(Physics) Roller Coaster Safety Design

Sweet! You've landed a job as a roller coaster engineer! One day, you and your co-worker come up with a nifty novel roller coaster design that utilizes small rocket boosters on the cart.

Human safety is a pretty important factor in roller coaster designing though so you need to make sure it's within healthy G-forces. The allowable G-forces can vary from region to region. The max acceleration this project permits is 5G. 

You run the design through a simulation program, and have narrowed down that the most extreme acceleration occurs in the region from x = 0 m to x = 1.15 m. The function below describes the position of the roller coast on the track during time t:

According to your calculations, will the maximum acceleration here meet the project specifications, or do you guys need to make some design modifications?

  • Yup, we're good...now to convince everybody else.
  • Nope, back to the drawing board!
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(Mechanics) Calculate Moment of Inertia

Qotd 5.png

Find the area (i.e. 2nd) moment of inertia of the triangle with respect to its base (i.e. the axis of rotation is the x-axis, which runs along the base of the triangle).

(Math) Second Degree Polynomials

Quadratic polynomials show up in history as far back as Babylonians taxing farmlands. Today they are used in any problem concerning parabolic trajectories from projectile motion to planetary orbits. As a simple tribute to the utility of second degree polynomials, try your hand at the problem below.


Find x in the following equation:




Try showing your work using latex equations.

(Fluid Mechanics) Superheroes Need Mach Numbers

A Quick Primer

Mach number...great for marketing, but is that all?

What does the number mean?

Why it's used? Perhaps to avoid potential shockwaves!

It turns out that when a really fast object like Superman, an aircraft, submarine, or vehicle travels through a fluid (like liquid or air), we need to be aware of whether its speed is near or over Mach 1 (the speed of sound through the fluid medium) to see if we have to deal with any additional loading conditions put on by the (faster-than-sound) travel speed. As a pilot, this helps ensure you're not putting the aircraft into dangerous speeds. As an engineer, this helps you determine the worst-case conditions you might have to design for.

where (for today's problem):

γ = specific heat ratio = 1.4
R = specific gas constant = 1716 ft*lb/(°R*slug) OR 287 J/(K*kg) 
T = temperature = [degrees Rankine °R ] OR [Kelvin K]

(The speed of sound is derived here.)

Read more:

A little more on shock waves (work-in-progress but great explanation)

Mach Angle


Taking a lead from your idol, you've become THE costume designer for superheroes.

You really have to ensure these outfits are rugged (and doesn't burn and leave your superhero flying stark-bare...after all, that's bad for business).

If your superhero client can fly at 2400 km/hr at an altitude of 1100 meters, and the surrounding temperature is -50°C, what kind of Mach condition do you need to ensure the suit is designed for?

  • Subsonic (lower than Mach 1)
  • Transonic (at or near Mach 1)
  • Supersonic (greater than Mach 1)
  • Hypersonic (greater than Mach 5)
  • Psh, flying's easy.

Use latex to write equations.

(Electrical) Mesh Current Analysis

We looked at how to analyze circuits using node voltage analysis. Now let's explore how to do the same using mesh current analysis.

Review

In a circuit, a mesh is a closed loop, such as $i_1$ and $i_2$ in the following figure. 

The first step of mesh analysis is to identify all the meshes and assign a clockwise current in each mesh where there is no current source. If there is a current source, then the mesh current should be in direction of the current source.

Kirchoff's Voltage Law (KVL) says that the sum of all the voltages across every element in a mesh is 0. For example, looking at the following circuit we can write KVL equations:
$v_s-v_1-v_2=0$, and
$v_2+v_3+v_4=0$

The voltage across an element shared by two meshes is expressed in terms of both mesh currents. For example, $R_2$ is $v_2=(i_1-i_2)R_2$.

KVL for mesh 1 is: $v_s-i_1R_1-(i_1-i_2)R_2=0$
... and for mesh 2 is: $(i_2-i_1)R_2+i_2R_3+i_2R_4=0$

Notice that the polarity of the voltage across $R_2$ for mesh 2 is now flipped. This is to keep consistent with the direction of $i_2$. The voltage across can be written $R_2$ $v_2=(i_2-i_1)R_2$.

Using KVL, we emerge with the following system of equations:
$(R_1+R_2)i_1-R_2i_2=v_s$
$-R_2i_1+(R_2+R_3+R4)i_2=0$

Question

Given DC voltage sources of $V_1=10V$,  $V_2=9V$, and $V_3=1V$ and resistors with resistances of $R_{1}=5\Omega$, $R_2=10\Omega$, $R_3=5\Omega$, and $R_4=5\Omega$, you connect them into the following circuit:

question_figure.png

Find mesh currents $i_1$ and $i_2$
Bonus: Write your KVL equations

 

Use latex to write equations.


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