3-Dimensional Position and Motion
Claimed by Benjamin Tasistro-Hart Fall 2016 Motion is defined as a change in position over a time interval. Objects exist in 3-Dimensional space, so it is necessary to know how these objects change position in 3-Dimensional space.
The Main Idea
Objects, exist and move in three dimensions, so it is necessary to understand how it is possible to predict their future position using one of three fundamental principles, in this case the momentum principle. We use an iterative process to calculate and predict the future position of objects. It may seem that using the momentum principle complicates the iterative calculation process, but very rarely do we know only the velocity and acceleration of an object. For this reason, using the momentum principle as the most general form of position update guarantees we can apply it to any situation.
A Mathematical Model
What are the mathematical equations that allow us to model this topic? Position is determined by the net force [math]\displaystyle{ \vec{F}_{net} }[/math] so every type of force, be it a spring, gravity, or electric force, affects the position of an object. It is possible that the net force is provided, and we can avoid the iterative calculation of the new magnitude of the spring and gravity force.
Spring Force: [math]\displaystyle{ \vec{F}_{spring} = k_sS\hat{L} }[/math], [math]\displaystyle{ S = \left\vert \vec{L} \right\vert- L_0 }[/math]
Gravity Force: [math]\displaystyle{ \vec{F}_{grav}= }[/math] [math]\displaystyle{ {-G}{m_{1}m_{2} \over\ {\left\vert \vec{r} \right\vert}^2}{\hat{r}} }[/math]
Electric Force: [math]\displaystyle{ \vec{F}_{elec}= }[/math] [math]\displaystyle{ {1 \over\ 4\pi\varepsilon_{0}}{q_{1}q_{2} \over\ {\left\vert \vec{r} \right\vert}^2}{\hat{r}} }[/math]
Momentum Principle: [math]\displaystyle{ {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} }[/math]
The general workflow to solving position-update problems by hand would be as follows:
1. Calculate the current net force [math]\displaystyle{ \vec{F}_{net} }[/math] acting on the system. For this step, remember to update forces which are distant-dependent such as the spring, gravity, and electric forces.
2. Update the new momentum [math]\displaystyle{ {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} }[/math]
3. Update the new position [math]\displaystyle{ {\vec{r}_{f}}= {\vec{r}_{i}+ \vec{v}_{avg}\Delta t} }[/math].
For this step, [math]\displaystyle{ \vec{v}_{avg} }[/math] can take many forms:
Constant net force: [math]\displaystyle{ \vec{v}_{avg} \approx }[/math] [math]\displaystyle{ \vec{v}_{f}+ \vec{v}_{i} \over\ 2 }[/math]
Non-constant net force: [math]\displaystyle{ \vec{v}_{avg} \approx }[/math] [math]\displaystyle{ \vec{p}_{f}\over\ m }[/math]
A Computational Model
The following is code for a simple computational model showing the effects of a constant force on a mass:
https://trinket.io/embed/glowscript/75acfdd1c6
Examples
Here are a few examples:
Simple
At t = 10.0 seconds a mass of 3 kg has velocity of [math]\displaystyle{ \langle 12, 27, −8 \rangle }[/math] m/s. At t = 10.1 seconds its velocity was [math]\displaystyle{ \langle 24, 19, 22 \rangle }[/math] m/s. What was the average (vector) net force acting on the object?
Begin from a fundamental principle:
[math]\displaystyle{ {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} }[/math]
[math]\displaystyle{ {\vec{p}_{f}-\vec{p}_{i} \over\ \Delta t}= \vec{F}_{net} }[/math]
[math]\displaystyle{ {\vec{p}_{f}-\vec{p}_{i} \over\ \Delta t}= \vec{F}_{net} }[/math]
Substitute the provided values into the symbolic expression and you should arrive at your final answer:
[math]\displaystyle{ \left\langle 360,-240, 900\right\rangle N = \vec{F}_{net} }[/math]
Middling
An acorn of mass 2.7 g is acted upon by the Earth, air resistance, and a strong wind. Below are recorded times and positions:
First Interval:
At t = 1.56 s, the position was [math]\displaystyle{ \langle 4.22, 2.45, −9.63 \rangle }[/math] m.
At t = 1.59 s, the position was [math]\displaystyle{ \langle 4.26, 2.37, −10.35 \rangle }[/math] m.
Second Interval:
At t = 3.56 s, the position was [math]\displaystyle{ \langle 8.09, 6.18, -58.35 \rangle }[/math] m.
At t = 3.59 s, the position was [math]\displaystyle{ \langle 8.17, 6.37, -59.07 \rangle }[/math] m.
(a) What is the average momentum in first interval?
(b) the second interval?
(c) What was the average force applied during these two intervals?
(a) Begin with the definition of momentum:
[math]\displaystyle{ {\vec{p}_{avg,1}}= {m\vec{v}_{avg}} }[/math]
[math]\displaystyle{ {\vec{p}_{avg,1}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} }[/math]
[math]\displaystyle{ {\vec{p}_{avg,1}}= \langle 0.0036, -0.0072, -0.0648 \rangle }[/math]
(b) Begin with the definition of momentum:
[math]\displaystyle{ {\vec{p}_{avg,2}}= {m\vec{v}_{avg}} }[/math]
[math]\displaystyle{ {\vec{p}_{avg,2}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} }[/math]
[math]\displaystyle{ {\vec{p}_{avg,2}}= \langle 0.0072, 0.0171, -0.0648 \rangle }[/math]
(c) Begin from a fundamental principle
[math]\displaystyle{ {\vec{p}_{2}}= {\vec{p}_{1}+ \vec{F}_{net}\Delta t} }[/math]
[math]\displaystyle{ {\vec{p}_{2}-\vec{p}_{1} \over\ \Delta t}= {\vec{F}_{net}} }[/math]
[math]\displaystyle{ \langle 0.0018, 0.01215, 0 \rangle = {\vec{F}_{net}} }[/math]
Difficult
A cat toy is built from a clump of feathers with mass of 0.014 kg attached to the end of a spring with stiffness 0.915 N/m and relaxed length 0.265 m. You hold the toy such that the feathers hang suspended from the spring when your cat swats it around, setting it in motion. At a particular instant the momentum of the ball is [math]\displaystyle{ \langle −0.02, −0.01, −0.02 \rangle }[/math] kg · m/s, and the moving ball is at location [math]\displaystyle{ \langle −0.2, −0.61, 0 \rangle }[/math] m relative to an origin located at the base of the spring. What is the position of the ball 0.1 s later with a time-step [math]\displaystyle{ \Delta t }[/math] of 0.1 s?
Remember that since this problem involves the spring force: [math]\displaystyle{ \vec{v}_{avg} \approx }[/math] [math]\displaystyle{ \vec{p}_{f}\over\ m }[/math]
1. Identify a coordinate system. We choose [math]\displaystyle{ +\hat{x} }[/math] to be to the right [math]\displaystyle{ +\hat{y} }[/math] to point up, and [math]\displaystyle{ +\hat{z} }[/math] to point out of the page.
2. Choose a system and its surroundings. For this example, we choose the system to be the clump of feathers and the surroundings to be the spring and earth.
3. Identify external forces which contribute to the net force. In this example, the force of earth and a spring force are the only objects which exert force on the feathers.
Now we can begin to solve this problem. As always, begin from a fundamental principle:
[math]\displaystyle{ {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} }[/math]
- We know that [math]\displaystyle{ \vec{F}_{net} }[/math] is comprised of the spring force [math]\displaystyle{ \vec{F}_{spring} }[/math] and the force of earth [math]\displaystyle{ \vec{F}_{grav} }[/math].
- Recall that [math]\displaystyle{ \vec{F}_{spring} = k_sS\hat{L} }[/math]
- Since the distance between the earth and the feathers is small compared to the radius of earth, we can approximate the magnitude of the force of earth as [math]\displaystyle{ mg }[/math]
Connectedness
- How is this topic connected to something that you are interested in?
Everything we do involves three dimensional position and motion, so it is useful to understand how matter moves in three dimensions. I'm interested in parametric design. so understanding how things change in three dimensions is critical for good design.
- How is it connected to your major?
I study architecture, so an awareness of three dimensional space is critical to designing well. As mentioned before, I'm interested in parametric design which when used to create complex shapes (like those buildings which have a flowy diagrid [see Norman Foster's British Museum]) demands an understanding of the way matter moves through space.
- Is there an interesting industrial application?
Parametric design is related to fabrication, and modern fabrication techniques typically involved creating geometrically complex shapes which an ability to visualize parts in three dimensions.