User:Prof McCarthy/simple machine

Mechanical advantage
The mechanical advantage of an inclined plane is the ratio of the weight of the load on the ramp to the force required to pull it up the ramp. Assuming that no energy is dissipated or stored in the movement of the load, then this mechanical advantage can be computed from the dimensions of the ramp.

In order to show this, let the position of a rail car A on along ramp be given by the coordinates


 * $$\mathbf{r}_A = R (\cos\theta, \sin\theta),$$

where R is the distance along the ramp. The velocity of the car up the ramp is given by


 * $$\mathbf{v}_A = V (\cos\theta, \sin\theta).$$

Because the movement of the load does not dissipate or store energy, the power into load generated by the force F must equal the power out which is the lift of the weight W of the load.

The input power pulling the car up the ramp is given by


 * $$P_{\mathrm{in}} = FV,\!$$

and the power out is


 * $$P_{\mathrm{out}} = \mathbf{W}\cdot\mathbf{v}_A = (0, W)\cdot V (\cos\theta, \sin\theta) = WV\sin\theta$$

Equate the power in to the power out in order to obtain the mechanical advantage as


 * $$ MA = \frac{W}{F} = \frac{1}{\sin\theta}.$$

To calculate the forces on an object placed on an inclined plane, consider the three forces acting on it.


 * 1) The normal force (N) exerted on the body by the plane due to the force of gravity i.e. mg cos θ
 * 2) the force due to gravity (mg, acting vertically downwards) and
 * 3) the frictional force (f) acting parallel to the plane.

We can decompose the gravitational force into two vectors, one perpendicular to the plane and one parallel to the plane. Since there is no movement perpendicular to the plane, the component of the gravitational force in this direction (mg cos θ) must be equal and opposite to normal force exerted by the plane, N. If the remaining component of the gravitational force parallel to the surface (mg sin θ) is greater than the static frictional force fs – then the body will slide down the inclined plane with acceleration (g sin θ − fk/m), where fk is the kinetic friction force – otherwise it will remain stationary.

When the slope angle (θ) is zero, sin θ is also zero so the body does not move.

The MA or Mechanical advantage(ratio of load to effort) of the inclined plane equals to length of the plane over the height of the plane, in an ideal case where efficiency is 100%.

To calculate the MA (Mechanical Advantage) of an inclined plane, divide the length by the height of the ramp.

M.A=Length of the sloping surface / Height of the plane.(or) 1 / sin θ

Example: The height of the ramp = 1 meter The length of the ramp = 5 meters Divide 5 by 1=5 ma= 5

Velocity Ratio= height of the inclined plane / Length of the sloping surface

A lever is modeled as a rigid bar connected a ground frame by a hinged joint called a fulcrum. The lever is operated by applying an input force FA at a point A located by the coordinate vector rA on the bar. The lever then exerts an output force FB at the point B located by rB.  The rotation of the lever about the fulcrum P is defined by the rotation angle θ.

Let the coordinate vector of the point P that defines the fulcrum be rP, and introduce the lengths
 * $$ a = |\mathbf{r}_A - \mathbf{r}_P|, \quad  b = |\mathbf{r}_B -  \mathbf{r}_P|, $$

which are the distances from the fulcrum to the input point A and to the output point B, respectively.

Now introduce the unit vectors eA and eB from the fulcrum to the point A and B, so
 * $$ \mathbf{r}_A -  \mathbf{r}_P = a\mathbf{e}_A, \quad \mathbf{r}_B -  \mathbf{r}_P = b\mathbf{e}_B.$$

This notation allows us to define the velocity of the points A and B as
 * $$ \mathbf{v}_A = \dot{\theta} a \mathbf{e}_A^\perp, \quad \mathbf{v}_B = \dot{\theta} b \mathbf{e}_B^\perp,$$

where eA⊥ and eB⊥ are unit vectors perpendicular to eA and eB, respectively.

The angle θ is the generalized coordinate that defines the configuration of the lever, therefore using the formula above for forces applied to a one degree-of-freedom mechanism, the generalized force is given by
 * $$ F_\theta = \mathbf{F}_A \cdot \frac{\partial\mathbf{v}_A}{\partial\dot{\theta}} - \mathbf{F}_B \cdot \frac{\partial\mathbf{v}_B}{\partial\dot{\theta}}= a(\mathbf{F}_A \cdot \mathbf{e}_A^\perp) - b(\mathbf{F}_B \cdot \mathbf{e}_B^\perp).$$

Now, denote as FA and FB the components of the forces that are perpendicular to the radial segments PA and PB. These forces are given by
 * $$ F_A = \mathbf{F}_A \cdot \mathbf{e}_A^\perp, \quad F_B = \mathbf{F}_B \cdot \mathbf{e}_B^\perp.$$

This notation and the principle of virtual work yield the formula for the generalized force as
 * $$ F_\theta = a F_A - b F_B = 0. \,\!$$

The ratio of the output force FB to the input force FA is the mechanical advantage of the lever, and is obtained from the principle of virtual work as
 * $$ MA = \frac{F_B}{F_A} = \frac{a}{b}.$$

This equation shows that if the distance a from the fulcrum to the point A where the input force is applied is greater than the distance b from fulcrum to the point B where the output force is applied, then the lever amplifies the input force. If the opposite is true that the distance from the fulcrum to the input point A is less than from the fulcrum to the output point B, then the lever reduces the magnitude of the input force.

This is the law of the lever, which was proven by Archimedes using geometric reasoning.

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Block and tackle
A set of pulleys assembled so they rotate independently on the same axle form a block. Two blocks with a rope is attached to one of the blocks and threaded through its pulleys form a block and tackle. A block and tackle is assembled so one block is attached to fixed mounting point and the other is attached to the moving load. The mechanical advantage of the block and tackle is equal to the number of parts of the rope that support the moving block.

In the diagram on the right the mechanical advantage of each of the block and tackle assemblies shown is as follows:
 * Gun Tackle: 2
 * Luff Tackle: 3
 * Double Tackle: 4
 * Gyn Tackle: 5
 * Threefold purchase: 6

These are different types of pulley systems:
 * Fixed: A fixed pulley has an axle mounted in bearings attached to a supporting structure. A fixed pulley is change the direction of the force on a rope or belt that moves along its circumference.  Mechanical advantage is gained by combining a fixed pulley with a movable pulley or another fixed pulley of a different diameter.
 * Movable: A movable pulley has an axle in a movable block. A single movable pulley is supported by two parts of the same rope and has a mechanical advantage of 2.
 * Compound: A combination of fixed and a movable pulleys forms a block and tackle. A block and tackle can have several pulleys are mounted on each axle, further increasing the mechanical advantage.

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A  simple machine is a mechanical device that changes the direction or magnitude of a force. In general, they can be defined as the simplest mechanisms that provide mechanical advantage (also called leverage).

Usually the term refers to the six classical simple machines which were defined by Renaissance scientists: These simple machines fall into two classes; those dependent on the vector resolution of forces (inclined plane, wedge, screw) and those in which there is an equilibrium of torques (lever, pulley, wheel).
 * Lever
 * Wheel and axle
 * Pulley
 * Inclined plane
 * Wedge
 * Screw

A simple machine is an elementary device that has a specific movement (often called a mechanism), which can be combined with other devices and movements to form a machine. The view of machines as decomposable into simple machines arose in the Renaissance as an interpretation of Greek texts on technology.

Simple machines are the elementary "building blocks" of more complicated machines. For example, wheels, levers, and pulleys are all used in the mechanism of a bicycle. The mechanical advantage of a compound machine is just the product of the mechanical advantages of the simple machines of which it is composed.

A simple machine uses a single applied force to do work against a single load force. Ignoring friction losses, the work done on the load is equal to the work done by the applied force. They can be used to increase the amount of the output force, at the cost of a proportional decrease in the distance moved by the load. The ratio of the output to the input force is called the mechanical advantage.

A page from a 1728 text by Ephraim Chambers (see figure to the right) shows more simple machines. By the late 1800's Franz Reuleaux identified hundreds of simple machines. Models of these devices can be found at Cornell's KMODDL site.

History
The idea of a "simple machine" originated with the Greek philosopher Archimedes around the 3rd century BC, who studied the "Archimedean" simple machines: lever, pulley, and screw. He discovered the principle of mechanical advantage in the lever. Later Greek philosophers defined the classic five simple machines (excluding the inclined plane) and were able to roughly calculate their mechanical advantage. Heron of Alexandria (ca. 10–75 AD) in his work Mechanics lists five mechanisms that can "set a load in motion"; lever, windlass, pulley, wedge, and screw, and describes their fabrication and uses. However the Greeks' understanding was limited to the statics of simple machines; the balance of forces, and did not include dynamics; the tradeoff between force and distance, or the concept of work.

During the Renaissance the dynamics of the Mechanical Powers, as the simple machines were called, began to be studied from the standpoint of how much useful work they could perform, leading eventually to the new concept of mechanical work. In 1586 Flemish engineer Simon Stevin derived the mechanical advantage of the inclined plane, and it was included with the other simple machines. The complete dynamic theory of simple machines was worked out by Italian scientist Galileo Galilei in 1600 in Le Meccaniche ("On Mechanics"). He was the first to understand that simple machines do not create energy, only transform it.

The classic rules of sliding friction in machines were discovered by Leonardo Da Vinci (1452–1519), but remained unpublished in his notebooks. They were rediscovered by Guillaume Amontons (1699) and were further developed by Charles-Augustin de Coulomb (1785).

Alternate definitions
Any list of simple machines is somewhat arbitrary; the central idea is that every mechanism that manipulates force should be able to be understood as a combination of devices on the list. Some variations that have been proposed to the classical list of six simple machines:
 * Some exclude the wedge from the list of simple machines, as it is a moving inclined plane.
 * The screw, being a helical inclined plane, is sometimes also excluded. This position is less accepted because a screw converts a rotational force (torque) to a linear force.
 * It has been said that the pulley, and wheel and axle can be viewed as unique forms of levers, leaving only the lever and the inclined plane as simple machines from which all others can be derived.
 * Hydraulic systems can also provide amplification of force, so some say they should be added to the list.

Mechanical advantage
An ideal simple machine does not dissipate or store energy, which means there is no friction in its movement or bending of its components. The result is that the power into the device equals the power out.

A simple machine is often described as having the property that it changes the direction or magnitude of a force. In general, they can be defined as the simplest mechanisms that use mechanical advantage (also called leverage) to multiply force. A simple machine uses a single applied force to do work against a single load force. Ignoring friction losses, the work done on the load is equal to the work done by the applied force. They can be used to increase the amount of the output force, at the cost of a proportional decrease in the distance moved by the load. The ratio of the output to the input force is called the mechanical advantage.

Although each machine works differently, the way they function is similar mathematically. In each machine, a force $$F_{in}\,$$ is applied to the device at one point, and it does work moving a load, $$F_{out}\,$$ at another point. Although some machines only change the direction of the force, such as a stationary pulley, most machines multiply (or divide) the magnitude of the force by a factor, the mechanical advantage, that can be calculated from the machine's geometry. For example, the mechanical advantage of a lever is equal to the ratio of its lever arms.

Simple machines do not contain a source of energy, so they cannot do more work than they receive from the input force. A simple machine with no friction or elasticity is called an ideal machine. Due to conservation of energy, in an ideal simple machine, the work output $$W_{out}\,$$  (that is done on the load) is equal to the work input  $$W_{in}\,$$  (from the applied force). The work is defined as the force multiplied by the distance it moves. So the applied force, times the distance the input point moves, $$d_{in}\,$$, must be equal to the load force, times the distance the load moves, $$d_{out}\,$$ :
 * $$W_{in} = W_{out} \,$$
 * $$F_{in}d_{in} = F_{out}d_{out}\,$$

So the ratio of output to input force, the mechanical advantage, of a frictionless machine is equal to the "distance ratio"; the ratio of input distance to output distance moved:


 * $$\frac{F_{out}}{F_{in}} = \frac{d_{in}}{d_{out}} \,$$      (Ideal Mechanical Advantage)

In the screw, which uses rotational motion, the input force should be replaced by the torque, and the distance by the angle the shaft is turned.

Friction and efficiency
All actual machines have some friction. When friction is included, the mechanical advantage of a simple machine is no longer equal to the "distance ratio" $$d_{in}/d_{out}\,$$  but also depends on the machine's efficiency. Due to conservation of energy, in a machine with friction all the work done on the machine by the input force, $$W_{in}$$ goes into either moving the load $$W_{out}$$ or is dissipated as heat by friction $$W_{fric}$$.


 * $$W_{in} = W_{out} + W_{fric} \, $$

The efficiency η of a machine is a number between 0 and 1 defined as the ratio of output work to input work


 * $$\eta \equiv W_{out} / W_{in} \, $$


 * $$W_{out} = \eta W_{in} \, $$

Work is defined as the force multiplied by the distance moved, so $$W_{in} = F_{in} d_{in} \, $$ and $$W_{out} = F_{out} d_{out} \, $$, and thus


 * $$F_{out} d_{out} = \eta F_{in} d_{in} \, $$


 * $$\frac {F_{out} }{F_{in} } = \eta \frac {d_{in} }{d_{out} } \, $$        (Actual Mechanical Advantage)

So in all practical machines, the mechanical advantage is always less than the distance ratio, and equal to the distance ratio din/dout multiplied by the efficiency η. So a real machine, with friction, will not be able to move as large a load as a corresponding ideal frictionless machine using the same input force.

Self-locking machines
In many simple machines, if the load force Fout on the machine is high enough in relation to the input force Fin, the machine will move backwards, with the load force doing work on the input force. So these machines can be used in either direction, with the driving force applied to either input point. For example, if the load force on a lever is high enough, the lever will move backwards, moving the input arm backwards against the input force. These are called "reversible", "non-locking" or "overhauling" machines, and the backward motion is called "overhauling". However in some machines, if the frictional forces are high enough, no amount of load force can move it backwards, even if the input force is zero. This is called a "self-locking", "nonreversible", or "non-overhauling" machine. These machines can only be set in motion by a force at the input, and when the input force is removed will remain motionless, "locked" by friction at whatever position they were left.

Self-locking occurs mainly in those machines which have large areas of sliding contact and therefore large frictional losses: the screw,  inclined plane, and wedge: A machine will be self-locking if and only if its efficiency η is below 50%:
 * The most common example is a screw. In most screws, applying torque to the shaft can cause it to turn, moving the shaft linearly to do work against a load, but no amount of axial load force against the shaft will cause it to turn backwards.
 * In an inclined plane, a load can be pulled up the plane by a sideways input force, but if the plane is not too steep and there is enough friction between load and plane, when the input force is removed the load will remain motionless and will not slide down the plane, regardless of its weight.
 * A wedge can be driven into a block of wood by force on the end, such as from hitting it with a sledge hammer, forcing the sides apart, but no amount of compression force from the wood walls will cause it to pop back out of the block.


 * $$\eta \equiv \frac {F_{out}/F_{in} }{d_{in}/d_{out} } < 0.50  \,$$

Whether a machine is self-locking depends on both the friction forces (coefficient of static friction) between its parts, and the distance ratio din/dout (ideal mechanical advantage). If both the friction and ideal mechanical advantage are high enough, it will self-lock.

Derivation
When a machine moves in the forward direction from point 1 to point 2, with the input force doing work on a load force, from conservation of energy
 * $$W_{i1,2} = W_{load} + W_{fric} \qquad \qquad (1)\,$$

When it moves backward from point 2 to point 1 with the load force doing work on the input force, the work lost to friction Wfric is the same
 * $$W_{load} = W_{i2,1} + W_{fric} \,$$

When the input force is removed, the machine will self-lock if the work dissipated in friction is greater than the work done by the load force moving it backwards
 * $$W_{load} < W_{fric} \,$$

From (1)
 * $$W_{load} < W_{i1,2} - W_{load} \,$$


 * $$2W_{load} < W_{i1,2} \,$$


 * $$\eta \equiv \frac {W_{load}}{W_{i1,2}} < \frac {1}{2} \,$$

Compound machines
A compound machine is a machine made up of a number of simple machines connected in series, with the output force of each providing the input force for the next. For example a bench vise consists of a lever (the vise's handle) in series with a screw, and a car's transmission consists of a number of gears (wheels and axles) in series. The mechanical advantage of the compound machine MAcompound is defined as the output force applied to the load by the last machine, divided by the input force applied to the first machine. As the force propagates through the machine, each simple machine scales the force by its own mechanical advantage, so the mechanical advantage of the compound machine is equal to the product of the mechanical advantages of each simple machine of which it is composed


 * $$\mathrm{MA}_{compound} =    \mathrm{MA}_1 \mathrm{MA}_2 \mathrm{MA}_3 \mathrm{MA}_4 \mathrm{MA}_5   \,$$

Proof:


 * $$\mathrm{MA}_{compound} =  \frac {F_{out1}} {F_{in1}} \frac {F_{out2}} {F_{in2}} \frac {F_{out3}} {F_{in3}}  \frac {F_{out4}} {F_{in4}} \frac {F_{out5}} {F_{in5}}   \,$$

Since the output force of each machine is the input of the next: $$F_{in(N+1)} = F_{outN}   \,$$,  so


 * $$\mathrm{MA}_{compound}  =  \frac {F_{out1}} {F_{in1}} \frac {F_{out2}} {F_{out1}} \frac {F_{out3}} {F_{out2}}  \frac {F_{out4}} {F_{out3}} \frac {F_{out5}} {F_{out4}}   \,$$


 * $$\mathrm{MA}_{compound}  =  \frac {F_{out5}} {F_{in1}}  \,$$

Similarly, the efficiency of the compound machine is equal to the product of the efficiencies of the simple machines


 * $$\eta_{compound} = \eta_1 \eta_2  \eta_3  \eta_4  \eta_5 \,$$

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Lever
A lever is constructed from a beam attached to ground by a hinge, or fulcrum. It is one of the six simple machines identified by Renaissance scientists. The word comes from the French lever, "to raise", cf. a levant. It amplifies an input force to provide a greater output force, and is said to provide leverage.  The ratio of the output force to the input force is the mechanical advantage of the lever.