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| − | raxu is currently a [[Division C]] competitor at [[Scarsdale High School]] and Captain of the team in [[2017]] and [[2018]]. He also competed in Division B for the [[Scarsdale Middle School]] team in [[2014]]. | + | raxu (Richard Xu) was a [[Division C]] competitor at [[Scarsdale High School]] and Captain of the team in [[2017]] and [[2018]]. He also competed in Division B for the [[Scarsdale Middle School]] team in [[2014]]. |
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| | ==Sandbox== | | ==Sandbox== |
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| − | '''Compound Machines''' was an event for [[Division C]] in [[2014]] and [[2015]] in which students answer questions on simple and compound machines and use a compound lever to determine the ratios of 3 unknown masses. Students should bring their device and any other supplies for impound, a binder of [[notes]], and any calculator.
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| − | Compound Machines can be considered the Division C equivalent of the Division B event, [[Simple Machines]].
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| − | ==General Overview==
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| − | In Compound Machines, teams take a written test over compound and simple machines, and create and use a device to accurately and quickly determine the ratios of some unknown masses.
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| − | The written test for Compound Machines tests knowledge over the different types of simple and compound machines and making calculations about the forces involved in these machines at static equilibrium. This test can include simple and compound machine terminology, mechanical advantage, load, effort, energy, and friction. Free response answers must include metric units and significant figures. The test covers all 6 simple machines: lever, pulley, inclined plane, wedge, screw, and wheel and axle, often put together into compound machines.
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| − | For the written test, teams are allowed one binder of reference materials that does not need to be impounded.
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| − | The device testing for Compound Machines involves using a series of 2 levers to determine the mass ratios of 3 weights. Teams construct the device before competition, and impound the device together with necessary tools.
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| − | ==Terms and Vocabulary==
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| − | ''This section is similar to the section 2 of [[Simple Machines]]. However, it goes into some material in greater depth.
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| − | ===Force===
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| − | A force, intuitively a push or a pull, is an interaction between two objects that, when unopposed, causes the object to accelerate. Forces are represented by the symbol [math]F[/math]. The SI unit of force is Newton, [math]N=\frac{kg\cdot m}{s^2}[/math]. Forces are vectors, having both magnitude and direction. The net force on an object is the sum of all the forces acting on the object. An object's acceleration is given by Newton's Second Law
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| − | <div align="center">[math]F=ma,[/math] </div>
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| − | where [math]m[/math] is the mass of the object.
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| − | ===Energy===
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| − | The energy of an object quantifies its ability to affect its environment. The SI unit of energy is Joule [math]J=N\cdot m[/math].
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| − | There are many forms of energy. In this event, we primarily consider an object's kinetic energy, the energy it possess from motion, and potential energy, the energy it posses from its location in a field (in this event, the gravitational field). The total mechanical energy of an object is given by the sum of its kinetic and potential energy.
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| − | ===Work===
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| − | A force [math]F[/math] acting on an object is doing work if the object experiences displacement [math]\Delta s[/math] under the force. Doing work changes the energy of the object. The SI unit of work is Joule [math]J=N\cdot m[/math].
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| − | If the angle formed by the force and the displacement is [math]\theta[/math], then the work that [math]F[/math] does on the object is given by
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| − | <div align="center">[math]W=F\cdot \Delta s = |F|\cdot |\Delta s|\cdot\cos \theta, [/math] </div>
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| − | where [math]F, \Delta s[/math] are vectors, while work [math]W[/math] is a scalar.
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| − | ===Power===
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| − | Power is the rate of energy transfer from one object to another, or the rate of work being done. The SI unit of power is Watt [math]W=\frac{J}{s}.[/math]
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| − | Power can be computed using the formula [math]P=\frac{W}{t},[/math] where [math]W[/math] is the work done to the object, and [math]t[/math] is the time over which the work is done.
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| − | ===Conservation of Mechanical Energy===
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| − | The law of 'Conservation of Mechanical Energy' states that the total mechanical energy of an object remains the same if there is no non-conservative forces doing work. In other words, if all forces other than gravity are acting perpendicular to the displacement of the object, the mechanical energy is conserved.
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| − | ===Torque===
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| − | [[image:LeverTorque.png|150px|thumb|Force and the corresponding moment arm.]]
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| − | A torque, also known as a moment of force, is the rotation equivalent of force. It is denoted by either [math]\tau[/math] or [math]M[/math]. The SI unit of torque is [math]N\cdot m[/math].
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| − | Torque is caused by a force, and is defined as [math]\tau=F\times r[/math], where [math]r[/math] is the vector from the axis of rotation to the force. In simple machines, it can be calculated by the formula <div align=center>[math]\tau =F\cdot d,[/math]</div> where [math]d[/math], known as a moment arm, is calculated by drawing a perpendicular from the center to the force, as shown in the figure to the right.
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| − | Although the unit of torque is dimensionally equilvalent to Joule, Joule is not used for torque because torque and energy are different concepts. The rotation equivalent formula for energy and power, used in calculations with motors, is <div align = center>[math]E = \tau\cdot\theta,\ P=\tau\cdot\omega,[/math]</div> where [math]\theta[/math] is the angular displacement and [math]\omega[/math] is the angular velocity.
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| − | ===Simple and Compound Machine===
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| − | A simple machine is a device that changes the direction or magnitude of a force. There are 6 Simple Machines: Lever, Inclined plane, Wedge, Pulley, Wheels and axle, and Screw.
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| − | A compound machine is made of more than one simple machine. A compound Machine can allow more complex machines and more complex outputs and functions.
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| − | For example, a scissor combines three different simple machines: the handle is a lever; the pin to attach both sides is a wheel and axle; the blade is a wedge.
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| − | ===Mechanical Advantage and Efficiency===
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| − | Mechanical advantage is a measure of force amplification done by the machine. It is the ratio between the output and input force. In equations, <div align=center>[math]MA=\frac{F_{out}}{F_{in}}.[/math]</div>
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| − | The Ideal Mechanical Advantage (IMA) is the mechanical advantage under ideal conditions: when no friction or air resistance is present. The ideal condition is equivalent to that the machine does not dissipate energy: the energy input is equal to the energy output. Then, using the formula for work, [math]F_{out}d_{out}=F_{in}d_{in},[/math] so we have <div align=center>[math]IMA=\frac{F_{out}}{F_{in}}=\frac{d_{in}}{d_{out}}[/math].</div>
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| − | It is often easier to calculate IMA using the ratio of distances, because the ratio can be derived from the geometry of the machine.
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| − | The Actual Mechanical Advantage (AMA) is the mechanical advantage under real conditions: when friction and air resistance are present. It is always lower than the IMA due to energy losses associated with non-ideal conditions. AMA is calculated with the formula <div align=center>[math]AMA=\frac{F_{out}}{F_{in}},[/math]</div> where [math]F_{in}, F_{out}[/math] are experimentally determined.
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| − | The efficiency of a machine is a measure of how close to ideal a machine is. Efficiency is denoted by the Greek letter [math]\eta[/math], is at most 1, and can be computed using <div align=center>[math]\eta = \frac{AMA}{IMA}.[/math]</div>
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| − | ==Types of Simple Machines==
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| − | ===Lever===
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| − | [[Image:LeverIMAs.jpg|thumb|300px|Three classes of levers.]]
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| − | A lever is a rigid rod that can rotate around a fixed pivot point known as the fulcrum. The input force is called the effort and the output force is called the load or the resistance. There are three classes of levers based on the locations of the fulcrum, effort and load:
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| − | *Class 1: Effort and Load are on different sides of the fulcrum. Examples: seesaw, beam balance, scissor (handle).
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| − | *Class 2: Effort and load are on the same side of the fulcrum, with effort being farther away. Examples: wheel barrow, stapler, nail clipper. Class 2 levers always amplify the input force, at the cost of more distance by the input force. [math]IMA > 1.[/math]
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| − | *Class 3: Effort and load are on the same side of the fulcrum, with load being farther away. Examples: Fishing rod, tweezers. Class 3 levers always reduce the input force, but reduces the distance by the input force. [math]IMA < 1.[/math]
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| − | The lever is balanced if it is at rest or rotating at a constant rate. When a lever is balanced, the net torque is zero, so the effort torque is equal to the load toque, <div align = center>[math]F_{in}d_{in}=F_{out}d_{out},\ IMA = \frac{d_{in}}{d_{out}}.[/math]</div>
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| − | ===Pulley===
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| − | [[Image:Fixed_pulley_vs_movable_pulley.jpg|thumb|200px|Fixed and Movable pulley.]]
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| − | A pulley is a wheel on an axle that either changes either direction or magnitude of of force:
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| − | * A fixed pulley has an axle attached to a structure. A fixed pulley changes the direction of the force on a rope or belt that moves along its circumference. It has an IMA of 1.
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| − | * A movable pulley has an axle in a movable block. A movable pulley is supported by two parts of the same rope. It has an IMA of 2.
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| − | ===Wheels and axle===
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| − | ===Inclined Plane===
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| − | ===Screw===
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| − | ===Wedge===
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| − | ==Common Compound Machines==
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| − | ===Differential Pulley===
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| − | ==History of Simple Machines==
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| − | * Archimedes studied the lever, pulley and the screw around 3rd century BC, and discovered the principle of mechanical advantage in the lever. He also invented the Archimedes Screw, a device to transfer water to higher elevations.
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| − | * Heron of Alexandria listed five devices in his book ''Mechanics'' that can "set a load in motion", the simple machines excluding the inclined plane, and with wheel and axle replaced by the windlass.
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| − | * The inclined plane was included as a simple machine after Simon Stevin derived its mechanical advantage in 1586.
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| − | * Galileo Galilei published the book ''Le Meccaniche'' (''On Mechanics'') in 1600, in which he expanded the theory behind simple machines. He was the first scientist to know that simple machines do not create energy, but only transform it.
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| − | * Sir Isaac Newton stated the Laws of Motion in his book ''Philosophiæ Naturalis Principia Mathematica'' in 1687.
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| − | Amontons’ Laws of friction, rediscovered by Amontons after da Vinci and expanded by Coulomb, explained the role of friction in simple machines.
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| − | ==Device Testing==
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| − | The device testing for Compound Machines involves using a series of 2 levers to determine the mass ratios of 3 weights as quickly and accurately as possible.
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| − | ====Construction Restrictions====
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| − | The device must be made of a Class 1 and Class 2 lever connected in series. The device must fit inside a box of size 100cm * 100cm * 50cm during impound, and the beams must have length at most 50cm. The device can be made out of anything except anything electronic and must not include springs. Students are not allowed to bring any masses of one's own into the competition to determine the weights of the unknown masses.
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| − | ====During Competition====
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| − | The supervisors will provide 3 masses, labeled A, B, and C. Teams have a maximum of 4 minutes to determine the mass ratios A/B and B/C using the device.
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| − | The device testing is a tradeoff between speed and accuracy: The total score for device testing is a sum of time and accuracy scores, where
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| − | Time Score = [math]\frac{240 - \text{Elapsed Time in Seconds}}{240}\cdot10,[/math]
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| − | Mass (Accuracy) Score = [math]\left(1-\frac{|\text{Actual Ratio} - \text{Calculated Ratio}|}{\text{Actual Ratio}}\right)\cdot20.[/math]
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| − | Then, for every 1% of improvement in accuracy, one should take at most 3.6 more seconds.
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