Question Video: Finding the Mass of the Sun Given the Magnitude of the Gravitational Force Between Earth and the Sun and the Distance Between Them Physics • 9th Grade

Earth has a mass of 5.97 × 10²⁴ kg and orbits the Sun at a distance of 1.50 × 10¹¹ m. The magnitude of the gravitational force between Earth and the Sun is 3.54 × 10²² N. What is the mass of the Sun? Use a value of 6.67 × 10⁻¹¹ m³/kg ⋅ s² for the universal gravitational constant. Give your answer to 3 significant figures.

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Video Transcript

Earth has a mass of 5.97 times 10 to the 24 kilograms and orbits the Sun at a distance of 1.50 times 10 to the 11 meters. The magnitude of the gravitational force between Earth and the Sun is 3.54 times 10 to the 22 newtons. What is the mass of the Sun? Use a value of 6.67 times 10 to the minus 11 meters cubed over kilogram-second squared for the universal gravitational constant. Give your answer to three significant figures.

In this problem, we’re considering the Sun and Earth. Both the Sun and Earth have mass. And therefore, according to Newton’s law of gravity, there is a gravitational force that acts to attract them towards each other, which acts along the line that connects their centers of mass. We’re given the mass of the Earth, which is 5.97 times 10 to the 24 kilograms. We’ll denote that using 𝑚 with this subscript of a plus inside a circle, which is commonly used for values specific to Earth, but we could just as easily call it 𝑚 one or 𝑚 subscript 𝑒. We’re also given the distance between the Sun and Earth of 1.50 times 10 to the 11 meters, so we can add that to our diagram. And we’re given the magnitude of the gravitational force between Earth and the Sun of 3.54 times 10 to the 22 newtons.

The mass of the Sun is the quantity we’re trying to find. We’ll denote the mass of the Sun using a subscript of a circle with a dot in it, which is commonly used for values specific to the Sun. But again, we could use any notation we like so long as we can distinguish it from that of the Earth. Now, we need to recall the equation that links the gravitational force between two objects with their masses and the distance between them. That is, the force 𝐹 is equal to the universal gravitational constant 𝐺 times the mass of the first object 𝑚 sub one times the mass of the second object 𝑚 sub two divided by the distance between them squared.

In our case, we’re looking at Earth and the Sun. So this becomes 𝐹 is equal to 𝐺 times the mass of the Earth times the mass of the Sun divided by distance squared. The value we’re trying to find is the mass of the Sun. So, we need to rearrange this in terms of that quantity. We’ll start by multiplying both sides of this equation by distance squared. And then we’ll divide both sides by 𝐺 and by the mass of the Earth. What we end up with is that the mass of the Sun is equal to the Earth–Sun distance squared times the gravitational force between them divided by the universal gravitational constant times the mass of the Earth.

When we put numbers into this, we have the distance of 1.50 times 10 to the 11, which is squared. And we do need to remember to square all of this, including the times 10 to the 11, which is why we’ve put it in brackets. That’s multiplied by the gravitational force of 3.54 times 10 to the 22 and then divided by the universal gravitational constant of 6.67 times 10 to the minus 11, which is given in the problem statement, times the mass of the Earth, 5.97 times 10 to the 24.

And when we evaluate this, we find 2.000256 et cetera times 10 to the 30. We’re asked to give this to three significant figures. So that becomes 2.00 times 10 to the 30. Now, for units, we need to make sure that we’ve used SI units throughout the problem. So, we have force in newtons, distance in meters, the mass of the Earth in kilograms, and the universal gravitational constant in meters cubed over kilogram-second squared. So that means the mass of the Sun will be in kilograms. So, the mass of the Sun is 2.00 times 10 to the 30 kilograms.

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