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Question Video: Finding the Mass of a Planet Using Newton’s Law of Universal Gravitation Mathematics

Find the mass of a planet, given that the acceleration due to gravity at its surface is 6.003 m/s², its radius is 2,400 km, and the universal gravitational constant is 6.67 × 10⁻¹¹ N ⋅ m²/kg².

03:39

Video Transcript

Find the mass of a planet, given that the acceleration due to gravity at its surface is 6.003 meters per square second, its radius is 2,400 kilometers, and the universal gravitational constant is 6.67 times 10 to the power of negative 11 newton square meters per square kilogram.

When we think about the acceleration due to gravity at the surface of some object, we can think about that as gravitational field strength. This is the gravitational force per unit mass exerted by a mass on some body. And there’s a formula that links the gravitational field strength 𝑔 with the mass of the body and the universal gravitational constant capital 𝐺.

For an object such as a planet with mass 𝑚 and radius 𝑟, where the universal gravitational constant is capital 𝐺, the gravity at its surface, the gravitational field strength lowercase 𝑔, is given by capital 𝐺 times 𝑚 over 𝑟 squared. So, with this in mind, let’s define each of these variables. We’re told that the acceleration due to gravity at the surface of the planet is 6.003 meters per square second. We’re also told that its radius is 2,400 kilometers. Since, however, we’re working in meters with our other variables, we’re going to multiply this value by 1,000 to find that the radius of the planet is 2,400,000 meters.

Finally, we’re given the universal gravitational constant. It’s 6.67 times 10 to the power of negative 11 newton square meters per square kilogram. If we define the mass of the planet that we’re trying to find to be equal to 𝑚, we can substitute everything into our formula. We get 6.003 is equal to 6.67 times 10 to the power of negative 11 times the mass 𝑚 divided by 2,400,000 squared.

To solve for the mass 𝑚, which we note is now going to be in kilograms, we’re going to multiply through by 2,400,000 squared and divide by 6.67 times 10 to the power of negative 11. So, the mass is 6.003 times 2,400,000 squared over 6.67 times 10 to the power of negative 11. And whilst not entirely necessary, we might choose to rewrite 2,400,000 squared using standard form. 2,400,000 is equivalent to 2.4 times 10 to the sixth power. Then, we square the number in standard form by squaring 2.4 to get 5.76 and then multiplying the exponent of the power of 10 by two. So, we get 6.003 times 5.76 times 10 to the 12th power over 6.67 times 10 to the negative 11th power. Multiplying 6.003 by 5.76 and then dividing by 6.67 gives us 5.184.

Let’s now deal with the powers of 10. We have times 10 to the 12th power divided by 10 to the power of negative 11. And of course when we divide two numbers whose base is the same, we subtract the exponents. So, we do 12 minus negative 11, which is 12 plus 11, which gives us 10 to the 23rd power.

And so we’ve calculated the mass of the planet in kilograms. It’s 5.184 times 10 to the power of 23 kilograms.

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