/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 58 A satellite \(A\) of mass \(m\) ... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 58

A satellite \(A\) of mass \(m\) is at a distance of \(r\) from the centre of the earth. Another satellite \(B\) of mass \(2 m\) is at distance of \(2 r\) from the earth's centre. Their time periods are in the ratio of (a) \(1: 2\) (b) \(1: 16\) (c) \(1: 32\) (d) \(1: 2 \sqrt{2}\)

Short Answer

Expert verified
The ratio of the periods of the two satellites is \(1: 2\sqrt{2}\).

Step by step solution

01

Recall Kepler鈥檚 third law

The third Kepler鈥檚 law states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. This law can be represented mathematically as \(T^2 \propto r^3\) or \(T = k r^{3/2}\), where \(T\) is the period, \(r\) is the distance, and \(k\) is the constant of proportionality.
02

Use the law to calculate the period of satellite A

Let's use this law for satellite A. \(T_A = k r^{3/2}\), where \(T_A\) is the period of satellite A, and \(r\) is its distance from the center of the earth. As the proportionality factor \(k\) remains the same for all objects orbiting the same central body, it can be neglected in this problem because we are asked to find the ratio of two time periods.
03

Use the law to calculate the period of satellite B

Now, similarly for satellite B. \(T_B = k (2r)^{3/2}\), where \(T_B\) is the period of satellite B, and \(2r\) is its distance from the center of the earth. Again, the proportionality constant \(k\) can be neglected.
04

Calculate the ratio of their periods

Now we have to find the ratio \(T_A / T_B = (k r^{3/2}) / (k (2r)^{3/2})\). The constant \(k\) in both the numerator and the denominator cancels out, and we get a simpler equation that we can solve.
05

Solve for the ratio

After simplifying, we find that \(T_A / T_B = 1/2\sqrt{2}\), which simplifies to \(1: 2\sqrt{2}\).

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