4.8:
Uniform Circular Motion
An object moving along a circular path is said to exhibit circular motion, which can be either uniform or non-uniform. In uniform circular motion, the object moves at a constant speed.
For an orbiting satellite, the Earth's gravitational force provides the centripetal force required for circular motion.
At any instant, the linear velocity of the satellite is tangential to the circular path, such that its magnitude remains constant, but its direction varies.
Since the direction of the velocity changes, the satellite accelerates radially inwards at any instant. This acceleration is called centripetal acceleration or radial acceleration.
Dividing the force equation by m, the magnitude of the centripetal acceleration is the square of the velocity divided by the satellite's distance from the Earth's center.
As speed is equal to the orbital circumference, 2πr, divided by the period T, a relation between the centripetal acceleration and time period of the satellite is obtained.
4.8:
Uniform Circular Motion
Uniform circular motion is a specific type of motion in which an object travels in a circle with a constant speed. For example, any point on a propeller spinning at a constant rate is undergoing uniform circular motion. The second, minute, and hour hands of a watch also undergo uniform circular motion. It is hard to believe that points on these rotating objects are actually accelerating, even though the rotation rate is constant. To understand this, we must analyze the motion in terms of vectors.
In one-dimensional kinematics, objects with a constant speed have zero acceleration. 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. The direction of the acceleration vector is toward the center of the circle. This is a radial acceleration and is called the centripetal acceleration. The word centripetal comes from the Latin words centrum meaning “center” and petere meaning “to seek,” and thus takes the meaning “center seeking.” Centripetal acceleration can have a wide range of values, depending on the speed and radius of curvature of the circular path.
This text is adapted from Openstax, University Physics Volume 1, Section 4.4: Uniform Circular Motion.