Video Transcript
Two objects, object A with a mass
of 55 kilograms and object B with a mass of 12 kilograms, are near an even larger
object with a mass of 10 to the 22 kilograms. Object A and object B are at an
equal distance, 1,000 kilometers, away from the center of mass of the very large
object. Which of the objects A and B will
have the greater acceleration towards the very large object?
Looking at the diagram, we have two
small spherical objects some distance away from a large spherical object. We can see that one of these
objects has a mass of 55 kilograms. So we know that this is object
A. And the second one has a mass of 12
kilograms, so this must be object B. The very large object is shown as a
rectangle but labeled as spherical. So we know it must be very, very
large on this scale, too large to show here. So we’re just seeing a small
portion of it.
We’re given its mass of 10 to the
22 kilograms. And we’re also told that the
smaller objects A and B are both the same distance away from the very large object
at 1,000 kilometers. Now that distance is measured to
the center of mass of the very large object. So the dotted line here indicates
that this is actually measured to a point off the screen. We know that both objects A and B
experience acceleration towards the center of mass of the very large object because
of the force of gravity. And we’re asked to determine which
one of those experiences the greater acceleration.
So we need to recall the equation
for acceleration due to gravity, which is 𝑎 is equal to 𝐺𝑀 over 𝑟 squared, where
𝑎 is the acceleration due to gravity. 𝐺 is the universal gravitational
constant equal to 6.67 times 10 to the minus 11 meters cubed per kilogram second
squared. 𝑀 is the mass of the object we’re
accelerating towards. And 𝑟 is the distance between the
centers of mass of the object experiencing the acceleration and the object it’s
accelerating towards.
So let’s first consider the
acceleration of object A, which we’ll call 𝑎 subscript 𝐴. This is equal to 𝐺𝑀 over 𝑟
squared, where 𝐺 is a constant. 𝑀 is the mass of the large object,
which is 10 to the 22 kilograms. And 𝑟 is the distance between the
center of mass of object A and the center of mass of the large spherical object,
which is 1,000 kilometers. Now, let’s compare that to the
acceleration of object B, which we’ll call 𝑎 subscript B. This is also equal to 𝐺𝑀 over 𝑟
squared, where 𝐺 is the same constant, 𝑀 is the same mass of 10 to the 22
kilograms, and 𝑟 is the same distance, 1,000 kilometers. These are identical, so we can
write 𝑎 subscript A is equal to 𝑎 subscript B.
And this is true because both
objects are the same distance away from the center of the large spherical object and
because the masses of objects A and B do not appear in this equation. That is, acceleration due to
gravity is independent of the mass of the object experiencing the acceleration. Therefore, the answer to the
question “Which of the objects A and B will have the greater acceleration towards
the very large object?” is that objects A and B have the same acceleration.
