1, \infty$ to $0, i, 2$

The hydrogen sulfide in a $80.0-\mathrm{g}$ sample of crude petroleum was removed by distillation and uncollected in a solution of $\mathrm{CdCl}_2$. The precipitated $\mathrm{CdS}$ was then filtered, washed, and ignited to $\mathrm{CdSO}_4$. Calculate the percentage of $\mathrm{H}_2 \mathrm{~S}$ in the sample if $0.125 \mathrm{~g}$ of $\mathrm{CdSO}_4$ was recovered.

Simplify. Express all answers in terms of positive exponents. Rationalize the denominator where necessary to avoid fractional exponents in the denominator. $\frac{3^{0}}{\left(3^{-3} x^{1 / 3} y^{-3}\right)^{2}}$

Construct a linear fractional transformation that maps the given triple $z_{1}, z_{2}$, and $z_{3}$ to the triple $w_{1}, w_{2}$, and $w_{3}$. $0,1, \infty$ to $1+i, 0,1-i$

Construct a linear fractional transformation that maps the given triple $z_{1}, z_{2}$, and $z_{3}$ to the triple $w_{1}, w_{2}$, and $w_{3}$. $i, 0,-i$ to $0,1, \infty$