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Chapter 6: Problem 766

The period of a satellite in circular orbit around a planet is independent of (A) the mass of the planet (B) the radius of the planet (C) mass of the satellite (D) all the three parameters (A), (B) and (C)

Short Answer

Expert verified
The period of a satellite in circular orbit around a planet is independent of (C) mass of the satellite. This can be concluded by analyzing the equation \(T^2= \frac{4\pi^2r^3}{GM_{planet}}\), where T is the period, r is the orbit radius, G is the gravitational constant, and M_planet is the mass of the planet. The mass of the satellite does not appear in the equation, thus making it independent of the period.

Step by step solution

01

Write down the formula for gravitational force.

The gravitational force acting between two masses (m1 and m2) separated by a distance r can be expressed by Newton's law of gravitation \(F = G \frac{m_1m_2}{r^2}\), where G is the gravitational constant.
02

Write down the formula for centripetal force.

The centripetal force acting on a mass (m) in a circular orbit of radius r with a speed v can be expressed as \(F_c = m\frac{v^2}{r}\).
03

Relate gravitational force and centripetal force for a satellite in orbit.

In the case of a satellite in a circular orbit around a planet, the gravitational force provides the centripetal force necessary to keep the satellite in that orbit. So, we can set these two equal: \(G \frac{M_{planet}m_{satellite}}{r^2} = m_{satellite}\frac{v^2}{r}\), where M_planet and m_satellite are the masses of the planet and the satellite, respectively.
04

Solve for v in terms of r.

To find the dependency of the period on M_planet, m_satellite, and r, we need to solve for the speed of the satellite (v) in the orbit. From Step 3, we have: \( G \frac{M_{planet}}{r} = v^2 \). Now, taking the square root of both sides, we have: \( v = \sqrt{\frac{GM_{planet}}{r}} \).
05

Relate speed (v), the radius of the orbit (r), and the period (T).

In a circular orbit, the distance traveled by satellite in one full rotation is the circumference of the orbit, which is 2πr. The speed of the satellite is the distance traveled divided by the time, in this case the period (T) of the orbit, so: \( v = \frac{2\pi r}{T} \).
06

Solve for the period (T).

To find the relationship between period T and other parameters, we substitute the expression for v from step 4 into the equation in step 5: \( \sqrt{\frac{GM_{planet}}{r}} = \frac{2\pi r}{T} \). Now, solving for T: \( T = \frac{2\pi r}{\sqrt{\frac{GM_{planet}}{r}}} \). Now, simplify the expression: \( T^2= \frac{4\pi^2r^3}{GM_{planet}}\). Here, we can see that T^2 is dependent on the mass of the planet (M_planet) and the radius of the orbit (r) only.
07

Determine the independence of the parameters.

By analyzing the final equation from step 6, we can determine the parameters that have no influence on the period of the satellite: - The mass of the planet (M_planet) is part of the equation for T, so it is not independent. - The radius of the orbit (r) is also part of the equation, so it is not independent. - The mass of the satellite (m_satellite) does not appear in the final equation for T, so it is independent of the period. Therefore, the correct answer is (C) mass of the satellite.

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