An object with a mass of 10
kilograms is positioned 15 metres above the surface of an unknown planet. The object has 1800 joules of
gravitational potential energy. What is the acceleration due to
gravity at the surface of the planet?
Okay, so we have an object with a
mass of 10 kilograms at certain distance above the surface of an unknown planet. Let’s draw that surface this
way. And as the problem statement
explained, this is indeed an unknown planet.
Let’s say further that this is our
mass. And we’re told that that mass exist
at height of 15 metres above the surface of this planet. And in addition to all that, the
mass of this object, we’ll call it 𝑚, is 10 kilograms. We want to solve for the
acceleration due to gravity at the surface of our unknown planet.
To start working in this direction,
let’s write up a mathematical equation for the gravitational potential energy of a
mass of object. We typically write that as capital
GPE. And this is equal to the object’s
mass multiplied by its height above some reference point multiplied by the
acceleration due to gravity of the gravitational field that’s in. Now often, we shorthand this as
lowercase 𝑔, indicating that we’re on the surface of the Earth. That’s often a safe assumption to
make. But in this case of course, it’s
not a good one. After all, we’re not on Earth. We’re on some other planet.
So let’s write our expression for
the gravitational potential energy of this mass, this way. We’ll write it as 𝑚, the mass of
our object, multiplied by its height above some reference point multiplied by 𝑔 sub
p, where we understand that 𝑔 sub p is the acceleration due to gravity of this
Okay, knowing that, let’s refer to
another bit of information revealed to us. We’re told that this object has
1800 joules of gravitational potential energy. Remember that that’s what 𝑚 times
ℎ times 𝑔 is equal to, gravitational potential energy, which means that we can take
our expression, 𝑚 times ℎ times 𝑔 sub p, and set it equal to 1800 joules.
And then, since we’re interested in
solving for 𝑔 sub p, the acceleration due to gravity on the surface of this planet,
let’s divide both sides of this equation by 𝑚 times ℎ. When we do that, these factors
cancel out on the left-hand side of our equation. And now, it’s a matter of
substituting in the values for 𝑚 and ℎ. Recall that 𝑚 is 10 kilograms. And ℎ is 15 metres.
So the acceleration due to gravity
on planet unknown is equal to 1800 joules divided by 10 kilograms multiplied by 15
metres. This comes out to 12 metres per
second squared. So the acceleration due to gravity
on this planet is about 20 percent greater than that on planet Earth.