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.
