Non-uniform circular motion

Non-uniform circular motion is any case in which an object moving in a circular path has a varying speed. Some examples of non-uniform circular motion include a roller coaster, a vertical pendulum, a car riding over a hill, and much more. All of these situations include an object traveling at different speeds in a circular path. For the roller coaster, its cars travel faster during its trip down than its trip up. The pendulum’s mass swings in a semicircle in which its trip up slows down to 0m/s and comes back down. A car riding over a hill will experience different speeds at different points of the hill.

The way to express non-uniform circular motion is not that different from the techniques used in calculating uniform circular motion. In uniform circular motion, the magnitude of the tangential acceleration is always equal to zero assuming speed remains constant. The radial acceleration in uniform circular motion is equal to the centripetal acceleration, which is v^2/r towards the center of the circle. In non-uniform circular motion, the radial acceleration is the same and equal to v^2/r towards the center of the circle. What’s different is the tangential acceleration, since speed is non-zero and changing.

Since there is a non-zero tangential acceleration, which means there are forces that act tangentially on an object in addition to its centripetal force (composed of the mass and radial acceleration). These tangential forces on the object include forces such as weight, normal force, and other forces acted on the object due to the environment it is in such as friction.

An interesting characteristic of the additional forces that non-uniform circular motion takes into account is that they do not always follow traditional force concepts. For example, in many situations weight is countered by its normal force. When weight points down, normal force points down. In non-uniform circular motion, normal force does not always point in the opposite direction of weight. Here is an example with a person sitting inside of a capsule traveling in a straight path then travels in a circle then back into a straight path again:

                            File:Freebody circular.JPG

This diagram shows the normal force pointing in other directions rather than opposite to the weight force. The normal force is actually the sum of the radial and tangential forces that help to counteract the weight force and contribute to the centripetal force. The normal force can be broken down into horizontal and vertical components such as in the diagram below:

                                     File:Freebody object.JPG

From this diagram, the horizontal component of normal force is what contributes to the centripetal force. The vertical component of the normal force is what counteracts the weight force of the object.

One characteristic that is in non-uniform circular motion that may seem illogical is the fact that normal force and weight can point in the same direction. Both forces can point down, yet the object will remain in a circular path without falling straight down. First let’s see why normal force can point down in the first place. In the first diagram, with the person sitting inside the capsule, the two forces point down only when it reaches the top of the circle. The reason for this is because the normal force is the sum of the weight and centripetal force. Since both weight and centripetal force points down at the top of the circle, normal force will point down as well. From a logical standpoint, a person who is traveling in the capsule will be upside down at the top of the circle. At that moment, the person’s seat is actually pushing down on the person, which is the normal force.

The reason why the object does not fall down when subjected to only downward forces is a simple one. Think about what keeps any object up after it is thrown. Once an object is thrown into the air, there is only the downward force of earth’s gravity that acts on the object. That does not mean that once an object is thrown in the air that it will fall instantly. What keeps these objects up in the air is its velocity. Newton’s first law states that an object’s inertia keeps it in motion, and since the objects in the air has velocity, it will tend to keep moving in that direction.

                             File:Normal and weight.JPG