Which of the lines on the graph shows how the acceleration due to gravity around a massive object varies with the distance away from the center of mass of that object?
Looking at the graph, we can see that we have acceleration due to gravity on the vertical axis and distance on the horizontal axis. We have a variety of different lines to choose from. And we need to select which one best represents the acceleration due to gravity around a massive object varying with the distance away from the center of mass of that object.
So let’s first think about how we expect these two quantities to relate to each other. And for that, we need to recall the equation for acceleration due to gravity around a massive object. That is, 𝑎 is equal to 𝐺𝑀 over 𝑟 squared, where 𝑎 is the acceleration due to gravity, 𝐺 is the universal gravitational constant, which is, as its name suggests, a constant, 𝑀 is the mass of the massive object, which, for the purpose of this question, we can also assume is staying constant, and 𝑟 is the distance away from the center of mass of the massive object.
So looking at how the acceleration due to gravity 𝑎 relates to the distance 𝑟, we can say that 𝑎 is proportional to one over 𝑟 squared. Now, since these two are inversely proportional to one another, which means that 𝑟 is on the denominator of this fraction, we know that as 𝑟 increases, 𝑎 decreases. That is, the larger the distance 𝑟 is, the smaller the acceleration due to gravity. That means we’re looking for a line that always has a negative gradient. So we can immediately exclude the green and blue lines which have positive gradients.
That leaves the red, purple, and black lines, which all have negative gradients. So let’s think about what happens at the extreme case when 𝑟 is zero. Then, the acceleration due to gravity will be equal to 𝐺𝑀 divided by zero squared and zero squared is equal to zero. So this presents us with a problem. We know we can’t divide by zero because the answer will be ∞. So let’s look at what happens to our remaining three lines when the distance is zero.
The black line and the purple line both have some finite value of acceleration. Only the red line tends to ∞ as we get closer to zero. Therefore, the line on the graph that shows how the acceleration due to gravity around a massive object varies with the distance away from the center of mass of that object must be the red line.