If the velocity at that distance is too small in magnitude, the moon will eventually collide with the Earth. If the velocity v of the moon is too large in magnitude for a given distance from the Earth, the moon's course curves slightly, but it continues to move away from the Earth. If the moon already has velocity v as shown above, the acceleration due to gravity causes the path of the moon to bend inward, as shown below. But because the Earth exerts a gravitational force on the moon, the moon is accelerated toward the Earth. Obviously, the moon would just continue on its course, regardless of the presence of the Earth. But what if the moon had an initial velocity? Consider what would happen if the moon had some velocity v tangent to the surface of the Earth but no forces act on either body. If both the Earth and moon were at rest initially, then the gravitational force would cause the moon to accelerate toward the Earth (with catastrophic results). The gravitational force on the moon is shown as F g. Let's illustrate this situation by way of a diagram. In the case of the moon orbiting the Earth (or any object orbiting another object to which it is attracted by some force), the net force on the moon is always directed toward the Earth. We know from Newton's second law of motion that an object experiencing a net force undergoes acceleration. Let's take a look at why the moon moves in a circle around the Earth (and why objects in other similar situations behave as they do). This situation is one example of circular motion (or nearly circular-we will assume it is sufficiently close that we can neglect any deviations from perfect circularity), where an object experiences a net force yet does not move linearly as a result. Because of gravity, the moon is "pulled" toward the Earth but (thankfully) doesn't ever collide with it. Obviously, the moon is in motion around the Earth. If we look up into the sky (at least at certain times of the day and month), we can see the moon as it orbits the Earth. We will consider how to approach such problems (such as the moon orbiting the Earth) and how to understand them in terms of forces, acceleration, and vectors. In addition, however, cases of circular motion are also common. Cases of linear motion, such as an object that is released from some height above the ground and is allowed to fall down under the influence of gravity, are common to our daily experience. O Use centripetal acceleration to solve problems involving objects in uniform circular motionĪccording to Newton's second law of motion, the net force acting on an object causes the object to accelerate in the direction of that net force. O Derive a formula for centripetal acceleration of objects in uniform circular motion We derive the acceleration of such objects as well as, by Newton's second law of motion, the force acting upon them. Examples of such motion include the orbits of celestial objects, such as planets and stars. In this case the velocity vector is changing, or d\mathbf(t)|, which is also the radius of the circle, and \omega is the angular frequency.In this article, we look at how to apply both vectors and the geometry of circles and triangles to uniform circular motion. However, in two- and three-dimensional kinematics, even if the speed is a constant, a particle can have acceleration if it moves along a curved trajectory such as a circle. In one-dimensional kinematics, objects with a constant speed have zero acceleration.
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