Suppose the radius of the Earth doubled but its mass stayed the same. What would be the...

Question:

Suppose the radius of the Earth doubled but its mass stayed the same. What would be the approximate new value for the free fall acceleration at the surface of the Earth?

Free Fall:

Freefall is when a falling object experiences only a gravitational force (and thus, only a gravitational acceleration as well). That is, there is no air resistance or friction force to impede the motion of the object. In free fall, an object will just keep accelerating instead of reaching a terminal velocity. On Earth, the freefall acceleration near the ground is {eq}\displaystyle 9.8\ \rm m/s^2 {/eq}.

Answer and Explanation: 1

Recall that the freefall acceleration on Earth is {eq}a = 9.8\ \rm m/s^2 {/eq}. The freefall acceleration is derived using Newton's law of gravitation. We can write the law of gravitation as:

$$F = \frac{GMm}{r^2} $$

where {eq}M {/eq} is the mass of the Earth, {eq}m {/eq} is the mass of some generic object, and {eq}r {/eq} is the radius of the Earth. Moreover, recall that Newton's second law states that the net force is:

$$F =mg $$

where {eq}g {/eq} is the acceleration. In freefall, the only force acting on a falling object is the gravitational force due to Earth. So, in this case, we can equate both forces:

$$mg = \frac{GMm}{r^2} $$

Cancelling both {eq}m{/eq} terms here gives us:

$$g = \frac{GM}{r^2}\\ $$

From here, we now double the radius of the Earth. We set:

  • {eq}r\ \to\ 2r {/eq}
  • {eq}g\ \to\ a {/eq}

So, we will have:

$$a = \frac{GM}{(2r)^2} = \frac{GM}{4r^2}\\ $$

Let us take out the new factor here:

$$a = \frac{1}{4} \left(\frac{GM}{r^2} \right) \\ $$

The term inside the parenthesis is just the original value of {eq}g {/eq}. Thus, the approximate new value for the free fall acceleration is

$$a = \rm \frac{1}{4} (9.8\ m/s^2) = \boxed{2.45\ \rm m/s^2}. $$


Learn more about this topic:

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Gravitational Pull of the Earth: Definition & Overview

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Chapter 15 / Lesson 17
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Earth's gravitational pull is often misunderstood, but without it, life on Earth would be impossible. In this lesson, we'll define the gravitational pull and give some examples of how it is used. A quiz is provided to test your understanding.


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