Video: Acceleration Due to Gravity at Half the Radius of Earth

What is the acceleration due to gravity at a distance from Earth’s center of mass that is equal to half the Earth’s radius? Assume that Earth is a perfect sphere with a radius of 6370 km and a constant density of 5510 kg/m³. Give your answer to 3 significant figures.

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Video Transcript

What is the acceleration due to gravity at a distance from Earth’s center of mass that is equal to half the Earth’s radius? Assume that Earth is a perfect sphere with a radius of 6370 kilometers and a constant density of 5510 kilograms per meters cubed. Give your answer to three significant figures.

In the problem, we are told that we’re within Earth’s surface. Therefore, we must use an equation where we find acceleration due to gravity inside a planet. The equation to find the acceleration due to gravity inside a planet is 𝑎, the acceleration due to gravity, equals four-thirds 𝐺, the universal gravitational constant, 𝜌, the density of the planet, 𝜋𝑟, where 𝑟 is the distance from the center of the planet to the position specified. Four-thirds, 𝐺, and 𝜋 are all constants, so that leaves 𝜌 and 𝑟 as variables we need to define from our problem.

For 𝐺, we use 6.67 times 10 to the negative 11th newtons times meter squared per kilogram squared. 𝜌 is given to us in our problem as 5510 kilograms per meters cubed. For 𝜋, we’ll take it out to three significant figures as 3.14 as our problem asks us to give our answer to three significant figures. For the distance from the center, we’re told that it’s half of Earth’s radius, and we’re told that the radius of Earth is 6370 kilometers. When we multiply one-half by 6370 kilometers, we get 3185 kilometers. Before we multiply out our values, we need to first check our units. Our distance is given in kilometers, but our density has meters in it, and so does our universal gravitational constant. Therefore, we need to convert from kilometers to meters.

We need to recall that kilo- is a prefix that means 1000 or 10 to the third. Therefore, we can replace 3185 kilometers with 3185 times 10 to the third meters. Now that all of our units are in agreement, we can multiply out our values to get our final answer. When we multiply out our values and round to three significant figures, we get an acceleration due to gravity of 4.90 meters per second squared. The acceleration due to gravity at a distance from Earth’s center of mass that is equal to half the Earth’s radius is 4.90 meters per second squared.

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