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, and it is possible to reduce the complexities of 3d motion into 3 directions [math]\displaystyle{ \hat{x}, \hat{y}, \hat{z} }[/math]. An object in motion has properties along each axis which are independent of other axes.
A Mathematical Model
The fundamental equations of motion allow us to observe motion in three dimensions.
[math]\displaystyle{ d= d_0+ v_0t+ }[/math] [math]\displaystyle{ at^2 \over\ 2 }[/math]
[math]\displaystyle{ v= v_0+at }[/math]
[math]\displaystyle{ v^2= v_0^2 + 2a(d-d_0) }[/math]
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?
[math]\displaystyle{ \vec{v}= \vec{v}_0+\vec{a}t }[/math]
[math]\displaystyle{ \vec{v}-\vec{v}_0 \over\ t }[/math] [math]\displaystyle{ = \vec{a} }[/math]
Remember Newton's second law [math]\displaystyle{ \vec{F}_{net}= m\vec{a} }[/math]
[math]\displaystyle{ m(\vec{v}-\vec{v}_0) \over\ t }[/math] [math]\displaystyle{ = m(\vec{a}) }[/math]
[math]\displaystyle{ m(\vec{v}-\vec{v}_0) \over\ t }[/math] [math]\displaystyle{ = \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 velocity in first interval?
(b) the second interval?
(c) What was the average force applied during these two intervals?
(a) Begin with the definition of velocity:
[math]\displaystyle{ {\vec{v}_{avg,1}}= {{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} }[/math]
[math]\displaystyle{ {\vec{v}_{avg,1}}= \langle 0.0036, -0.0072, -0.0648 \rangle }[/math]
(b) Begin with the definition of velocity:
[math]\displaystyle{ {\vec{v}_{avg,2}}= {{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} }[/math]
[math]\displaystyle{ {\vec{v}_{avg,2}}= \langle 0.0072, 0.0171, -0.0648 \rangle }[/math]
(c) Begin with a fundamental equation of motion
[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].
[math]\displaystyle{ {\vec{p}_{f}}= {\vec{p}_{i}+ (\vec{F}_{spring}+ \vec{F}_{grav})\Delta t} }[/math]
Recall that [math]\displaystyle{ \vec{F}_{spring} = k_sS\hat{L} }[/math]. We need to compute the magnitude of the displacement of the feather clump in order to find S 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.
See also
http://farside.ph.utexas.edu/teaching/301/lectures/node33.html