Table of Contents

Engineering Principles
Spring Theory:
The most basic formula when describing a spring is Hooke's Law represented by:
where: F represents a force being applied to the spring,
 k represents the spring constant,
 and X represents the distance the spring has been stretched.
The Potential Energy stored by a spring can be expressed as:
where:
 PE represents the potential energy,
 k represents the spring constant as described above,
 and x represents the distance the spring has been displaced.
A third equation of springs, not necessarily shown in our Rube Goldberg Machine, is also important. This is the Harmonic Oscillator equation which is another way of solving for the spring constant k or the frequency of a spring oscillating when a mass has been placed on one end and the other end is stationary. This equation is defined as:
where:
 f represents the frequency of the spring,
 k is the spring constant,
 and m is the mass of the weight placed on the one end of the spring.
The spring launcher in our Rube Goldberg Machine was constructed of scrap materials including a steel pipe with an OD of 15/16" an ID of 3/4" and a length of 6", one lock nut, one normal nut secured with locktite, 1 washer welded to the pipe, and a spring (5.75" long with an ID of 4/10"). In order to know how fast the spring will launch we need to control the amount the spring can be compressed. The spring is capable of being compressed 3.5 cm or 0.035 m. The spring constant has been calculated to be 261.4 kg/s^{2}. After using the basic law of energy: KE_{initial} + PE_{initial} = KE_{final} + PE_{final} which expands out to 1/2 mv_{i}^{2} + 1/2 kx_{i}^{2} = 1/2 mv_{f}^{2} + mgh_{f}*cos(theta) we've calculated the spring launcher should be able to transfer the ball 29 mm up the track.
Electromagnetism:
Electromagnetism is the branch of science which relates the forces involved in an electrical current or electrically charged particles, with the magnetic properties of certain materials which will attract opposite poles and repel like poles.^{[1]} Electric fields are the cause of battery sources having a constant voltage, and electricity flowing from a source of electricity through a wire to a form of power output. Magnetic fields are the reasons a simple or more complicated magnet can attract materials with magnetic properties. Both Electric charges and Magnetic charges attract opposite charges and and oppose like charges/forces. When an electric current is passed through a wire it creates a circular magnetic field on the outside of the wire. When this conductive wire sure as copper is wrapped around a magnetic material such as a steel rod, the steal then becomes magnetic itself. The strength of this force depends on:
 The electrical current being passed through the conductive material (Copper)
 The diameter of the conductive material
 The surface area of the magnetic material (Steel)
 The number of coils wrapped around the steel (more coils = stronger magnetic force)
Two main contributors to the development of Electromagnetism are Michael Faraday,
and James Clerk Maxwell in the early 19th century. Faraday extended off of other scientists discoveries such as Johann Carl Friedrich Gauss, and Maxwell further expanded off of that. In the late 18th century to early 18th century, Gauss discovered a fundamental principle of magnetism which stated the magnetic monopoles do not exist. This means that any magnetic field (B) diverges to zero. dA represents the differential area enclosed by the magnetic field. Using their research, the MaxwellFaraday Equation (also known as the Faraday Law of Induction) was created which defines that the line integral of an electric field (E)on a closed surface is equal to the negative double integral of the partial derivative of its magnetic field with respect to time multiplied by the differential surface area. Essentially meaning the electric field produced is proportional and can be calculated based off the magnetic field emitted and vise versa.^{[2]}
Another important equation to calculate the electromagnetic force outputted by an electromagnet is the Lorentz Law:
where: F is the force, q is the electric charge, v is the instantaneous velocity, E is the electric field, and B is the magnetic field. ^{[3]}
In order to calculate how strong our electromagnet would be we followed a simple tutorial found on eHow.com. The initial equation is:
where: N = the number of coils in the solenoid
 I = the current in Amperes (A) running through the solenoid
 A = the crosssectional area (m^{2}) of the solenoidal magnet
 g = the distance (m) between the magnet and the piece of metal being attracted
 and k = 4 x pi x 10^{7}
Our electromagnet was constructed by wrapping (_X_) feet of 14 AWG insulate copper wire tightly around a iron core (_X_) times. This iron core is 13 inches in length with a hexagonal width of 10.1 mm. Each end of the copper wire (stripped bare) is connected to a power source running (_X_) amps into the electromagnet. In order to turn this electrical current on and off we have placed a limit switch on the ramp which is hit by the metal ball rolling on the track. This limit switch is connect to an Arduino which is programmed to turn the electrical current off after (_X_) seconds. To learn more about the Arduino read the information located in the Second Subsystem Information Page.
For information on how to develop simple electromagnet, here's a short 2 minute youtube video of a young boy showing how simple it can be
For a more complex electromagnet showing Faraday's Law, watch this youtube video which shows an electromagnet giving off both heat and electricity!
References
 ↑ http://en.wikipedia.org/wiki/Electromagnetism
 ↑ http://en.wikipedia.org/wiki/Electromagnetism
 ↑ http://en.wikipedia.org/wiki/Lorentz_force
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