Compound Machines

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Compound Machines
Physics & Study Event
Forum Threads 2015
2014
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Question Marathons 2015
2014
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Division C Champion West Windsor-Plainsboro High School North



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.

Compound Machines can be considered the Division C equivalent of the Division B event, Simple Machines.

General Overview

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.

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.

For the written test, teams are allowed one binder of reference materials that does not need to be impounded.

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.

Terms and Vocabulary

This section is similar to the section 2 of Simple Machines. However, it goes into some material in greater depth.

Force

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

[math]F=ma,[/math]

where [math]m[/math] is the mass of the object.

Energy

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].

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.

Work

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].

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

[math]W=F\cdot \Delta s = |F|\cdot |\Delta s|\cdot\cos \theta, [/math]

where [math]F, \Delta s[/math] are vectors, while work [math]W[/math] is a scalar.

Power

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]

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.

Conservation of Mechanical Energy

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.

Torque

Force and the corresponding moment arm.

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].

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
[math]\tau =F\cdot d,[/math]
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.


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
[math]E = \tau\cdot\theta,\ P=\tau\cdot\omega,[/math]
where [math]\theta[/math] is the angular displacement and [math]\omega[/math] is the angular velocity.

Simple and Compound Machine

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.

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.

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.

Mechanical Advantage and Efficiency

Mechanical advantage is a measure of force amplification done by the machine. It is the ratio between the output and input force. In equations,
[math]MA=\frac{F_{out}}{F_{in}}.[/math]
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
[math]IMA=\frac{F_{out}}{F_{in}}=\frac{d_{in}}{d_{out}}[/math].

It is often easier to calculate IMA using the ratio of distances, because the ratio can be derived from the geometry of the machine.

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
[math]AMA=\frac{F_{out}}{F_{in}},[/math]
where [math]F_{in}, F_{out}[/math] are experimentally determined. 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
[math]\eta = \frac{AMA}{IMA}.[/math]

Types of Simple Machines

This section is similar to the section 3 of Simple Machines. However, it goes into some material in greater depth.

Lever

Three classes of levers.

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:

  • Class 1: Effort and Load are on different sides of the fulcrum. Examples: seesaw, beam balance, scissor (handle).
  • 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]
  • 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]
Fixed and Movable pulley.
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,
[math]F_{in}d_{in}=F_{out}d_{out},\ IMA = \frac{d_{in}}{d_{out}}.[/math]

Pulley

A pulley is a wheel on an axle that either changes either direction or magnitude of of force:

  • 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.
  • 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.

Both types of pulleys are levers. A fixed pulley is a class 1 lever with the effort arm equal to the load arm; a movable pulley is a class 2 lever with the effort arm twice the length of the load arm.

A system of fixed and movable pulleys form a block and tackle or a compound pulley system, using multiple pulleys to reach a higher IMA.

Click on the image for solution to the compound system.

There are two methods to calculate the IMA of a compound system:

  • Use the formula [math]IMA = \frac{d_{in}}{d_{out}}[/math]. When the load is lifted by distance [math]d_{out}[/math], find the distance each pulley moves, then the distance the input force moves, using that the length of strings remain constant.
  • Use the formula [math]IMA = \frac{F_{out}}{F_{in}}[/math]. Then tension applied by the same string is equal, and since the system is balanced, one can also draw free body diagram for each pulley to calculate tension of the different strings.

Deciding which method to use comes from practice, although most often the two methods have similar difficulty. The solution to the compound system of a gun tackle using both of the methods can be found in the picture on the right.

Belt and Pulley System

A belt and pulley system has two or more fixed pulleys connected by a belt. The IMA is given by the ratio [math]\frac{r_{out}}{r_{in}}[/math], and the direction of rotation is the same between the pulleys, unless the belt is crossed in an X shape, in which case the directions of rotation are opposites.

Wheels and axle

Diagram of a wheel and axle.

A wheel and axle is a wheel or crank rigidly attached to an axle. Both parts of the system rotate together. Examples of a wheel and axle include screwdrivers and driving wheel.

The wheel and axle is a lever, where the center of rotation for both the wheel and the axle is the fulcrum, and the rigid bar is turned into a circle. The Mechanical Advantage of a wheel and axle is given by
[math]IMA = \frac{R}{r},[/math]
where [math]R[/math] is the radius of the wheel and [math]r[/math] is the radius of the axle.

Inclined Plane

Free body diagram for a wheel and axle.

An inclined plane, also known as a ramp, is a flat surface tilted at an angle. Objects may be rolled or slid to a higher level using an incline plane.

The mechanical advantage of an inclined plane is the ratio of the weight of the load to the minimum force required to pull it up the ramp. Let [math]l[/math] be the slant length of the plane, [math]h[/math] be the height difference between the top and bottom of the plane, and [math]\theta[/math] be the angle the plane forms with the ground. By drawing the free body diagram in the right, the input force is [math]\sin\theta=\frac{h}{l}[/math] of the weight of the box. Therefore, the mechanical advantage is
[math]MA = \frac{h}{l} = \sin\theta.[/math]

When friction is introduced, the minimum required force can be calculated using a free body diagram, with a frictional force towards the bottom of the plane with magnitude [math]|f|=|F_N|\cdot \mu = mg\mu\cos\theta[/math], where [math]\mu[/math] is the coefficient of friction.

Screw

A screw is an inclined plane wrapped around a cylinder. The curve formed by the inclined plane is known as the thread. The vertical distance between the threads is called the pitch of the screw. A screw is often combined with a screwdriver, which is a wheel and axle. The ISO regulates the sizes and shapes of the screws.

Let [math]r[/math] be the distance from the center to the thread, and [math]p[/math] be the pitch of the screw. Then, the ideal mechanical advantage of a screw is
[math]MA=\frac{2\pi r}{d}.[/math]

Wedge

Diagram of a wedge.

A wedge is a triangular shaped tool with two slanting sides ending in a sharp edge. It is used cut materials apart. Examples: knives, axes.

A zipper, another example of a wedge, consists of an upper triangular wedge and two lower wedges that close the teeth of the zipper.

Let [math]l[/math] be the length of the slanting side, and [math]w[/math] the width of the wedge, then the mechanical advantage is given by

[math]MA=\frac{l}{h}.[/math]

Common Compound Machines

Gear system

A diagram of gears.

A gear is a rotating machine part with teeth. Two gears with their teeth meshed together transmit torque. If the input gear has [math]n_{in}[/math] teeth, and the output gear has [math]n_{out}[/math] teeth, its mechanical advantage is given by [math]\frac{n_{out}}{n_{in}}[/math]. For most gears (ones with teeth on the outer surface), the rotation of the output gear are in opposite direction as the rotation of the input gear.

A gear system is similar to a belt and pulley system, where the mechanical advantage is given by the ratio of radii, instead of ratio of number of teeth.

Differential Pulley and Windlass

Diagram for a differential pulley and differential windlass.

A differential pulley is made of two pulleys and one string. It is known for its high mechanical advantage and relatively simple design. In the figure to the right, the left is a differential pulley, while the right is a differential windlass.

The axles in the middle are both fixed and rotate together. Let the outer one have radius [math]R[/math], and the inner one have radius [math]r[/math].

Then, when [math]F[/math] moves by [math]2\pi R[/math], both the larger and smaller axles rotate by one revolution. The movable pulley, and therefore the load, moves up by [math]\pi (R-r)[/math].

Therefore, the differential pulley has mechanical advantage [math]\frac{2}{1-\frac{r}{R}}[/math]. As the radius of the two axles become closer, the mechanical advantage becomes much bigger.

The mechanical advantage of a differential windlass can be calculated similarly, with the long handle providing even more mechanical advantage.

History of Simple Machines

This section is similar to the section 2 of Simple Machines.

  • 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.
  • 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.
  • The inclined plane was included as a simple machine after Simon Stevin derived its mechanical advantage in 1586.
  • 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.
  • Sir Isaac Newton stated the Laws of Motion in his book Philosophiæ Naturalis Principia Mathematica in 1687.

Amontons’ Laws of friction, rediscovered by Amontons after da Vinci and expanded by Coulomb, explained the role of friction in simple machines.

Device Testing

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.

Construction Restrictions

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.

During Competition

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.

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

Time Score = [math]\frac{240 - \text{Elapsed Time in Seconds}}{240}\cdot10,[/math]

Mass (Accuracy) Score = [math]\left(1-\frac{|\text{Actual Ratio} - \text{Calculated Ratio}|}{\text{Actual Ratio}}\right)\cdot20.[/math]

Then, for every 1% of improvement in accuracy, one should take at most 3.6 more seconds.

Resources

Soinc page on Compound Machines

Wikipedia page on Simple machine, Lever, Pulley, Wheel and Axle, Inclined Plane, Screw, and Wedge.

Physics Classroom on Work, Energy and Power.

Hyperphysics on simple machines.