Why Doesn t the oon Hit us? In analysis of this question, we ll look at the following things: i. How do we get the acceleration due to gravity out of the equation for the force of gravity? ii. How does the acceleration of the oon due to Earth s gravity differ from the centripetal acceleration of the oon in its orbit the Earth? iii. If Earth s gravity causes us to fall, shouldn t the moon be falling as well?
How do we get the acceleration due to gravity out of the equation for the force of gravity? The force of gravity causes our mass to accelerate towards the Earth. Therefore, we can state the following relationship: G m earth m me m a Remember, all r 2 = forces can be me equated to F=ma We can see that the two me masses cancel out and we re simply left with: a = G m earth r 2 For the value of r, you use the radius of the Earth (more on why tomorrow). mass of Earth = 5.9742 10 24 kilograms G = 6.672 10-11 N.m 2 /kg 2 Radius of Earth = 6.378 x 10 6 m a = 9.799 m/s 2
How does the acceleration of the oon due to Earth s gravity differ from the centripetal acceleration of the oon s circular orbit? We know how to calculate the acceleration due to the force of gravity that exists between the Earth and the oon. So let s do that real quick: a = G m earth r 2 In this case r will be mean distance that exists between the Earth and the oon (we used this value when calculating the period (T) of the oon s orbit). ean distance of earth to moon = 3.84 x 10 8 m a =.002703 m/s 2 Now let s see what the oon s centripetal acceleration is equal to: We know that a c = v 2 /r -- we need v. Do we have an expression that will give us the velocity of a satellite in circular orbit?
Using this expression, solve for the tangential velocity of the oon in its orbit about the Earth. Remember, the radius of orbit is the mean distance between the Earth and oon. v = 1,018.831316 m/s Now let s use this velocity to find the oon s centripetal acceleration: a c = v 2 /r a c =.002703 m/s 2 The acceleration of the oon due to Earth s gravity and the oon s centripetal acceleration are the same thing.
If Earth s gravity causes us to fall, shouldn t the oon be falling as well? We see that in the case of the moon, the centripetal acceleration is the gravitational acceleration. And we know what gravitational acceleration does to objects -- it causes them to fall and hit the ground. So why is the oon not falling? Why does the moon not hit us? The oon is falling. The oon falls.00135 m towards the Earth every second (approximately 1.4 mm). So why doesn t the oon hit us? It doesn t hit the Earth because of its tangential velocity. The following provides a simple explanation:
See this video for an basic explanation http://www.youtube.com/watch?v=p0cdopr0u
So we ve established that the oon is indeed falling towards the Earth and we ve mentioned that the reason it doesn t hit the Earth is connected to its tangential velocity. We should explain a bit further how the tangential velocity not only keeps the oon from striking the surface of the planet, but also enables the oon (or any satellite) to establish a circular orbit. Let s reestablish a couple of points: The moon falls a distance of.00135 m every second (we ve seen how that comes from the acceleration of the moon). We can call this the radial component of the moon s displacement. We also know that due to its tangential velocity the moon moves approximately 1,019 m in 1 second. We can call this the tangential component of the moon s displacement. In your notes, draw a diagram representing the oon s orbit. Draw the Earth in the middle, a circle about the Earth for the oon s orbit, the oon somewhere in its orbit and draw two displacement vectors on the oon representing the two components of its displacement.
oon s orbit.00135 m 1,019 m E *nothing here is drawn to scale
E
One final piece of information we gain from this activity: We found that the centripetal acceleration of the oon was the same thing as the gravitational acceleration of the oon. This tells us something about centripetal acceleration (and centripetal force). For any mass moving in a circle, what is the direction of the centripetal force (and hence, the centripetal acceleration)? F c and a c are directed towards the center of the circle. v v F c a c