Curving Motion

claimed by Aayush Kumar

Editing by Qianyi Qu (qqu3)

Main Idea

In the previous section, we introduced the idea of conservation of momentum. By choosing multiple particles as the system, we can assume that the forces exerted by the particles in the system are reciprocal of one another, based on Newton's Third Law of Motion. The total change in momentum is zero if the system's forces all canceled out and no forces in the surrounding are acting on the system. This special case, where the momentum of the system remains constant (deltap = 0), can be implicated in many situations that helps us to identify forces exerted on the system. However, in the case when the momentum of the system changes with time, which means velocity and/or direction of motion are nonconstant and additional terms are needed to define the magnitude and direction of the momentum of the objects in the system. The momentum of an object is nonconstant when it is traveling along a curved path, regardless of whether or not it's speed along the curve is changing because the direction also changes.

The Main Idea

Such cases of curving motion can be analyzed using the properties of the parallel and perpendicular components of net Force as well as understanding the motion's "Kissing Circle".

Parallel and Perpendicular Components of [math]\displaystyle{ {\frac{d\vec{p}}{dt}} }[/math]

Perpendicular Component and the Kissing Circle

This is perhaps the most crucial element of this topic, as a property of smoothly continuous curving motion is that the perpendicular component of the change in momentum is [math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{perpendicular}=\frac{m v^{2}}{R} }[/math] where m is the mass of the object, v is the magnitude of its velocity, and R is the radius of the kissing circle, which we will explain shortly. In essence, the kissing circle of the object is the imaginary circular arc that it temporarily follows along its curved motion, where the radius is the distance to the center of said imaginary circle.

In the diagram above, we see that for a curved motion C, the kissing circle when the object is at point P is shown with the blue circle, where the radius is indicated by the red arrow. Often, there are several types of kissing circles within an object's curving motion, so we have to look at the the instantaneous situation to derive instantaneous values. The tangential straight blue line segways into our next topic, the parallel component of the change in momentum.

Parallel Component and Tangential Properties of Curving Motion

There are two scenarios of curving motion that we will entertain here, one where an object is not changing speed along its curved path and another where the object speeds up or slows down along this curved path. We will start with the constant speed scenario:

[math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{parallel}=\vec(0) }[/math]

When an object's tangential velocity is constant throughout the entire motion, we can say that the parallel component of [math]\displaystyle{ {\frac{d\vec{p}}{dt}} }[/math] is zero, as any change in momentum is solely for altering the direction of the object's momentum as opposed to its magnitude. To understand this in more mathematical terms, the only way the magnitude of a momentum vector can be changed is if some component of the force acting on the object lies in the direction of the objects motion. Otherwise, the net force must be in the perpendicular direction towards the center of the kissing circle, like shown below:

Here we see that the tangential velocities indicated by the red arrows are all of the same magnitude, and the only thing altering them is the centripetal acceleration, which leads into our discussion of centripetal and centrifugal forces later. Now we will consider cases where [math]\displaystyle{ {\frac{d\vec{p}}{dt}} }[/math] is nonzero. In these cases, the speed of the object along it's curved path is changing and our net force vector has a component that is parallel to the object's tangential motion.

===Calculating [math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{parallel} }[/math] and [math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{perpendicular} }[/math] Given a instantaneous force acting on the object as well as it's initial momentum, we can calculate the final momentum using the Momentum Principle and then observe any change in magnitude between these two vector quantities. Here, [math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{parallel} }[/math] is equal to the difference between the magnitudes of the initial and final momentum divided by the change in time. From this, we can calculate [math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{perpendicular} }[/math] by subtracting [math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{parallel} }[/math] from the net change in momentum.

A Computational Model Example

Here's a glowscript animation utilizing this method of calculating the parallel and perpendicular components of [math]\displaystyle{ {\frac{d\vec{p}}{dt}} }[/math] is linked below:

Curving Motion with a Satellite Projectile

Centripetal and Centrifugal Forces

The Centripetal and Centrifugal Forces of an object in motion demonstrate reciprocity in that the Centripetal Force acts inward towards the center of the kissing circle whereas the centrifugal force is the object's tendency to fly outward.

Simple

It's fairly simple identifying cases of curving motion, in which case a person can easily apply [math]\displaystyle{ {\frac{m v^{2}}{R}} }[/math] to a situation immediately.

Middling

Occasionally it can be difficult to gauge any components that influence the object's tangential velocity and effectively speed up or slow down the object by altering the magnitude of the momentum vector.

Difficult

It can be fairly difficult to understand when there are other factors that influence the perpendicular component of the change of momentum, or when an individual needs to identify another influential force in that axis. For instance, in the Ferris Wheel example, when asked about having to find the necessary velocity for the rider to feel weightless, the individual needs to understand that there's a surface contact force that is also in play and in this case needs to be 0, in which case gravitational force is equal to [math]\displaystyle{ {\frac{m v^{2}}{R}} }[/math].

Connectedness

1. How is this topic connected to something that you are interested in?

Curving motion is heavily employed when observing planetary orbits, something I find highly fascinating. Although in reality these orbits aren't perfectly circular as we often make them out to be, they still heavily demonstrate the ideas of curving motion and especially [math]\displaystyle{ {\frac{m v^{2}}{R}} }[/math]!

1. How is it connected to your major?

As of now I am a CS major, and using vPython to model curving motion animations among many things helps apply programming ideas such as iteration and variable update.

1. Is there an interesting industrial application?

Centripetal and Centrifugal forces can be found in many industrial applications, as curved motion is a very frequent occurrence when observing interactions between various objects. For instance, a centrifuge using these mechanics helps astronauts train for higher effective gravity.