![]() In order to derive the Moon’s centripetal acceleration, we first need to find the circumference of its orbit.SfC Home > Physics > Gravity > Overview of the Force of Gravity We can get this from the circumference around its orbital path and the time period that it takes to go once around. In modern units it is 384,400 km, or 3.844 x 10 8 meters.Ĭoncept: in order to use the formula a c = v 2/ r, we are given r, but we will still need to find the orbital velocity of the Moon, v. The radius of the Moon’s orbit has long been known from observations. Worked Example: Calculating Orbital Accelerationsįind the centripetal acceleration of the Moon. Then you will have a chance to do a similar calculation for Earth’s orbit around the Sun. We can use the Moon’s orbital size and period to find the Moon’s speed, and then its acceleration, as we show in the next activity. For the Moon and Earth, you just plug in the Moon’s orbital speed and its distance from Earth. If you know the formula for centripetal acceleration, as Newton did, it is fairly easy to compute the value of the acceleration in a given situation. This acceleration points toward the center of the circular path, as the animation shows. Thus, there must be an acceleration that points in the direction of its change in motion. ![]() Since the direction of its motion is changing, the velocity of the object must also be changing. This object is moving in a circular path with a constant speed, but a constantly changing direction. See Figure 7.5 for an animated representation of this situation.Īnimated Figure 7.5: Object moving in a circular path. It continues to point in a direction tangent to the circular path, and the moving object travels farther along the circumference of the circle. The velocity vector never turns all the way toward the center of the circle, of course. For circular motion like this, where the only thing changing is the direction of the velocity vector, the vector is constantly turning around toward the center of the circle. For this reason, acceleration of this type is called centripetal (center-seeking) acceleration, hence the subscript “c.” It is sensible that the acceleration should point to the center if you recall that the acceleration is just the change in velocity over time. ![]() What is the direction of this acceleration? It is not obvious from this expression, but it turns out to be pointed along the radius of the circle, directly toward its center. We know from our earlier discussions that acceleration is a vector and has both size and direction. You might be surprised that the velocity is squared, but that is how this type of circular motion works. With this equation, we see that a larger velocity causes a larger acceleration, while a larger radius for the motion causes a smaller acceleration. Where \(a_c\) is the acceleration, \(v\) is the velocity, and \(r\) is the distance between the object and the center of the circle. The equation that describes the acceleration for circular motion is: Furthermore, if an object travels on a very big circle, say like the orbit of the Moon, its acceleration will be smaller than if it travels at the same speed on a smaller circle. Its velocity must change more quickly than the slower object. If we consider such motion, we realize that a faster object has a larger acceleration than a slower one. Why? Because as it travels along its path, the direction of its velocity is constantly changing, and changing velocity implies an acceleration. To keep an object moving in a circular path, it must be accelerated. (This is the time required to go 360 degrees around Earth, not the time from new moon to new moon.) Newton also knew that the Moon’s orbit around Earth is roughly circular, and that the orbital radius of the Moon is about 60 times larger than Earth’s own radius. Newton knew that the Moon circled Earth in 27.5 days. He used this insight and his new laws of motion to deduce the Law of Universal Gravitation. ![]() Figure 7.4: Isaac Newton had an insight that the same force that causes an apple to fall to the ground must also keep the Moon in its circular path about Earth. As he sat staring at the Moon, pondering what made it go around Earth, suddenly an apple is supposed to have fallen from the tree onto his head, causing him to exclaim “Eureka!” This story is probably untrue, but it does accurately relate the essence of Newton’s insight into gravity: The same force that causes apples to fall to the ground, and keeps people firmly on the ground, also causes the Moon to circle Earth (and Earth to circle the Sun). One legend suggests that Newton’s epiphany on gravity came as he was sitting under an apple tree on the Cambridge University campus where he was a professor.
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