Video Transcript
Which of the following relations shows how the acceleration due to gravity 𝑎 around a massive object varies with the distance away from the center of mass of that object 𝑟? (A) Acceleration is proportional to 𝑟. (B) Acceleration is proportional to one over 𝑟. (𝐶) Acceleration is proportional to one over 𝑟 squared. (D) Acceleration is proportional to one over 𝑟 cubed. (E) Acceleration is proportional to the square root of 𝑟.
Let’s imagine we have here a very massive object, like a planet or a star. Because this object is a sphere, we can model all of its mass as though it exists at the center of the object. We’ll call that mass capital 𝑀. In general, any mass will change the gravitational field around itself and therefore tend to accelerate other massive objects. Say, for example, we have a small mass, we’ll call it lowercase 𝑚, that is a distance 𝑟 away from the center of mass of our very massive object. In this case, there will be a gravitational force 𝐹 sub g between these masses. It’s equal to the universal gravitational constant, capital 𝐺, multiplied by the product of the two masses divided by the square of the distance between their centers of mass. So that’s the gravitational force that exists between these two masses. But what about the gravitational acceleration?
Let’s recall that Newton’s second law of motion tells us that the net force on an object equals that object’s mass times its net acceleration. If we assume that the only force acting on our smaller mass lowercase 𝑚 is the gravitational force, then we can write that that gravitational force is equal to the mass of our smaller object multiplied by its acceleration.
Notice something interesting about this equation. The mass of the smaller object appears on both sides of the equation, and therefore it cancels out. The acceleration due to gravity that our smaller mass experiences is actually independent of that mass’s value. It’s equal to a constant, the universal gravitational constant, multiplied by the mass of the much larger object divided by the distance between the two masses’ centers of mass squared.
We’re interested in understanding from this relationship the connection between gravitational acceleration and distance 𝑟. Both of the values in the numerator on the right-hand side are constants: capital 𝐺, because it’s a universal constant, and capital 𝑀, because it’s the constant mass of a large object. So we can effectively write this equation as a constant, we’ll call it capital 𝐶, multiplied by one over 𝑟 squared, being equal to 𝑎.
This implies something very specific about the acceleration 𝑎 and the distance 𝑟. 𝑎 is proportional to one over 𝑟 squared. This result matches up with answer option (C). The acceleration due to gravity 𝑎 around a massive object is proportional to one over the square of the distance away from the center of mass of that object, 𝑟.
