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# formula for mechanical advantage of a pulley

From Wikipedia

In physics and engineering, mechanical advantage (MA) is the factor by which a mechanism multiplies the force or torque applied to it. Generally, the mechanical advantage is defined as follows:

MA = \frac{\text{output force}}{\text{input force}}

For an ideal (frictionless) mechanism, it is also equal to:

MA = \frac{\text{distance over which effort is applied}}{\text{distance over which the load is moved}}

For an ideal machine, the two equations can be combined, indicating that the force exerted IN to such a machine (denominator of first ratio) multiplied by the distance moved IN (numerator of second ratio) will equal the force exerted OUT of the machine multiplied by the distance moved OUT (i.e., work IN equals work OUT).

As an ideal example, using a block and tackle with 6 ropes, and a 600 pound load, the operator would be required to pull the rope 6 feet, and exert 100 pounds of force to lift the load 1 foot. Both equations show that the MA is 6. In the first equation, 100 pounds of force IN results in 600 pounds of force OUT. The second equation calculates only the ideal mechanical advantage (IMA) and ignores real world energy losses due to friction and other causes. Subtracting those losses from the IMA or using the first equation yields the actual mechanical advantage (AMA). The ratio of AMA to IMA is the mechanical efficiency of the system.

## Types

The ideal mechanical advantage (IMA), or theoretical mechanical advantage, is the mechanical advantage of an ideal machine. It is calculated using physics principles because no ideal machine actually exists.

The IMA of a machine can be found with the following formula:

IMA = \frac {D_E} {D_R}

where

DEequals the 'effort distance' (for alever, the distance from the fulcrum to where the effort is applied)
DRequals theresistance distance (for a lever, the distance from the fulcrum to where the resistance is encountered)

The actual mechanical advantage (AMA) is the mechanical advantage of a real machine. Actual mechanical advantage takes into consideration real world factors such as energy lost in friction.

The AMA of a machine is calculated with the following formula:

AMA = \frac {R} {E_\text{actual}}

where

R = resistance force obtained from the machine
Eactual = actual effort force applied to the machine

## Simple machines

The following simple machines exhibit a mechanical advantage:

• The beam shown is in static equilibrium around the fulcrum. This is due to the moment created by vector force "A" counterclockwise (moment A*a) being in equilibrium with the moment created by vector force "B" clockwise (moment B*b). The relatively low vector force "B" is translated in a relatively high vector force "A". The force is thus increased in the ratio of the forces A : B, which is equal to the ratio of the distances to the fulcrum b : a. This ratio is called the mechanical advantage. This idealised situation does not take into account friction. For more explanation, see also lever.
• Wheel and axle motion (e.g. screwdrivers, doorknobs): A wheel is essentially a lever with one arm the distance between the axle and the outer point of the wheel, and the other the radius of the axle. Typically this is a fairly large difference, leading to a proportionately large mechanical advantage. This allows even simple wheels with wooden axles running in wooden blocks to still turn freely, because their friction is overwhelmed by the rotational force of the wheel multiplied by the mechanical advantage.
• Pulley: Pulleys change the direction of a tension force on a flexible material, e.g. a rope or cable. In addition, a block and tackle of multiple pulleys creates mechanical advantage, by having the flexible material looped over several pulleys in turn. Adding more loops and pulleys increases the mechanical advantage.
• Screw: A screw is essentially an inclined plane wrapped around a cylinder. The run over the rise of this inclined plane is the mechanical advantage of a screw.

### Pulleys

Consider lifting a weight with rope and pulleys. A rope looped through a pulley attached to a fixed spot, e.g. a barn roof rafter, and attached to the weight is called a single pulley. It has an MA = 1 (assuming frictionless bearings in the pulley), meaning no mechanical advantage (or disadvantage) however advantageous the change in direction may be.

A single movable pulley has an MA of 2 (assuming frictionless bearings in the pulley). Consider a pulley attached to a weight being lifted. A rope passes around it, with one end attached to a fixed point above, e.g. a barn roof rafter, and a pulling force is applied upward to the other end with the two lengths parallel. In this situation the distance the lifter must pull the rope becomes twice the distance the weight travels, allowing the force applied to be halved. Note: if an additional pulley is used to change the direction of the rope, e.g. the person doing the work wants to stand on the ground instead of on a rafter, the mechanical advantage is not increased.

By looping more ropes around more pulleys we can continue to increase the mechanical advantage. For example if we have two pulleys attached to the rafter, two pulleys attached to the weight, one end attached to the rafter, and someone standing on the rafter pulling the rope, we have a mechanical advantage of four. Again note: if we add another pulley so that someone may stand on the ground and pull down, we still have a mechanical a

Question:OMG, I can't take it anymore and I seriously need help from professional and if you will answer, please give an proper explanation of why and how. Go to the web page below and tell me the mechanical advantage of number C. I need to ask someone who did this experiment before.

Answers:Sorry no web page there. I probably couldn't do it anyway though. Sorry.

Question:1) Inclined plane 2) Wedge 3) One lever (any class) 4) Wheel and axle 5) Fixed pulley 6) Moveable pulley 7) Block and tackle (MA 3)

Question:I am using a a trailer winch and a single pulley. Pulley is attached to ceiling. I must use the winch and pulley to raise the load approximately 12'. The winch has a mechanical advantage of 8.68. The load is 50kg. Since there are 2 ropes does this pulley system have a total mechanical advantage of 17.36? Also, how do I calculate the time required to lift the load with this pulley system?

Answers:mechanical advantage of pulley system is derived from counting the number of ropes between the pulleys. In your case, 2. So multiplying the 8.68 by 2 sounds correct. And it would take twice as long to lift than without the pulley system.

Question:If the Ideal Mechanical Advantage is 1, the Actual has to be less because no machine has perfect Mechanical Advantage, right? Then how do I find the AMA? And if you can, could you please help me with my presentation question? http://answers.yahoo.com/question/index;_ylt=AsNL1IcXDWUB_qD7tXfFEW7sy6IX;_ylv=3?qid=20090404075208AAUM2zg

Answers:The easiest way I use is to simply count the number of strands of rope you have between the object and the anchor. Assume MA=1 is a simple rope tied to the object, up over a pulley, and then into your hand. That's 1 strand. Now loop it around to the object again, MA=3. Then again MA=5. Each time you add two strands to the object.