Съдържание

IAO 2002, Теория, Група α, английски — Задача 6

Оригинал на условието (изображение)Съобщи за грешка

Всички материали от IAO 2002

Theoretical round. Problems to solve — Group A · 22 октомври 2002 г.

Бележки от източника

The supplied key refers to a drawing for problem 2 but contains none. The requested polar-bear illustration for problem 4 is also absent.

Условие

An elevator operates between Earth’s surface and a geostationary satellite. Estimate (at least roughly), at what distance below the satellite will the effective gravity be equal to one hundredth of the usual value.

Решение

Finally necessary to find the distance $x$ where $g_{\mathrm{eff}}=0.01g_0$ .

g_{\mathrm{eff}}=GM/R^2-\omega^2R=g_0R_0^2/R^2-\omega^2R,

where $g_0$ and $R_0$ are values near the earth surface. But at first necessary to find the radius $R_S$ of the geostationary satellite's orbit; at this position $g_{\mathrm{eff}}=0$ :

GM/R_S^2=\omega^2R_S,

R_S^3=GM/\omega^2=g_0R_0^2/\omega^2=g_0(R_0T/2\pi)^2,

where $T$ is period of the Earth revolution.

R_S=\{g_0R_0^2/\omega^2\}^{1/3}\approx42\,200\ \mathrm{km}.

For small variations one may use formula

g_{\mathrm{eff}}(A+x)=g_{\mathrm{eff}}(A)+g_{\mathrm{eff}}'x.

So in measuring gravitation very near the satellite (the satellite is zero point and $g_{\mathrm{eff}}=0$ ), for the small variations of the distance $x$ directed from Earth,

g_{\mathrm{eff}}(x)=g_{\mathrm{eff}}'x=(-2GM/R_S^3-\omega^2)x.

So for $g_{\mathrm{eff}}=0.01g_0$ one can calculate

x=\frac{0.01g_0}{-2GM/R_S^3-\omega^2}=\frac{0.01g_0}{-2g_0R_0^2/R_S^3-(2\pi/T)^2}.

Taking into account that $R_S^3=g_0(R_0T/2\pi)^2$ ,

x=\frac{0.01g_0}{-3(2\pi/T)^2}\approx-6150\ \mathrm{km}.

The sign 'minus' means that these 6150 km are in a direction opposite to our axis direction: the position of the point before the satellite. Of course, the value 6150 km already is not a 'small variation', but it is enough to make desirable estimation: about 6000 km.

Оригинал в Архива: alphatheo.jpg · официални решения: commonsol.jpg · допълнителни данни: alphasol.jpg · допълнителни данни: alphatheodata.jpg