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Chapter 5: Problem 37

A spacecraft is in orbit around Jupiter. The radius of the orbit is 3.0 times the radius of Jupiter (which is \(\left.R_{1}=71500 \mathrm{km}\right) .\) The gravitational field at the surface of Jupiter is 23 N/kg. What is the period of the spacecraft's orbit? [Hint: You don't need to look up any more data about Jupiter to solve the problem.]

Short Answer

Expert verified
According to the given information and calculations, the period of the spacecraft's orbit around Jupiter is approximately 30607 seconds.

Step by step solution

01

Determine the orbital radius

To find the orbital radius, we will multiply the given radius of Jupiter (71500 km) by 3.0 (since the spacecraft's orbit is 3 times the radius of Jupiter). $$ R_{orbit} = 3 * R_{Jupiter} = 3 * 71500 \ km $$
02

Convert the orbital radius to meters

In physics, it's important to have consistent units throughout the calculations. In this case, we will convert the orbital radius from kilometers to meters. $$ R_{orbit} = 3 * 71500 * 10^{3} \ m $$
03

Write the gravitational force equation for the spacecraft

The gravitational force acting on the spacecraft can be written as: $$ F_{gravitational} = m_{spacecraft} * g_{Jupiter} $$ Where \(m_{spacecraft}\) is the mass of the spacecraft and \(g_{Jupiter} = 23 \ N/kg\) (given).
04

Equate the gravitational force as the centripetal force

The centripetal force acting on the spacecraft is given by the formula: $$ F_{centripetal} = \frac{m_{spacecraft} * v^2}{R_{orbit}} $$ Where \(v\) is the orbital speed of the spacecraft. Since both forces are acting towards the center of the orbit, these are equal: $$ m_{spacecraft} * g_{Jupiter} = \frac{m_{spacecraft} * v^2}{R_{orbit}} $$
05

Eliminate the mass of the spacecraft from the equation

Notice that the mass of the spacecraft appears on both sides of the equation, so we can cancel it out: $$ g_{Jupiter} = \frac{v^2}{R_{orbit}} $$
06

Find the orbital speed of the spacecraft

Rearrange the equation to find the speed \(v\): $$ v = \sqrt{g_{Jupiter} * R_{orbit}} $$ Now substitute the given values of \(g_{Jupiter}\) and the calculated value of \(R_{orbit}\): $$ v = \sqrt{23 * (3 * 71500 * 10^{3})} $$
07

Relate the orbital speed to the period

To derive the period \(T\) from the orbital speed, we can use the formula: $$ v = \frac{2 * \pi * R_{orbit}}{T} $$ Rearrange the equation to find \(T\): $$ T = \frac{2 * \pi * R_{orbit}}{v} $$
08

Calculate the period of the spacecraft's orbit

Now, substitute the values for \(R_{orbit}\) and \(v\) calculated earlier: $$ T = \frac{2 * \pi * (3 * 71500 * 10^{3})}{\sqrt{23 * (3 * 71500 * 10^{3})}} $$ Calculate the result and express it in seconds: $$ T \approx 30607s $$ The period of the spacecraft's orbit around Jupiter is approximately 30607 seconds.

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Most popular questions from this chapter

Find the orbital radius of a geosynchronous satellite. Do not assume the speed found in Example \(5.9 .\) Start by writing an equation that relates the period, radius, and speed of the orbiting satellite. Then apply Newton's second law to the satellite. You will have two equations with two unknowns (the speed and radius). Eliminate the speed algebraically and solve for the radius.
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