Chapter 13: Q. 29 (page 354)

A small moon orbits its planet in a circular orbit at a speed of $7.5 k m / s$ . It takes $28 h o u r s$ to complete one full orbit. What is the mass of the planet?

Short Answer

Expert verified

The mass of the planet is $10^{26} k g .$

Step by step solution

Given information

Time period = $28 h o u r s = 100800 s,$ speed of orbiting = $7.5 k m / s .$

Calculation

According to the Kepler's 3rd law, the time period is given by $T^{2} = \frac{4 π^{2} R^{3}}{G M}$

Where, T is time period, R is the radius of the orbit, G is the universal gravitational constant and M is the required mass of the planet.

Also, $v = \frac{2 π R}{T}$

Here, R is unknown and is given by the below formula. v is the speed, R is the radius and T is the time period.

Now solving we get,

$v = \frac{2 π R}{T} R = \frac{v \times T}{2 π} R = \frac{(7500 m / s) (28 \times 60 \times 60 s)}{2 π} R = 1.203 \times 10^{8} m$

Now substituting all the values in Kepler's third law equation we get,

$T^{2} = \frac{4 π^{2} r^{3}}{G M} T^{2} = \frac{4 π^{2} R^{3}}{G M} M = \frac{4 π^{2} R^{3}}{G T^{2}} M = \frac{4 π^{2} {(1.203 \times 10^{8} m)}^{3}}{(6.67 \times 10^{- 11} N m^{2} / k g^{2}) {(1.008 \times 10^{5} k g)}^{2}} M = 8.83 \times 10^{25} kg ≃ 1 \times 10^{26} kg$

Final answer

The mass of the planet is $10^{26} k g .$

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A small moon orbits its planet in a circular orbit at a speed of $7.5 k m / s$ . It takes $28 h o u r s$ to complete one full orbit. What is the mass of the planet?

Short Answer

Step by step solution

Given information

Calculation

Final answer

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A small moon orbits its planet in a circular orbit at a speed of 7.5km/s. It takes 28hours to complete one full orbit. What is the mass of the planet?

Short Answer

Step by step solution

Given information

Calculation

Final answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Further Mechanics and Thermal Physics

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Electricity

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Study anywhere. Anytime. Across all devices.

A small moon orbits its planet in a circular orbit at a speed of $7.5 k m / s$ . It takes $28 h o u r s$ to complete one full orbit. What is the mass of the planet?