Video: Finding the Distance between Two Celestial Objects Given Their Masses and the Force between Them

Titan is the largest moon of Saturn. It has a mass of 1.35 × 10²³ kg. Saturn has a mass of 5.68 × 10²⁶ kg. If the magnitude of the gravitational force between them is 3.43 × 10²¹ N, what is the distance between the centers of mass of Saturn and Titan? Use a value of 6.67 × 10⁻¹¹ m³/kg ⋅ s² for the universal gravitational constant. Give your answer in scientific notation to two decimal places.

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Video Transcript

Titan is the largest moon of Saturn. It has a mass of 1.35 times 10 to the 23 kilograms. Saturn has a mass of 5.68 times 10 to the 26 kilograms. If the magnitude of the gravitational force between them is 3.43 times 10 to the 21 newtons, what is the distance between the centers of mass of Saturn and Titan? Use a value of 6.67 times 10 to the minus 11 meters cubed per kilogram second squared for the universal gravitational constant. Give your answer in scientific notation to two decimal places.

Okay, so in this question, we’re being asked to find the distance between the centers of mass of Saturn and its largest moon, Titan. We’re told that the mass of Titan is 1.35 times 10 to the 23 kilograms. Let’s imagine that this pink blob here is Titan. And we’ll label the mass of Titan as 𝑚 subscript T. We’re also told that the mass of Saturn is 5.68 times 10 to the 26 kilograms. Let’s say that Saturn is represented by this orange blob here. And we’ll label its mass as 𝑚 subscript S.

We’re told that the magnitude of the gravitational force between Saturn and Titan is 3.43 times 10 to the 21 newtons. We’ll label this force as 𝐹 subscript ST, standing for the force between Saturn and Titan. The final number we’re given in the question is that we’re told to use a value of 6.67 times 10 to the minus 11 meters cubed per kilogram second squared for the universal gravitational constant. We’ll label this with its usual symbol of capital 𝐺. Let’s label the thing that we’re trying to find, which is the distance between the centers of mass of Saturn and Titan, as 𝑟 subscript ST.

Since we’re talking about the gravitational force between Saturn and Titan, then a formula that’s going to be very useful to us is Newton’s law of gravitation. This law says that the gravitational force between two objects, which we’ll call 𝐹, is equal to the universal gravitational constant, capital 𝐺, multiplied by the masses of the two objects, which we’ll call 𝑚 one and 𝑚 two, divided by the square of the distance between the objects’ centers of mass, which we’ll call 𝑟.

Now, in our case, we know the magnitude of the gravitational force and we know the masses of each of the two objects. What we’re trying to find is the value of the distance between their centers of mass. So let’s take this formula and rearrange it to make this distance 𝑟 the subject. We’ll begin by multiplying both sides of the equation by 𝑟 squared. Then, on the right-hand side, the 𝑟 squared in the numerator cancels with the one in the denominator. Then, we’ll divide both sides by 𝐹 so that the 𝐹 ’s in the numerator and denominator on the left-hand side cancel out.

Finally, we’ll take the square root of both sides of the equation. This gives us an equation that says that the distance between the centers of mass of two objects is equal to the square root of the universal gravitational constant times the masses of the two objects divided by the magnitude of the gravitational force between those objects.

Now, in our specific case, the two objects in question are the planet Saturn and its largest moon, Titan. And so we have that the distance 𝑟 subscript ST between the centers of mass of Saturn and Titan is equal to the square root of the universal gravitational constant times the mass of Saturn times the mass of Titan divided by the magnitude of the gravitational force between Saturn and Titan.

We’re now in a position to substitute these values into this equation to calculate this distance 𝑟 subscript ST. When we do that, we get this equation here, which says that the distance 𝑟 subscript ST is equal to the square root of 6.67 times 10 to the minus 11 meters cubed per kilogram second squared, the value of our universal gravitational constant, capital 𝐺, multiplied by 5.68 times 10 to the 26 kilograms, that’s the mass of Saturn, multiplied by 1.35 times 10 to the 23 kilograms, that’s the mass of Titan, divided by 3.43 times 10 to the 21 newtons, that’s the value of the gravitational force between Saturn and Titan.

Evaluating the multiplication in the numerator of this fraction gives us a result of 5.114556 times 10 to the 39 meters cubed kilogram per second squared. In the denominator, we’ll use the fact that units of newtons are equivalent to units of kilogram meters per second squared to rewrite the units of our force as kilogram meters per second squared. This will make it easier to see how the units work out between the numerator and denominator of this fraction.

When we do this division under the square root, we get a result of 1.4911 times 10 to the 18, where we’ve used ellipses to indicate that there are further decimal places. In terms of the units, we have kilograms in the numerator which cancel with kilograms in the denominator. And we have per second squared in the numerator which cancel with per second squared in the denominator. Then, we’re left with meters cubed divided by meters, which gives units of meters squared. Taking the square root gives us the distance 𝑟 subscript ST as 1.2211 times 10 to the nine meters, where again the ellipses indicate that there are further decimal places.

Looking back at the question, we see that we are asked to give our answer in scientific notation to two decimal places. Now, our result is already in scientific notation. We just need to round it to two decimal places. When we do this, we get our final answer to the question that the distance between the centers of mass of Saturn and Titan is equal to 1.22 times 10 to the nine meters.

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