4595:
40:
231:
378:
243:
270:
51:
350:
675:
330:
428:
362:
1004:
255:
2019:
285:
4048:
2982:
453:. If the string was heavier on one side than the other, and began to slide right or left under its own weight, when each bead had moved to the position of the previous bead the string would be indistinguishable from its initial position and therefore would continue to be unbalanced and slide. This argument could be repeated indefinitely, resulting in a circular
396:
2781:
476:
Since the beads are at equal intervals on the string, the total number of beads supported by each plane, the total load, is proportional to the length of the plane. Since the input supporting force, the tension in the string, is the same for both, the mechanical advantage of each plane is
4075:
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. If energy is not dissipated or stored in the movement of the load, then this mechanical advantage can be computed from the dimensions of the ramp.
1610:
from this equation is the force needed to hold the load motionless on the inclined plane, or push it up at a constant velocity. If the input force is greater than this, the load will accelerate up the plane. If the force is less, it will accelerate down the plane.
4405:
525:
to be recognised as a machine. This is probably because it is a passive and motionless device (the load is the moving part), and also because it is found in nature in the form of slopes and hills. Although they understood its use in lifting heavy objects, the
78:
defined by
Renaissance scientists. Inclined planes are used to move heavy loads over vertical obstacles. Examples vary from a ramp used to load goods into a truck, to a person walking up a pedestrian ramp, to an automobile or railroad train climbing a grade.
468:
The portion of the chain hanging below the inclined planes is symmetrical, with an equal number of beads on each side. It exerts an equal force on each side of the string. Therefore, this portion of the string can be cut off at the edges of the planes
3819:
441:
above). A loop of string with beads at equal intervals is draped over the inclined planes, with part of the string hanging down below. The beads resting on the planes act as loads on the planes, held up by the tension force in the string at point
497:
built by ancient civilizations such as the Romans are examples of early inclined planes that have survived, and show that they understood the value of this device for moving things uphill. The heavy stones used in ancient stone structures such as
1425:
3932:
3095:
2216:
the load will be stationary or in steady motion if the sum of the forces on it is zero. Since the direction of the frictional force is opposite for the case of uphill and downhill motion, these two cases must be considered separately:
485:
when let go. Even if the chain is released with a zero angular momentum, motion including oscillations is possible unless the chain is initially in its equilibrium configuration, a supposition which would make the argument circular.
1576:
1714:
Therefore, more input force is required, and the mechanical advantage is lower, than if friction were not present. With friction, the load will only move if the net force parallel to the surface is greater than the frictional force
2977:{\displaystyle {\frac {F_{w}}{F_{i}}}={\frac {1}{\sin \theta +\tan \phi \cos \theta }}={\dfrac {1}{\sin \theta +{\dfrac {\sin \phi }{\cos \phi }}\cos \theta }}={\frac {\cos \phi }{\sin \theta \cos \phi +\cos \theta \sin \phi }}\,}
756:
794:
of a simple machine as defined as the ratio of the output force exerted on the load to the input force applied.The inclined plane the output load force is just the gravitational force of the load object on the plane, its weight
3679:
2738:
561:(1570) proposed the incorrect solution that the input force is proportional to the angle of the plane. Then at the end of the 16th century, three correct solutions were published within ten years, by Michael Varro (1584),
1202:
Work is defined as the force multiplied by the displacement an object moves. The work done on the load is equal to its weight multiplied by the vertical displacement it rises, which is the "rise" of the inclined plane
437:(Stevinus) derived the mechanical advantage of the inclined plane by an argument that used a string of beads. He imagined two inclined planes of equal height but different slopes, placed back-to-back as in a prism (
4473:
903:
1495:
3598:
2657:
1352:
157:
The term may also refer to a specific implementation; a straight ramp cut into a steep hillside for transporting goods up and down the hill. This may include cars on rails or pulled up by a cable system; a
1709:
1263:
4720:
3262:
2321:
4589:
3335:
2394:
4214:
4146:
573:
is the most well-known, because of its originality and use of a string of beads (see box). In 1600, Italian scientist
Galileo Galilei included the inclined plane in his analysis of simple machines in
2010:
for which the load is stationary, neither sliding up or down the plane, whereas with a frictionless inclined plane there is only one particular value of input force for which the load is stationary.
4947:
3459:
2518:
206:
a railroad car is pulled up a steep inclined plane using cables. Inclined planes also allow heavy fragile objects, including humans, to be safely lowered down a vertical distance by using the
4998:
in the study of bodies sliding on plane surfaces, the angle between the perpendicular to the surface and the resultant force (between the body and the surface) when the body begins to slide.
4287:
4650:
4529:
1197:
553:
that the inclined plane was solved mathematically and classed with the other simple machines. The first correct analysis of the inclined plane appeared in the work of 13th century author
5737:
3513:
2572:
4276:
305:
have gentle ramps to limit their slope, to ensure that pedestrians can keep traction. Inclined planes are also used as entertainment for people to slide down in a controlled way, in
3142:
1978:
3723:
5089:
The Genius of
Archimedes – 23 Centuries of Influence on Mathematics, Science and Engineering: Proceedings of an International Conference Held at Syracuse, Italy, June 8–10, 2010
4725:
so a 100 lb tension force on the cable will lift a 4810 lb load. The grade of this incline is 2%, which means the angle θ is small enough that sin θ≈tan θ.
4034:
3972:
3716:
3383:
2775:
2442:
1787:
481:
As pointed out by
Dijksterhuis, Stevin's argument is not completely tight. The forces exerted by the hanging part of the chain need not be symmetrical because the hanging part
4001:
1648:
2202:
of the plane on the load acts parallel to the surface, and is always in a direction opposite to the motion of the object. It is equal to the normal force multiplied by the
1114:
993:
962:
4036:: The mechanical advantage is positive. In the absence of applied force the load will slide down the plane, and requires some positive (uphill) force to hold it motionless
3974:: The mechanical advantage is negative. In the absence of applied force the load will remain motionless, and requires some negative (downhill) applied force to slide down.
1141:
931:
792:
2200:
2166:
2133:
2103:
2008:
1817:
1740:
1363:
1293:
1059:
1032:
847:
820:
530:
philosophers who defined the other five simple machines did not include the inclined plane as a machine. This view persisted among a few later scientists; as late as 1826
1608:
5885:
3843:
3006:
1865:
1448:
669:
1623:
between the plane and the load, as for example with a heavy box being slid up a ramp, some of the work applied by the input force is dissipated as heat by friction,
1913:
1885:
1933:
1841:
1887:
the component of gravitational force parallel to the plane will be too small to overcome friction, and the load will remain motionless. This angle is called the
502:
are believed to have been moved and set in place using inclined planes made of earth, although it is hard to find evidence of such temporary building ramps. The
5824:
1506:
5710:
697:
297:
Other inclined planes are built into permanent structures. Roads for vehicles and railroads have inclined planes in the form of gradual slopes, ramps, and
90:
of an inclined plane, the factor by which the force is reduced, is equal to the ratio of the length of the sloped surface to the height it spans. Owing to
4007:'. The mechanical advantage is infinite. With no applied force, load will not slide, but the slightest negative (downhill) force will cause it to slide.
3606:
2665:
5429:
639:
or steepness. The smaller the slope, the larger the mechanical advantage, and the smaller the force needed to raise a given weight. A plane's slope
538:". The problem of calculating the force required to push a weight up an inclined plane (its mechanical advantage) was attempted by Greek philosophers
5680:
5569:
4955:
5653:
5022:
5039:
Angle of repose is the limiting angle of inclination of a plane when a body, placed on the inclined plane, just starts sliding down the plane.
301:
to allow vehicles to surmount vertical obstacles such as hills without losing traction on the road surface. Similarly, pedestrian paths and
4416:
5623:
4917:
855:
1460:
1581:
So the mechanical advantage of a frictionless inclined plane is equal to the reciprocal of the sine of the slope angle. The input force
3522:
2581:
510:
ramps enabled ancient armies to surmount fortress walls. The ancient Greeks constructed a paved ramp 6 km (3.7 miles) long, the
5941:
5052:
4978:
1301:
2224:
The total force on the load is toward the uphill side, so the frictional force is directed down the plane, opposing the input force.
1660:
1209:
74:, with one end higher than the other, used as an aid for raising or lowering a load. The inclined plane is one of the six classical
4664:
3182:
2241:
4540:
3268:
2327:
4161:
4093:
5002:
for any given granular material the steepest angle to the horizontal at which a heaped surface will stand in stated conditions.
5215:
5895:
5864:
5834:
5747:
5720:
5690:
5663:
5633:
5606:
5579:
5545:
5515:
5449:
5374:
5347:
5283:
5256:
5198:
5153:
5123:
5096:
5062:
5032:
4988:
4927:
4895:
4827:
4795:
4658:
The
Liverpool Minard inclined plane has the dimensions 1804 meters by 37.50 meters, which provides a mechanical advantage of
5854:
4885:
3392:
2451:
5300:
4400:{\displaystyle P_{\mathrm {out} }=\mathbf {W} \cdot \mathbf {v} =(0,W)\cdot V(\cos \theta ,\sin \theta )=WV\sin \theta .}
473:, leaving only the beads resting on the inclined planes, and this remaining portion will still be in static equilibrium.
4608:
1847:
between the two surfaces, which varies with the material. When no input force is applied, if the inclination angle
5439:
5596:
2213:
601:
4492:
1891:
and depends on the composition of the surfaces, but is independent of the load weight. It is shown below that the
1148:
4602:
Example: If the height of a ramp is H = 1 meter and its length is L = 5 meters, then the mechanical advantage is
1007:
Instrumented inclined plane used for physics education, around 1900. The lefthand weight provides the load force
31:
4853:
3468:
2527:
2203:
1844:
130:
110:, but the inclined plane allows the same work to be done with a smaller force exerted over a greater distance.
493:
Inclined planes have been used by people since prehistoric times to move heavy objects. The sloping roads and
5934:
4236:
5505:
3814:{\displaystyle {\frac {F_{w}}{F_{i}}}={\frac {\cos \phi }{\sin \theta \cos \phi -\cos \theta \sin \phi }}\,}
3104:
1940:
531:
4594:
4478:
The mechanical advantage of an inclined plane can also be calculated from the ratio of length of the ramp
3165:
The total force on the load is toward the downhill side, so the frictional force is directed up the plane.
593:
4012:
3950:
3688:
3344:
2747:
2403:
1748:
5794:
5535:
3979:
1626:
1420:{\displaystyle {\text{MA}}={\frac {F_{\rm {w}}}{F_{\rm {i}}}}={\frac {\text{Length}}{\text{Rise}}}\,}
143:
5764:
1092:
970:
939:
6010:
5927:
1119:
911:
849:
exerted on the object, parallel to the plane, to move it up the plane. The mechanical advantage is
772:
167:
5405:
2178:
2144:
2111:
2081:
1986:
1795:
1718:
1271:
1037:
1010:
825:
798:
5426:(c. 10 – 70 CE) consist of the five classical simple machines, excluding the inclined plane. –
5419:
4745:
3945:
is less than given in this equation, the load will slide down the plane. There are three cases:
1584:
609:
114:
4819:
3927:{\displaystyle \mathrm {MA} ={\frac {F_{w}}{F_{i}}}={\frac {\cos \phi }{\sin(\theta -\phi )}}\,}
3090:{\displaystyle \mathrm {MA} ={\frac {F_{w}}{F_{i}}}={\frac {\cos \phi }{\sin(\theta +\phi )}}\,}
577:("On Mechanics"), showing its underlying similarity to the other machines as a force amplifier.
5979:
3824:
2987:
1892:
1651:
1082:
147:
91:
5339:
5143:
5113:
4785:
39:
5273:
5188:
5084:
1850:
1433:
654:
24:
5472:
5364:
5246:
4223:
to move the load up the ramp equals the power out, which is the vertical lift of the weight
146:
can be considered a moving inclined plane or two inclined planes connected at the base. The
4735:
4072:
1898:
1870:
767:
626:
543:
87:
1918:
1826:
1654:, the sum of the output work and the frictional energy losses is equal to the input work
1571:{\displaystyle {\text{MA}}={\frac {F_{\rm {w}}}{F_{\rm {i}}}}={\frac {1}{\sin \theta }}\,}
557:, however his solution was apparently not communicated to other philosophers of the time.
457:, which is absurd. Therefore, it is stationary, with the forces on the two sides at point
142:
Two other simple machines are often considered to be derived from the inclined plane. The
8:
5423:
1357:
Substituting these values into the conservation of energy equation above and rearranging
539:
230:
211:
151:
5399:
4812:
751:{\displaystyle \theta =\tan ^{-1}{\bigg (}{\frac {\text{Rise}}{\text{Run}}}{\bigg )}\,}
589:
554:
515:
450:
121:, is the maximum angle at which a load can rest motionless on an inclined plane due to
71:
5891:
5882:
This derives slightly more general equations which cover force applied at any angle:
5860:
5830:
5743:
5716:
5686:
5659:
5629:
5602:
5575:
5541:
5511:
5445:
5370:
5343:
5332:
5279:
5252:
5194:
5149:
5119:
5092:
5058:
5028:
4984:
4923:
4891:
4823:
4791:
2071:
585:
503:
377:
103:
95:
242:
50:
5223:
3674:{\displaystyle {\frac {F_{w}}{F_{i}}}={\frac {1}{\sin \theta -\mu \cos \theta }}\,}
2733:{\displaystyle {\frac {F_{w}}{F_{i}}}={\frac {1}{\sin \theta +\mu \cos \theta }}\,}
2168:
of the inclined plane on the load, supporting it. This is directed perpendicular (
558:
454:
306:
269:
261:
218:
allow people to rapidly and safely reach the ground from the height of a passenger
215:
5466:
588:(1452-1519), but remained unpublished in his notebooks. They were rediscovered by
5537:
Thinking With
Objects: The Transformation of Mechanics in the Seventeenth Century
4004:
2236:
The equilibrium equations for forces parallel and perpendicular to the plane are
1888:
605:
566:
318:
275:
183:
118:
86:
than lifting it straight up, at a cost of an increase in the distance moved. The
43:
3936:
This is the condition for impending motion down the plane; if the applied force
349:
5989:
5950:
2169:
2138:
The force of the plane on the load. This can be resolved into two components:
1086:
679:
597:
527:
522:
99:
75:
20:
5308:
651:". It can also be expressed by the angle the plane makes with the horizontal,
6004:
4755:
4750:
4740:
674:
636:
203:
195:
163:
55:
4155:
is the distance along the ramp. The velocity of the car up the ramp is now
5914:
5565:
5222:. Prof. Steve Dutch's page, Univ. of Wisconsin at Green Bay. Archived from
4410:
Equate the power in to the power out to obtain the mechanical advantage as
4052:
2034:
1820:
570:
562:
434:
207:
2105:
exerted on the load to move it, which acts parallel to the inclined plane.
569:(1592). Although it was not the first, the derivation of Flemish engineer
329:
5501:
5491:
5244:
4598:
Layout of the cable drive system for the
Liverpool Minard inclined plane.
550:
310:
3157:
is greater than given by this equation, the load will move up the plane.
5085:"Simon Stevin and the rise of Archimedean mechanics in the Renaissance"
4468:{\displaystyle \mathrm {MA} ={\frac {W}{F}}={\frac {1}{\sin \theta }}.}
499:
254:
187:
126:
5141:
4954:. College of Engineering, Univ. of Colorado at Boulder. Archived from
1073:. This condition might be approached if the object is rolling like a
1069:
between the object being moved and the plane, the device is called an
368:
314:
199:
191:
159:
1823:
between the load and the plane, directed normal to the surface, and
898:{\displaystyle \mathrm {MA} ={\frac {F_{\text{w}}}{F_{\text{i}}}}\,}
427:
1620:
1490:{\displaystyle \sin \theta ={\frac {\text{Rise}}{\text{Length}}}\,}
1066:
1003:
581:
494:
361:
302:
298:
219:
122:
107:
5708:
5418:
for example, the lists of simple machines left by Roman architect
4655:
which means that a 20 lb force will lift a 100 lb load.
30:
This article is about the physical structure. For other uses, see
5919:
5507:
Mechanics and natural philosophy before the scientific revolution
4060:
3593:{\displaystyle F_{i}+\mu F_{w}\cos \theta -F_{w}\sin \theta =0\,}
2652:{\displaystyle F_{i}-\mu F_{w}\cos \theta -F_{w}\sin \theta =0\,}
2064:
511:
190:
to get over vertical obstacles without exceeding their strength.
19:"Ramp" redirects here. For the financial technology company, see
1295:
on the object times the diagonal length of the inclined plane.
933:
of an ideal inclined plane without friction is sometimes called
5974:
5190:
Not etched in stone: essays on ritual memory, soul, and society
5142:
Feynman, Richard P.; Robert B. Leighton; Matthew Sands (1963).
4760:
2018:
1347:{\displaystyle W_{\rm {in}}=F_{\rm {i}}\cdot {\text{Length}}\,}
1078:
1074:
643:
is equal to the difference in height between its two ends, or "
340:
5245:
Moffett, Marian; Michael W. Fazio; Lawrence
Wodehouse (2003).
1704:{\displaystyle W_{\text{in}}=W_{\text{fric}}+W_{\text{out}}\,}
1258:{\displaystyle W_{\rm {out}}=F_{\rm {w}}\cdot {\text{Rise}}\,}
5984:
5969:
5401:
The kinematics of machinery: Outlines of a theory of machines
4715:{\displaystyle \mathrm {MA} ={\frac {W}{F}}=1804/37.50=48.1,}
4064:
3257:{\displaystyle \sum F_{\|}=F_{i}+F_{f}-F_{w}\sin \theta =0\,}
2316:{\displaystyle \sum F_{\|}=F_{i}-F_{f}-F_{w}\sin \theta =0\,}
631:
507:
284:
83:
5186:
4584:{\displaystyle \mathrm {MA} ={\frac {W}{F}}={\frac {L}{H}}.}
4047:
5334:
Sticks, stones, and shadows: building the
Egyptian pyramids
5111:
4056:
3330:{\displaystyle \sum F_{\perp }=F_{n}-F_{w}\cos \theta =0\,}
2389:{\displaystyle \sum F_{\perp }=F_{n}-F_{w}\cos \theta =0\,}
2060:
521:
However the inclined plane was the last of the six classic
70:, is a flat supporting surface tilted at an angle from the
4209:{\displaystyle \mathbf {v} =V(\cos \theta ,\sin \theta ).}
4141:{\displaystyle \mathbf {r} =R(\cos \theta ,\sin \theta ),}
2070:
A load resting on an inclined plane, when considered as a
395:
5682:
Nanoscience: friction and rheology on the nanometer scale
5148:. USA: California Inst. of Technology. pp. 4.4–4.5.
1983:
With friction, there is always some range of input force
5115:'Magic is no magic': The wonderful world of Simon Stevin
4230:
The input power pulling the car up the ramp is given by
5434:. London: Walton and Maberly; John Murray. p. 722.
5050:
546:(c. 290 - 350 CE), but their solutions were incorrect.
536:...is no more a machine than is the slope of a mountain
5709:
Handley, Brett; David M. Marshall; Craig Coon (2011).
5628:. London: Cambridge University Press. pp. 47–48.
4486:
because the sine of the angle of the ramp is given by
3174:
Derivation of mechanical advantage for downhill motion
182:
to load and unload goods on trucks, ships and planes.
5444:. USA: Courier Dover Publications. pp. 98, 120.
4948:"Lesson 04:Slide Right on By Using an Inclined Plane"
4667:
4611:
4543:
4495:
4419:
4290:
4239:
4219:
Because there are no losses, the power used by force
4164:
4096:
4015:
3982:
3953:
3846:
3726:
3691:
3609:
3525:
3471:
3454:{\displaystyle F_{i}+\mu F_{n}-F_{w}\sin \theta =0\,}
3395:
3347:
3271:
3185:
3107:
3009:
2874:
2855:
2784:
2750:
2668:
2584:
2530:
2513:{\displaystyle F_{i}-\mu F_{n}-F_{w}\sin \theta =0\,}
2454:
2406:
2330:
2244:
2181:
2147:
2114:
2084:
1989:
1943:
1921:
1901:
1873:
1853:
1829:
1798:
1751:
1742:
opposing it. The maximum friction force is given by
1721:
1663:
1629:
1587:
1509:
1463:
1436:
1366:
1304:
1274:
1212:
1151:
1122:
1095:
1040:
1013:
973:
964:
while the MA when friction is included is called the
942:
914:
858:
828:
801:
775:
700:
657:
150:
consists of a narrow inclined plane wrapped around a
5771:. Dept. of Physics and Astronomy, Georgia State Univ
5563:
4915:
4055:
that is perpendicular to the plane, W=mg, where m =
2233:
Derivation of mechanical advantage for uphill motion
82:
Moving an object up an inclined plane requires less
5915:
An interactive simulation of
Physics inclined plane
4042:
5829:. New Delhi: Pearson Education India. p. 69.
5795:"Lab Mech14:The Inclined Plane - A Simple Machine"
5651:
5331:
4941:
4939:
4811:
4714:
4644:
4583:
4523:
4467:
4399:
4270:
4208:
4140:
4028:
3995:
3966:
3926:
3813:
3710:
3673:
3592:
3507:
3453:
3377:
3329:
3256:
3136:
3089:
2976:
2769:
2732:
2651:
2566:
2512:
2436:
2388:
2315:
2194:
2160:
2127:
2097:
2002:
1972:
1927:
1907:
1879:
1859:
1835:
1811:
1781:
1734:
1703:
1642:
1602:
1570:
1489:
1442:
1419:
1346:
1287:
1257:
1191:
1135:
1108:
1053:
1026:
987:
956:
925:
897:
841:
814:
786:
750:
663:
383:Burma Road, Assam, India, through Burma to China,
5187:Conn, Marie A.; Therese Benedict McGuire (2007).
4267:
1430:To express the mechanical advantage by the angle
742:
725:
125:without sliding down. This angle is equal to the
6002:
5499:
5112:Devreese, Jozef T.; Guido Vanden Berghe (2008).
4890:. USA: Heinemann-Raintree Classroom. p. 7.
4814:Merriam-Webster's collegiate dictionary, 11th Ed
1116:, is equal to the work done by the input force,
1034:. The righthand weight provides the input force
102:) is required to lift a given object by a given
54:Demonstration inclined plane used in education,
5856:Engineering Mechanics and Strength of Materials
5362:
4936:
4848:
4846:
4844:
4842:
4645:{\displaystyle \mathrm {MA} ={\frac {W}{F}}=5,}
4083:of a rail car on along the ramp with an angle,
1614:
178:Inclined planes are widely used in the form of
5788:
5786:
5338:. USA: University of Oklahoma Press. pp.
1450:of the plane, it can be seen from the diagram
998:
355:Pedestrian ramp, Palacio do Planalto, Brasilia
5935:
5356:
3148:up the inclined plane. If the applied force
1867:of the plane is less than some maximum value
5826:Physics class 10 - The IIT Foundation Series
5756:
5647:
5645:
4839:
4524:{\displaystyle \sin \theta ={\frac {H}{L}},}
3194:
2253:
1650:, so less work is done on the load. Due to
1192:{\displaystyle W_{\rm {out}}=W_{\rm {in}}\,}
682:. The horizontal length is sometimes called
678:The inclined plane's geometry is based on a
5783:
5601:. Courier Dover Publications. p. 106.
5388:
5386:
5193:. University Press of America. p. 23.
5078:
5076:
5074:
4879:
4877:
4875:
5942:
5928:
5704:
5702:
5137:
5135:
5054:Encyclopedia of Physical Science, Volume 1
4916:Ortleb, Edward P.; Richard Cadice (1993).
5878:
5876:
5848:
5846:
5642:
5431:Dictionary of Greek and Roman antiquities
5016:
5014:
5012:
5010:
4922:. Lorenz Educational Press. pp. iv.
4887:Simple Machines: Forces in Action, 4th Ed
4883:
4025:
3992:
3963:
3923:
3810:
3707:
3670:
3589:
3508:{\displaystyle F_{n}=F_{w}\cos \theta \,}
3504:
3450:
3374:
3326:
3253:
3133:
3086:
2973:
2766:
2729:
2648:
2567:{\displaystyle F_{n}=F_{w}\cos \theta \,}
2563:
2509:
2433:
2385:
2312:
1969:
1778:
1700:
1567:
1486:
1416:
1343:
1254:
1188:
894:
747:
647:", divided by its horizontal length, or "
198:are also forms of an inclined plane.In a
5859:. Laxmi Publications. pp. 165–167.
5621:
5615:
5559:
5557:
5486:
5484:
5482:
5397:
5383:
5275:Rough Stone Monuments and Their Builders
5105:
5082:
5071:
4911:
4909:
4907:
4872:
4593:
4079:In order to show this, let the position
4046:
3515:and substituting into the above equation
2574:and substituting into the above equation
2017:
1085:, for a frictionless inclined plane the
1002:
673:
584:on an inclined plane were discovered by
506:were constructed using inclined planes,
49:
38:
5822:
5816:
5699:
5529:
5527:
5422:(c. 80 – 15 BCE) and Greek philosopher
5369:. USA: Barnes & Noble. p. 43.
5251:. Laurence King Publishing. p. 9.
5238:
5145:The Feynman Lectures on Physics, Vol. I
5132:
5051:Rosen, Joe; Lisa Quinn Gothard (2009).
5020:
4271:{\displaystyle P_{\mathrm {in} }=FV,\!}
761:
6003:
5883:
5873:
5852:
5843:
5792:
5735:
5729:
5468:A History of Greek Mathematics, Vol. 2
5298:
5292:
5007:
4945:
4779:
4777:
3137:{\displaystyle \phi =\tan ^{-1}\mu \,}
1973:{\displaystyle \phi =\tan ^{-1}\mu \,}
580:The first elementary rules of sliding
248:Loading a truck on a ship using a ramp
5923:
5739:The Complete Idiot's Guide to Physics
5678:
5672:
5594:
5588:
5554:
5479:
5464:
5437:
5427:
5329:
5323:
5307:. world-mysteries.com. Archived from
5213:
5207:
4976:
4904:
4834:inclined plane definition dictionary.
4067:) = Angle of inclination of the plane
1268:The input work is equal to the force
608:on an inclined plane is equal to the
592:(1699) and were further developed by
5890:. Firewall Media. pp. 275–277.
5762:
5715:. Cengage Learning. pp. 71–73.
5533:
5524:
5471:. UK: The Clarendon Press. pp.
5271:
5265:
5175:
5057:. Infobase Publishing. p. 375.
5044:
4946:Reilly, Travis (November 24, 2011).
4783:
2037:that is perpendicular to the plane,
629:of an inclined plane depends on its
514:, to drag ships overland across the
446:. Stevin's argument goes like this:
401:Inclined planes in a skateboard park
236:Using ramps to load a car on a truck
5510:. USA: Springer. pp. 195–221.
4774:
4087:, above the horizontal be given by
339:built by Romans in 72 AD to invade
46:, Hotel Montescot, Chartres, France
13:
5949:
5652:Armstrong-Hélouvry, Brian (1991).
5625:The Cambridge Companion to Galileo
5598:A History of Mechanical Inventions
5441:A History of Mechanical Inventions
5301:"Transport and the Inclined Plane"
4983:. Chapman & Hill. p. 14.
4818:. Merriam-Webster. 2003. pp.
4790:. Pearson Education. p. 178.
4672:
4669:
4616:
4613:
4548:
4545:
4424:
4421:
4303:
4300:
4297:
4249:
4246:
4029:{\displaystyle \theta >\phi \,}
3967:{\displaystyle \theta <\phi \,}
3851:
3848:
3014:
3011:
1594:
1538:
1526:
1395:
1383:
1329:
1314:
1311:
1240:
1225:
1222:
1219:
1182:
1179:
1164:
1161:
1158:
981:
978:
975:
950:
947:
944:
919:
916:
863:
860:
780:
777:
449:The string must be stationary, in
426:
14:
6022:
5908:
5655:Control of machines with friction
5534:Meli, Domenico Bertoloni (2006).
5305:Construction of the Giza Pyramids
3711:{\displaystyle \mu =\tan \phi \,}
3378:{\displaystyle F_{f}=\mu F_{n}\,}
2770:{\displaystyle \mu =\tan \phi \,}
2437:{\displaystyle F_{f}=\mu F_{n}\,}
2135:, which acts vertically downwards
1782:{\displaystyle F_{f}=\mu F_{n}\,}
5571:A History of Mathematics, 3rd Ed
5496:The enigma of the inclined plane
5278:. Echo Library. pp. 11–12.
4858:Math and science activity center
4321:
4313:
4166:
4098:
4043:Mechanical advantage using power
2059:= weight of the load, where m =
1061:pulling the roller up the plane.
686:, the vertical change in height
477:proportional to its slant length
415:
394:
376:
360:
348:
328:
283:
268:
253:
241:
229:
5685:. World Scientific. p. 7.
5458:
5412:
5248:A world history of architecture
5162:
5118:. WIT Press. pp. 136–139.
4980:Dictionary of Civil Engineering
3996:{\displaystyle \theta =\phi \,}
3465:Solving second equation to get
2524:Solving second equation to get
2074:has three forces acting on it:
1643:{\displaystyle W_{\text{fric}}}
822:. The input force is the force
32:Inclined plane (disambiguation)
4970:
4804:
4373:
4349:
4340:
4328:
4200:
4176:
4132:
4108:
3917:
3905:
3177:The equilibrium equations are
3168:
3080:
3068:
2227:
2204:coefficient of static friction
1845:coefficient of static friction
1109:{\displaystyle W_{\text{out}}}
988:{\displaystyle \mathrm {AMA} }
957:{\displaystyle \mathrm {IMA} }
615:
534:wrote that an inclined plane "
131:coefficient of static friction
16:Tilted flat supporting surface
1:
5742:. Penguin. pp. 116–117.
5658:. USA: Springer. p. 10.
5595:Usher, Abbott Payson (1988).
5540:. JHU Press. pp. 35–39.
5465:Heath, Thomas Little (1921).
5438:Usher, Abbott Payson (1988).
5027:. PHI Learning. p. 446.
4767:
3144:. This is the condition for
2214:Newton's second law of motion
1136:{\displaystyle W_{\text{in}}}
1089:done on the load lifting it,
926:{\displaystyle \mathrm {MA} }
787:{\displaystyle \mathrm {MA} }
384:
5363:Sprague de Camp, L. (1990).
5091:. Springer. pp. 94–99.
5024:Mechanism and Machine Theory
2195:{\displaystyle F_{\text{f}}}
2161:{\displaystyle F_{\text{n}}}
2128:{\displaystyle F_{\text{w}}}
2098:{\displaystyle F_{\text{i}}}
2003:{\displaystyle F_{\text{i}}}
1812:{\displaystyle F_{\text{n}}}
1735:{\displaystyle F_{\text{f}}}
1615:Inclined plane with friction
1288:{\displaystyle F_{\text{i}}}
1077:, or supported on wheels or
1054:{\displaystyle F_{\text{i}}}
1027:{\displaystyle F_{\text{w}}}
842:{\displaystyle F_{\text{i}}}
815:{\displaystyle F_{\text{w}}}
367:Johnstown Inclined Plane, a
186:are used to allow people in
117:, also sometimes called the
7:
5736:Dennis, Johnnie T. (2003).
5622:Machamer, Peter K. (1998).
5220:Legacy of the Ancient World
5216:"Pre-Greek Accomplishments"
4728:
3838:The mechanical advantage is
3001:The mechanical advantage is
2206:μ between the two surfaces.
2013:
1603:{\displaystyle F_{\rm {i}}}
999:Frictionless inclined plane
966:actual mechanical advantage
594:Charles-Augustin de Coulomb
210:of the plane to reduce the
106:, disregarding losses from
10:
6029:
5392:Karl von Langsdorf (1826)
935:ideal mechanical advantage
433:In 1586, Flemish engineer
411:
29:
18:
5958:Classical simple machines
5957:
5712:Principles of Engineering
4884:Silverman, Buffy (2009).
483:need not retain its shape
5398:Reuleaux, Franz (1876).
3718:and simplifying as above
2108:The weight of the load,
1895:of the angle of repose
620:
168:Johnstown Inclined Plane
5574:. John Wiley and Sons.
5428:Smith, William (1848).
5083:Koetsier, Teun (2010).
5021:Ambekar, A. G. (2007).
4977:Scott, John S. (1993).
4787:Explore science, 2nd Ed
4746:Inclined plane railroad
1860:{\displaystyle \theta }
1443:{\displaystyle \theta }
664:{\displaystyle \theta }
610:coefficient of friction
600:(1750) showed that the
290:Loading ramp on a truck
173:
5763:Nave, Carl R. (2010).
5404:. MacMillan. pp.
5330:Isler, Martin (2001).
5299:Thomas, Burke (2005).
5272:Peet, T. Eric (2006).
5214:Dutch, Steven (1999).
5183:Light on Sacred Stones
4784:Cole, Matthew (2005).
4716:
4646:
4599:
4585:
4525:
4469:
4401:
4272:
4210:
4142:
4068:
4030:
3997:
3968:
3928:
3825:trigonometric identity
3815:
3712:
3675:
3594:
3509:
3455:
3379:
3331:
3258:
3138:
3091:
2988:trigonometric identity
2986:Using a sum-of-angles
2978:
2771:
2734:
2653:
2568:
2514:
2438:
2390:
2317:
2196:
2175:The frictional force,
2162:
2129:
2099:
2067:
2004:
1974:
1929:
1909:
1881:
1861:
1837:
1813:
1783:
1736:
1705:
1652:conservation of energy
1644:
1604:
1572:
1491:
1444:
1421:
1348:
1289:
1259:
1193:
1137:
1110:
1083:conservation of energy
1062:
1055:
1028:
989:
958:
927:
899:
843:
816:
788:
752:
691:
665:
431:
139:between the surfaces.
92:conservation of energy
59:
47:
23:. For other uses, see
5887:Engineering Mechanics
5884:Gujral, I.S. (2008).
5804:. Westminster College
5793:Martin, Lori (2010).
5679:Meyer, Ernst (2002).
5366:The Ancient Engineers
4860:. Edinformatics. 1999
4717:
4647:
4597:
4586:
4526:
4470:
4402:
4281:and the power out is
4273:
4211:
4143:
4050:
4031:
3998:
3969:
3929:
3816:
3713:
3676:
3595:
3510:
3456:
3380:
3332:
3259:
3139:
3092:
2979:
2772:
2735:
2654:
2569:
2515:
2439:
2391:
2318:
2197:
2163:
2130:
2100:
2021:
2005:
1975:
1930:
1910:
1908:{\displaystyle \phi }
1882:
1880:{\displaystyle \phi }
1862:
1838:
1814:
1784:
1737:
1706:
1645:
1605:
1573:
1492:
1445:
1422:
1349:
1290:
1260:
1194:
1138:
1111:
1056:
1029:
1006:
990:
959:
928:
900:
844:
817:
789:
753:
677:
666:
549:It was not until the
430:
94:, the same amount of
53:
42:
25:Ramp (disambiguation)
5853:Bansal, R.K (2005).
4854:"The Inclined Plane"
4736:Canal inclined plane
4665:
4609:
4541:
4493:
4417:
4288:
4237:
4162:
4094:
4073:mechanical advantage
4013:
3980:
3951:
3844:
3724:
3689:
3607:
3523:
3469:
3393:
3345:
3269:
3183:
3105:
3007:
2782:
2748:
2666:
2582:
2528:
2452:
2404:
2328:
2242:
2179:
2145:
2112:
2082:
1987:
1941:
1928:{\displaystyle \mu }
1919:
1899:
1871:
1851:
1836:{\displaystyle \mu }
1827:
1796:
1749:
1719:
1661:
1627:
1585:
1507:
1461:
1434:
1364:
1302:
1272:
1210:
1149:
1120:
1093:
1071:ideal inclined plane
1038:
1011:
971:
940:
912:
856:
826:
799:
773:
768:mechanical advantage
762:Mechanical advantage
698:
655:
627:mechanical advantage
544:Pappus of Alexandria
542:(c. 10 - 60 CE) and
88:mechanical advantage
5500:Laird, Walter Roy;
5424:Heron of Alexandria
3827:on the denominator,
3385:into first equation
2990:on the denominator,
2444:into first equation
2078:The applied force,
540:Heron of Alexandria
260:Aircraft emergency
212:gravitational force
5226:on August 21, 2016
5168:E.J.Dijksterhuis:
4712:
4642:
4600:
4581:
4521:
4465:
4397:
4268:
4206:
4138:
4069:
4026:
3993:
3964:
3924:
3811:
3708:
3671:
3590:
3505:
3451:
3375:
3327:
3254:
3134:
3087:
2974:
2912:
2899:
2767:
2730:
2649:
2564:
2510:
2434:
2386:
2313:
2192:
2158:
2125:
2095:
2068:
2000:
1970:
1925:
1905:
1877:
1857:
1833:
1809:
1779:
1732:
1701:
1640:
1600:
1568:
1487:
1440:
1417:
1344:
1285:
1255:
1189:
1133:
1106:
1063:
1051:
1024:
985:
954:
923:
895:
839:
812:
784:
748:
692:
661:
590:Guillaume Amontons
555:Jordanus de Nemore
532:Karl von Langsdorf
516:Isthmus of Corinth
451:static equilibrium
432:
72:vertical direction
66:, also known as a
60:
48:
5998:
5997:
5897:978-81-318-0295-3
5866:978-81-7008-094-7
5836:978-81-317-2843-7
5802:Science in Motion
5749:978-1-59257-081-2
5722:978-1-4354-2836-2
5692:978-981-238-062-3
5665:978-0-7923-9133-3
5635:978-0-521-58841-6
5608:978-0-486-25593-4
5581:978-0-470-63056-3
5547:978-0-8018-8426-9
5517:978-1-4020-5966-7
5490:Egidio Festa and
5451:978-0-486-25593-4
5376:978-0-88029-456-0
5349:978-0-8061-3342-3
5311:on March 13, 2012
5285:978-1-4068-2203-8
5258:978-1-85669-371-4
5200:978-0-7618-3702-2
5181:Therese McGuire,
5155:978-0-465-02493-3
5125:978-1-84564-391-1
5098:978-90-481-9090-4
5064:978-0-8160-7011-4
5034:978-81-203-3134-1
4996:angle of friction
4990:978-0-412-98421-1
4952:Teach Engineering
4929:978-1-55863-060-4
4919:Machines and Work
4897:978-1-4329-2317-4
4829:978-0-87779-809-5
4797:978-981-06-2002-8
4687:
4631:
4576:
4563:
4516:
4460:
4439:
3921:
3880:
3835:
3834:
3808:
3749:
3668:
3632:
3084:
3043:
2998:
2997:
2971:
2911:
2898:
2849:
2807:
2727:
2691:
2189:
2172:) to the surface.
2155:
2141:The normal force
2122:
2092:
1997:
1806:
1729:
1697:
1684:
1671:
1637:
1565:
1544:
1513:
1484:
1483:
1480:
1414:
1413:
1410:
1401:
1370:
1341:
1282:
1252:
1130:
1103:
1048:
1021:
892:
889:
879:
836:
809:
738:
737:
734:
586:Leonardo da Vinci
504:Egyptian pyramids
491:
490:
307:playground slides
216:evacuation slides
115:angle of friction
104:vertical distance
96:mechanical energy
6018:
5944:
5937:
5930:
5921:
5920:
5902:
5901:
5880:
5871:
5870:
5850:
5841:
5840:
5823:Pearson (2009).
5820:
5814:
5813:
5811:
5809:
5799:
5790:
5781:
5780:
5778:
5776:
5760:
5754:
5753:
5733:
5727:
5726:
5706:
5697:
5696:
5676:
5670:
5669:
5649:
5640:
5639:
5619:
5613:
5612:
5592:
5586:
5585:
5564:Boyer, Carl B.;
5561:
5552:
5551:
5531:
5522:
5521:
5488:
5477:
5476:
5462:
5456:
5455:
5435:
5416:
5410:
5409:
5390:
5381:
5380:
5360:
5354:
5353:
5337:
5327:
5321:
5320:
5318:
5316:
5296:
5290:
5289:
5269:
5263:
5262:
5242:
5236:
5235:
5233:
5231:
5211:
5205:
5204:
5179:
5173:
5166:
5160:
5159:
5139:
5130:
5129:
5109:
5103:
5102:
5080:
5069:
5068:
5048:
5042:
5041:
5018:
5005:
5004:
4974:
4968:
4967:
4965:
4963:
4943:
4934:
4933:
4913:
4902:
4901:
4881:
4870:
4869:
4867:
4865:
4850:
4837:
4836:
4817:
4808:
4802:
4801:
4781:
4721:
4719:
4718:
4713:
4699:
4688:
4680:
4675:
4651:
4649:
4648:
4643:
4632:
4624:
4619:
4590:
4588:
4587:
4582:
4577:
4569:
4564:
4556:
4551:
4530:
4528:
4527:
4522:
4517:
4509:
4474:
4472:
4471:
4466:
4461:
4459:
4445:
4440:
4432:
4427:
4406:
4404:
4403:
4398:
4324:
4316:
4308:
4307:
4306:
4277:
4275:
4274:
4269:
4254:
4253:
4252:
4215:
4213:
4212:
4207:
4169:
4147:
4145:
4144:
4139:
4101:
4035:
4033:
4032:
4027:
4002:
4000:
3999:
3994:
3973:
3971:
3970:
3965:
3933:
3931:
3930:
3925:
3922:
3920:
3897:
3886:
3881:
3879:
3878:
3869:
3868:
3859:
3854:
3820:
3818:
3817:
3812:
3809:
3807:
3766:
3755:
3750:
3748:
3747:
3738:
3737:
3728:
3717:
3715:
3714:
3709:
3685:Substituting in
3680:
3678:
3677:
3672:
3669:
3667:
3638:
3633:
3631:
3630:
3621:
3620:
3611:
3599:
3597:
3596:
3591:
3573:
3572:
3551:
3550:
3535:
3534:
3514:
3512:
3511:
3506:
3494:
3493:
3481:
3480:
3460:
3458:
3457:
3452:
3434:
3433:
3421:
3420:
3405:
3404:
3384:
3382:
3381:
3376:
3373:
3372:
3357:
3356:
3336:
3334:
3333:
3328:
3310:
3309:
3297:
3296:
3284:
3283:
3263:
3261:
3260:
3255:
3237:
3236:
3224:
3223:
3211:
3210:
3198:
3197:
3169:
3163:Downhill motion:
3146:impending motion
3143:
3141:
3140:
3135:
3126:
3125:
3096:
3094:
3093:
3088:
3085:
3083:
3060:
3049:
3044:
3042:
3041:
3032:
3031:
3022:
3017:
2983:
2981:
2980:
2975:
2972:
2970:
2929:
2918:
2913:
2910:
2900:
2897:
2886:
2875:
2856:
2850:
2848:
2813:
2808:
2806:
2805:
2796:
2795:
2786:
2776:
2774:
2773:
2768:
2739:
2737:
2736:
2731:
2728:
2726:
2697:
2692:
2690:
2689:
2680:
2679:
2670:
2658:
2656:
2655:
2650:
2632:
2631:
2610:
2609:
2594:
2593:
2573:
2571:
2570:
2565:
2553:
2552:
2540:
2539:
2519:
2517:
2516:
2511:
2493:
2492:
2480:
2479:
2464:
2463:
2443:
2441:
2440:
2435:
2432:
2431:
2416:
2415:
2395:
2393:
2392:
2387:
2369:
2368:
2356:
2355:
2343:
2342:
2322:
2320:
2319:
2314:
2296:
2295:
2283:
2282:
2270:
2269:
2257:
2256:
2228:
2201:
2199:
2198:
2193:
2191:
2190:
2187:
2167:
2165:
2164:
2159:
2157:
2156:
2153:
2134:
2132:
2131:
2126:
2124:
2123:
2120:
2104:
2102:
2101:
2096:
2094:
2093:
2090:
2048:= input force,
2009:
2007:
2006:
2001:
1999:
1998:
1995:
1979:
1977:
1976:
1971:
1962:
1961:
1934:
1932:
1931:
1926:
1914:
1912:
1911:
1906:
1886:
1884:
1883:
1878:
1866:
1864:
1863:
1858:
1842:
1840:
1839:
1834:
1818:
1816:
1815:
1810:
1808:
1807:
1804:
1788:
1786:
1785:
1780:
1777:
1776:
1761:
1760:
1741:
1739:
1738:
1733:
1731:
1730:
1727:
1710:
1708:
1707:
1702:
1699:
1698:
1695:
1686:
1685:
1682:
1673:
1672:
1669:
1649:
1647:
1646:
1641:
1639:
1638:
1635:
1609:
1607:
1606:
1601:
1599:
1598:
1597:
1577:
1575:
1574:
1569:
1566:
1564:
1550:
1545:
1543:
1542:
1541:
1531:
1530:
1529:
1519:
1514:
1511:
1496:
1494:
1493:
1488:
1485:
1481:
1478:
1477:
1449:
1447:
1446:
1441:
1426:
1424:
1423:
1418:
1415:
1411:
1408:
1407:
1402:
1400:
1399:
1398:
1388:
1387:
1386:
1376:
1371:
1368:
1353:
1351:
1350:
1345:
1342:
1339:
1334:
1333:
1332:
1319:
1318:
1317:
1294:
1292:
1291:
1286:
1284:
1283:
1280:
1264:
1262:
1261:
1256:
1253:
1250:
1245:
1244:
1243:
1230:
1229:
1228:
1198:
1196:
1195:
1190:
1187:
1186:
1185:
1169:
1168:
1167:
1142:
1140:
1139:
1134:
1132:
1131:
1128:
1115:
1113:
1112:
1107:
1105:
1104:
1101:
1060:
1058:
1057:
1052:
1050:
1049:
1046:
1033:
1031:
1030:
1025:
1023:
1022:
1019:
994:
992:
991:
986:
984:
963:
961:
960:
955:
953:
932:
930:
929:
924:
922:
904:
902:
901:
896:
893:
891:
890:
887:
881:
880:
877:
871:
866:
848:
846:
845:
840:
838:
837:
834:
821:
819:
818:
813:
811:
810:
807:
793:
791:
790:
785:
783:
757:
755:
754:
749:
746:
745:
739:
735:
732:
731:
729:
728:
719:
718:
670:
668:
667:
662:
559:Girolamo Cardano
471:(points S and V)
455:perpetual motion
416:
398:
389:
386:
380:
364:
352:
332:
319:skateboard parks
287:
272:
262:evacuation slide
257:
245:
233:
184:Wheelchair ramps
6028:
6027:
6021:
6020:
6019:
6017:
6016:
6015:
6011:Simple machines
6001:
6000:
5999:
5994:
5953:
5948:
5911:
5906:
5905:
5898:
5881:
5874:
5867:
5851:
5844:
5837:
5821:
5817:
5807:
5805:
5797:
5791:
5784:
5774:
5772:
5761:
5757:
5750:
5734:
5730:
5723:
5707:
5700:
5693:
5677:
5673:
5666:
5650:
5643:
5636:
5620:
5616:
5609:
5593:
5589:
5582:
5566:Uta C. Merzbach
5562:
5555:
5548:
5532:
5525:
5518:
5489:
5480:
5463:
5459:
5452:
5417:
5413:
5391:
5384:
5377:
5361:
5357:
5350:
5328:
5324:
5314:
5312:
5297:
5293:
5286:
5270:
5266:
5259:
5243:
5239:
5229:
5227:
5212:
5208:
5201:
5180:
5176:
5167:
5163:
5156:
5140:
5133:
5126:
5110:
5106:
5099:
5081:
5072:
5065:
5049:
5045:
5035:
5019:
5008:
5000:angle of repose
4991:
4975:
4971:
4961:
4959:
4944:
4937:
4930:
4914:
4905:
4898:
4882:
4873:
4863:
4861:
4852:
4851:
4840:
4830:
4810:
4809:
4805:
4798:
4782:
4775:
4770:
4765:
4731:
4695:
4679:
4668:
4666:
4663:
4662:
4623:
4612:
4610:
4607:
4606:
4568:
4555:
4544:
4542:
4539:
4538:
4508:
4494:
4491:
4490:
4449:
4444:
4431:
4420:
4418:
4415:
4414:
4320:
4312:
4296:
4295:
4291:
4289:
4286:
4285:
4245:
4244:
4240:
4238:
4235:
4234:
4165:
4163:
4160:
4159:
4097:
4095:
4092:
4091:
4045:
4014:
4011:
4010:
4005:angle of repose
3981:
3978:
3977:
3952:
3949:
3948:
3942:
3898:
3887:
3885:
3874:
3870:
3864:
3860:
3858:
3847:
3845:
3842:
3841:
3831:
3830:
3767:
3756:
3754:
3743:
3739:
3733:
3729:
3727:
3725:
3722:
3721:
3690:
3687:
3686:
3642:
3637:
3626:
3622:
3616:
3612:
3610:
3608:
3605:
3604:
3568:
3564:
3546:
3542:
3530:
3526:
3524:
3521:
3520:
3489:
3485:
3476:
3472:
3470:
3467:
3466:
3429:
3425:
3416:
3412:
3400:
3396:
3394:
3391:
3390:
3368:
3364:
3352:
3348:
3346:
3343:
3342:
3305:
3301:
3292:
3288:
3279:
3275:
3270:
3267:
3266:
3232:
3228:
3219:
3215:
3206:
3202:
3193:
3189:
3184:
3181:
3180:
3175:
3154:
3118:
3114:
3106:
3103:
3102:
3061:
3050:
3048:
3037:
3033:
3027:
3023:
3021:
3010:
3008:
3005:
3004:
2994:
2993:
2930:
2919:
2917:
2887:
2876:
2873:
2860:
2854:
2817:
2812:
2801:
2797:
2791:
2787:
2785:
2783:
2780:
2779:
2749:
2746:
2745:
2701:
2696:
2685:
2681:
2675:
2671:
2669:
2667:
2664:
2663:
2627:
2623:
2605:
2601:
2589:
2585:
2583:
2580:
2579:
2548:
2544:
2535:
2531:
2529:
2526:
2525:
2488:
2484:
2475:
2471:
2459:
2455:
2453:
2450:
2449:
2427:
2423:
2411:
2407:
2405:
2402:
2401:
2364:
2360:
2351:
2347:
2338:
2334:
2329:
2326:
2325:
2291:
2287:
2278:
2274:
2265:
2261:
2252:
2248:
2243:
2240:
2239:
2234:
2186:
2182:
2180:
2177:
2176:
2152:
2148:
2146:
2143:
2142:
2119:
2115:
2113:
2110:
2109:
2089:
2085:
2083:
2080:
2079:
2054:
2043:
2028:
2016:
1994:
1990:
1988:
1985:
1984:
1954:
1950:
1942:
1939:
1938:
1920:
1917:
1916:
1900:
1897:
1896:
1889:angle of repose
1872:
1869:
1868:
1852:
1849:
1848:
1828:
1825:
1824:
1803:
1799:
1797:
1794:
1793:
1772:
1768:
1756:
1752:
1750:
1747:
1746:
1726:
1722:
1720:
1717:
1716:
1694:
1690:
1681:
1677:
1668:
1664:
1662:
1659:
1658:
1634:
1630:
1628:
1625:
1624:
1619:Where there is
1617:
1593:
1592:
1588:
1586:
1583:
1582:
1554:
1549:
1537:
1536:
1532:
1525:
1524:
1520:
1518:
1510:
1508:
1505:
1504:
1476:
1462:
1459:
1458:
1435:
1432:
1431:
1406:
1394:
1393:
1389:
1382:
1381:
1377:
1375:
1367:
1365:
1362:
1361:
1338:
1328:
1327:
1323:
1310:
1309:
1305:
1303:
1300:
1299:
1279:
1275:
1273:
1270:
1269:
1249:
1239:
1238:
1234:
1218:
1217:
1213:
1211:
1208:
1207:
1178:
1177:
1173:
1157:
1156:
1152:
1150:
1147:
1146:
1127:
1123:
1121:
1118:
1117:
1100:
1096:
1094:
1091:
1090:
1065:If there is no
1045:
1041:
1039:
1036:
1035:
1018:
1014:
1012:
1009:
1008:
1001:
974:
972:
969:
968:
943:
941:
938:
937:
915:
913:
910:
909:
886:
882:
876:
872:
870:
859:
857:
854:
853:
833:
829:
827:
824:
823:
806:
802:
800:
797:
796:
776:
774:
771:
770:
764:
741:
740:
730:
724:
723:
711:
707:
699:
696:
695:
656:
653:
652:
623:
618:
606:angle of repose
567:Galileo Galilei
523:simple machines
414:
408:
406:
405:
404:
403:
402:
399:
391:
390:
387:
381:
373:
372:
365:
357:
356:
353:
345:
344:
333:
295:
294:
293:
292:
291:
288:
280:
279:
278:on Japanese bus
276:Wheelchair ramp
273:
265:
264:
258:
250:
249:
246:
238:
237:
234:
176:
137:
119:angle of repose
76:simple machines
44:Wheelchair ramp
35:
28:
17:
12:
11:
5:
6026:
6025:
6014:
6013:
5996:
5995:
5993:
5992:
5990:Wheel and axle
5987:
5982:
5977:
5972:
5967:
5965:Inclined plane
5961:
5959:
5955:
5954:
5951:Simple machine
5947:
5946:
5939:
5932:
5924:
5918:
5917:
5910:
5909:External links
5907:
5904:
5903:
5896:
5872:
5865:
5842:
5835:
5815:
5782:
5755:
5748:
5728:
5721:
5698:
5691:
5671:
5664:
5641:
5634:
5614:
5607:
5587:
5580:
5553:
5546:
5523:
5516:
5478:
5457:
5450:
5411:
5382:
5375:
5355:
5348:
5322:
5291:
5284:
5264:
5257:
5237:
5206:
5199:
5174:
5161:
5154:
5131:
5124:
5104:
5097:
5070:
5063:
5043:
5033:
5006:
4989:
4969:
4958:on May 8, 2012
4935:
4928:
4903:
4896:
4871:
4838:
4828:
4803:
4796:
4772:
4771:
4769:
4766:
4764:
4763:
4758:
4753:
4748:
4743:
4738:
4732:
4730:
4727:
4723:
4722:
4711:
4708:
4705:
4702:
4698:
4694:
4691:
4686:
4683:
4678:
4674:
4671:
4653:
4652:
4641:
4638:
4635:
4630:
4627:
4622:
4618:
4615:
4592:
4591:
4580:
4575:
4572:
4567:
4562:
4559:
4554:
4550:
4547:
4532:
4531:
4520:
4515:
4512:
4507:
4504:
4501:
4498:
4482:to its height
4476:
4475:
4464:
4458:
4455:
4452:
4448:
4443:
4438:
4435:
4430:
4426:
4423:
4408:
4407:
4396:
4393:
4390:
4387:
4384:
4381:
4378:
4375:
4372:
4369:
4366:
4363:
4360:
4357:
4354:
4351:
4348:
4345:
4342:
4339:
4336:
4333:
4330:
4327:
4323:
4319:
4315:
4311:
4305:
4302:
4299:
4294:
4279:
4278:
4266:
4263:
4260:
4257:
4251:
4248:
4243:
4217:
4216:
4205:
4202:
4199:
4196:
4193:
4190:
4187:
4184:
4181:
4178:
4175:
4172:
4168:
4149:
4148:
4137:
4134:
4131:
4128:
4125:
4122:
4119:
4116:
4113:
4110:
4107:
4104:
4100:
4044:
4041:
4040:
4039:
4038:
4037:
4024:
4021:
4018:
4008:
3991:
3988:
3985:
3975:
3962:
3959:
3956:
3940:
3934:
3919:
3916:
3913:
3910:
3907:
3904:
3901:
3896:
3893:
3890:
3884:
3877:
3873:
3867:
3863:
3857:
3853:
3850:
3839:
3833:
3832:
3829:
3828:
3823:Using another
3821:
3806:
3803:
3800:
3797:
3794:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3770:
3765:
3762:
3759:
3753:
3746:
3742:
3736:
3732:
3719:
3706:
3703:
3700:
3697:
3694:
3682:
3681:
3666:
3663:
3660:
3657:
3654:
3651:
3648:
3645:
3641:
3636:
3629:
3625:
3619:
3615:
3601:
3600:
3588:
3585:
3582:
3579:
3576:
3571:
3567:
3563:
3560:
3557:
3554:
3549:
3545:
3541:
3538:
3533:
3529:
3517:
3516:
3503:
3500:
3497:
3492:
3488:
3484:
3479:
3475:
3462:
3461:
3449:
3446:
3443:
3440:
3437:
3432:
3428:
3424:
3419:
3415:
3411:
3408:
3403:
3399:
3387:
3386:
3371:
3367:
3363:
3360:
3355:
3351:
3341:Substituting
3338:
3337:
3325:
3322:
3319:
3316:
3313:
3308:
3304:
3300:
3295:
3291:
3287:
3282:
3278:
3274:
3264:
3252:
3249:
3246:
3243:
3240:
3235:
3231:
3227:
3222:
3218:
3214:
3209:
3205:
3201:
3196:
3192:
3188:
3176:
3173:
3172:
3167:
3166:
3159:
3158:
3152:
3132:
3129:
3124:
3121:
3117:
3113:
3110:
3098:
3097:
3082:
3079:
3076:
3073:
3070:
3067:
3064:
3059:
3056:
3053:
3047:
3040:
3036:
3030:
3026:
3020:
3016:
3013:
3002:
2996:
2995:
2992:
2991:
2984:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2945:
2942:
2939:
2936:
2933:
2928:
2925:
2922:
2916:
2909:
2906:
2903:
2896:
2893:
2890:
2885:
2882:
2879:
2872:
2869:
2866:
2863:
2859:
2853:
2847:
2844:
2841:
2838:
2835:
2832:
2829:
2826:
2823:
2820:
2816:
2811:
2804:
2800:
2794:
2790:
2777:
2765:
2762:
2759:
2756:
2753:
2741:
2740:
2725:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2700:
2695:
2688:
2684:
2678:
2674:
2660:
2659:
2647:
2644:
2641:
2638:
2635:
2630:
2626:
2622:
2619:
2616:
2613:
2608:
2604:
2600:
2597:
2592:
2588:
2576:
2575:
2562:
2559:
2556:
2551:
2547:
2543:
2538:
2534:
2521:
2520:
2508:
2505:
2502:
2499:
2496:
2491:
2487:
2483:
2478:
2474:
2470:
2467:
2462:
2458:
2446:
2445:
2430:
2426:
2422:
2419:
2414:
2410:
2400:Substituting
2397:
2396:
2384:
2381:
2378:
2375:
2372:
2367:
2363:
2359:
2354:
2350:
2346:
2341:
2337:
2333:
2323:
2311:
2308:
2305:
2302:
2299:
2294:
2290:
2286:
2281:
2277:
2273:
2268:
2264:
2260:
2255:
2251:
2247:
2235:
2232:
2231:
2226:
2225:
2222:Uphill motion:
2210:
2209:
2208:
2207:
2185:
2173:
2151:
2136:
2118:
2106:
2088:
2052:
2041:
2026:
2015:
2012:
1993:
1981:
1980:
1968:
1965:
1960:
1957:
1953:
1949:
1946:
1924:
1904:
1876:
1856:
1832:
1802:
1790:
1789:
1775:
1771:
1767:
1764:
1759:
1755:
1725:
1712:
1711:
1693:
1689:
1680:
1676:
1667:
1633:
1616:
1613:
1596:
1591:
1579:
1578:
1563:
1560:
1557:
1553:
1548:
1540:
1535:
1528:
1523:
1517:
1498:
1497:
1475:
1472:
1469:
1466:
1439:
1428:
1427:
1405:
1397:
1392:
1385:
1380:
1374:
1355:
1354:
1337:
1331:
1326:
1322:
1316:
1313:
1308:
1278:
1266:
1265:
1248:
1242:
1237:
1233:
1227:
1224:
1221:
1216:
1200:
1199:
1184:
1181:
1176:
1172:
1166:
1163:
1160:
1155:
1126:
1099:
1044:
1017:
1000:
997:
983:
980:
977:
952:
949:
946:
921:
918:
906:
905:
885:
875:
869:
865:
862:
832:
805:
782:
779:
763:
760:
759:
758:
744:
727:
722:
717:
714:
710:
706:
703:
680:right triangle
660:
635:, meaning its
622:
619:
617:
614:
598:Leonhard Euler
489:
488:
479:
478:
474:
466:
423:
422:
420:Stevin's proof
413:
410:
400:
393:
392:
382:
375:
374:
366:
359:
358:
354:
347:
346:
334:
327:
326:
325:
324:
323:
289:
282:
281:
274:
267:
266:
259:
252:
251:
247:
240:
239:
235:
228:
227:
226:
225:
224:
196:conveyor belts
175:
172:
166:, such as the
135:
64:inclined plane
21:Ramp (company)
15:
9:
6:
4:
3:
2:
6024:
6023:
6012:
6009:
6008:
6006:
5991:
5988:
5986:
5983:
5981:
5978:
5976:
5973:
5971:
5968:
5966:
5963:
5962:
5960:
5956:
5952:
5945:
5940:
5938:
5933:
5931:
5926:
5925:
5922:
5916:
5913:
5912:
5899:
5893:
5889:
5888:
5879:
5877:
5868:
5862:
5858:
5857:
5849:
5847:
5838:
5832:
5828:
5827:
5819:
5803:
5796:
5789:
5787:
5770:
5766:
5765:"The Incline"
5759:
5751:
5745:
5741:
5740:
5732:
5724:
5718:
5714:
5713:
5705:
5703:
5694:
5688:
5684:
5683:
5675:
5667:
5661:
5657:
5656:
5648:
5646:
5637:
5631:
5627:
5626:
5618:
5610:
5604:
5600:
5599:
5591:
5583:
5577:
5573:
5572:
5567:
5560:
5558:
5549:
5543:
5539:
5538:
5530:
5528:
5519:
5513:
5509:
5508:
5503:
5497:
5493:
5487:
5485:
5483:
5474:
5470:
5469:
5461:
5453:
5447:
5443:
5442:
5433:
5432:
5425:
5421:
5415:
5407:
5403:
5402:
5396:, quoted in
5395:
5394:Machinenkunde
5389:
5387:
5378:
5372:
5368:
5367:
5359:
5351:
5345:
5341:
5336:
5335:
5326:
5310:
5306:
5302:
5295:
5287:
5281:
5277:
5276:
5268:
5260:
5254:
5250:
5249:
5241:
5225:
5221:
5217:
5210:
5202:
5196:
5192:
5191:
5184:
5178:
5171:
5165:
5157:
5151:
5147:
5146:
5138:
5136:
5127:
5121:
5117:
5116:
5108:
5100:
5094:
5090:
5086:
5079:
5077:
5075:
5066:
5060:
5056:
5055:
5047:
5040:
5036:
5030:
5026:
5025:
5017:
5015:
5013:
5011:
5003:
5001:
4997:
4992:
4986:
4982:
4981:
4973:
4957:
4953:
4949:
4942:
4940:
4931:
4925:
4921:
4920:
4912:
4910:
4908:
4899:
4893:
4889:
4888:
4880:
4878:
4876:
4859:
4855:
4849:
4847:
4845:
4843:
4835:
4831:
4825:
4821:
4816:
4815:
4807:
4799:
4793:
4789:
4788:
4780:
4778:
4773:
4762:
4759:
4757:
4756:Schiefe Ebene
4754:
4752:
4751:Ramp function
4749:
4747:
4744:
4742:
4741:Grade (slope)
4739:
4737:
4734:
4733:
4726:
4709:
4706:
4703:
4700:
4696:
4692:
4689:
4684:
4681:
4676:
4661:
4660:
4659:
4656:
4639:
4636:
4633:
4628:
4625:
4620:
4605:
4604:
4603:
4596:
4578:
4573:
4570:
4565:
4560:
4557:
4552:
4537:
4536:
4535:
4518:
4513:
4510:
4505:
4502:
4499:
4496:
4489:
4488:
4487:
4485:
4481:
4462:
4456:
4453:
4450:
4446:
4441:
4436:
4433:
4428:
4413:
4412:
4411:
4394:
4391:
4388:
4385:
4382:
4379:
4376:
4370:
4367:
4364:
4361:
4358:
4355:
4352:
4346:
4343:
4337:
4334:
4331:
4325:
4317:
4309:
4292:
4284:
4283:
4282:
4264:
4261:
4258:
4255:
4241:
4233:
4232:
4231:
4228:
4227:of the load.
4226:
4222:
4203:
4197:
4194:
4191:
4188:
4185:
4182:
4179:
4173:
4170:
4158:
4157:
4156:
4154:
4135:
4129:
4126:
4123:
4120:
4117:
4114:
4111:
4105:
4102:
4090:
4089:
4088:
4086:
4082:
4077:
4074:
4066:
4062:
4058:
4054:
4049:
4022:
4019:
4016:
4009:
4006:
3989:
3986:
3983:
3976:
3960:
3957:
3954:
3947:
3946:
3944:
3943:
3935:
3914:
3911:
3908:
3902:
3899:
3894:
3891:
3888:
3882:
3875:
3871:
3865:
3861:
3855:
3840:
3837:
3836:
3826:
3822:
3804:
3801:
3798:
3795:
3792:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3768:
3763:
3760:
3757:
3751:
3744:
3740:
3734:
3730:
3720:
3704:
3701:
3698:
3695:
3692:
3684:
3683:
3664:
3661:
3658:
3655:
3652:
3649:
3646:
3643:
3639:
3634:
3627:
3623:
3617:
3613:
3603:
3602:
3586:
3583:
3580:
3577:
3574:
3569:
3565:
3561:
3558:
3555:
3552:
3547:
3543:
3539:
3536:
3531:
3527:
3519:
3518:
3501:
3498:
3495:
3490:
3486:
3482:
3477:
3473:
3464:
3463:
3447:
3444:
3441:
3438:
3435:
3430:
3426:
3422:
3417:
3413:
3409:
3406:
3401:
3397:
3389:
3388:
3369:
3365:
3361:
3358:
3353:
3349:
3340:
3339:
3323:
3320:
3317:
3314:
3311:
3306:
3302:
3298:
3293:
3289:
3285:
3280:
3276:
3272:
3265:
3250:
3247:
3244:
3241:
3238:
3233:
3229:
3225:
3220:
3216:
3212:
3207:
3203:
3199:
3190:
3186:
3179:
3178:
3171:
3170:
3164:
3161:
3160:
3156:
3155:
3147:
3130:
3127:
3122:
3119:
3115:
3111:
3108:
3100:
3099:
3077:
3074:
3071:
3065:
3062:
3057:
3054:
3051:
3045:
3038:
3034:
3028:
3024:
3018:
3003:
3000:
2999:
2989:
2985:
2967:
2964:
2961:
2958:
2955:
2952:
2949:
2946:
2943:
2940:
2937:
2934:
2931:
2926:
2923:
2920:
2914:
2907:
2904:
2901:
2894:
2891:
2888:
2883:
2880:
2877:
2870:
2867:
2864:
2861:
2857:
2851:
2845:
2842:
2839:
2836:
2833:
2830:
2827:
2824:
2821:
2818:
2814:
2809:
2802:
2798:
2792:
2788:
2778:
2763:
2760:
2757:
2754:
2751:
2743:
2742:
2723:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2698:
2693:
2686:
2682:
2676:
2672:
2662:
2661:
2645:
2642:
2639:
2636:
2633:
2628:
2624:
2620:
2617:
2614:
2611:
2606:
2602:
2598:
2595:
2590:
2586:
2578:
2577:
2560:
2557:
2554:
2549:
2545:
2541:
2536:
2532:
2523:
2522:
2506:
2503:
2500:
2497:
2494:
2489:
2485:
2481:
2476:
2472:
2468:
2465:
2460:
2456:
2448:
2447:
2428:
2424:
2420:
2417:
2412:
2408:
2399:
2398:
2382:
2379:
2376:
2373:
2370:
2365:
2361:
2357:
2352:
2348:
2344:
2339:
2335:
2331:
2324:
2309:
2306:
2303:
2300:
2297:
2292:
2288:
2284:
2279:
2275:
2271:
2266:
2262:
2258:
2249:
2245:
2238:
2237:
2230:
2229:
2223:
2220:
2219:
2218:
2215:
2205:
2183:
2174:
2171:
2149:
2140:
2139:
2137:
2116:
2107:
2086:
2077:
2076:
2075:
2073:
2066:
2062:
2058:
2051:
2047:
2040:
2036:
2032:
2025:
2020:
2011:
1991:
1966:
1963:
1958:
1955:
1951:
1947:
1944:
1937:
1936:
1935:
1922:
1902:
1894:
1890:
1874:
1854:
1846:
1830:
1822:
1800:
1773:
1769:
1765:
1762:
1757:
1753:
1745:
1744:
1743:
1723:
1691:
1687:
1678:
1674:
1665:
1657:
1656:
1655:
1653:
1631:
1622:
1612:
1589:
1561:
1558:
1555:
1551:
1546:
1533:
1521:
1515:
1503:
1502:
1501:
1473:
1470:
1467:
1464:
1457:
1456:
1455:
1453:
1437:
1403:
1390:
1378:
1372:
1360:
1359:
1358:
1335:
1324:
1320:
1306:
1298:
1297:
1296:
1276:
1246:
1235:
1231:
1214:
1206:
1205:
1204:
1174:
1170:
1153:
1145:
1144:
1143:
1124:
1097:
1088:
1084:
1080:
1076:
1072:
1068:
1042:
1015:
1005:
996:
967:
936:
883:
873:
867:
852:
851:
850:
830:
803:
769:
720:
715:
712:
708:
704:
701:
694:
693:
689:
685:
681:
676:
672:
658:
650:
646:
642:
638:
634:
633:
628:
613:
611:
607:
603:
599:
595:
591:
587:
583:
578:
576:
575:Le Meccaniche
572:
568:
564:
560:
556:
552:
547:
545:
541:
537:
533:
529:
528:ancient Greek
524:
519:
517:
513:
509:
505:
501:
496:
487:
484:
475:
472:
467:
464:
460:
456:
452:
448:
447:
445:
440:
436:
429:
425:
424:
421:
418:
417:
409:
397:
379:
370:
363:
351:
342:
338:
331:
322:
320:
316:
312:
308:
304:
300:
286:
277:
271:
263:
256:
244:
232:
223:
221:
217:
213:
209:
205:
204:cable railway
201:
197:
193:
189:
185:
181:
180:loading ramps
171:
169:
165:
164:cable railway
161:
155:
153:
149:
145:
140:
138:
132:
128:
124:
120:
116:
111:
109:
105:
101:
97:
93:
89:
85:
80:
77:
73:
69:
65:
57:
56:Museo Galileo
52:
45:
41:
37:
33:
26:
22:
5964:
5886:
5855:
5825:
5818:
5808:September 8,
5806:. Retrieved
5801:
5775:September 8,
5773:. Retrieved
5769:Hyperphysics
5768:
5758:
5738:
5731:
5711:
5681:
5674:
5654:
5624:
5617:
5597:
5590:
5570:
5536:
5506:
5495:
5467:
5460:
5440:
5430:
5414:
5400:
5393:
5365:
5358:
5333:
5325:
5313:. Retrieved
5309:the original
5304:
5294:
5274:
5267:
5247:
5240:
5228:. Retrieved
5224:the original
5219:
5209:
5189:
5182:
5177:
5170:Simon Stevin
5169:
5164:
5144:
5114:
5107:
5088:
5053:
5046:
5038:
5023:
4999:
4995:
4994:
4979:
4972:
4962:September 8,
4960:. Retrieved
4956:the original
4951:
4918:
4886:
4862:. Retrieved
4857:
4833:
4813:
4806:
4786:
4724:
4657:
4654:
4601:
4533:
4483:
4479:
4477:
4409:
4280:
4229:
4224:
4220:
4218:
4152:
4150:
4084:
4080:
4078:
4070:
4053:Normal force
3938:
3937:
3162:
3150:
3149:
3145:
2221:
2211:
2069:
2056:
2049:
2045:
2038:
2035:Normal force
2030:
2023:
1982:
1915:is equal to
1821:normal force
1791:
1713:
1618:
1580:
1499:
1451:
1429:
1356:
1267:
1201:
1070:
1064:
965:
934:
907:
765:
687:
683:
648:
644:
640:
630:
624:
579:
574:
571:Simon Stevin
565:(1586), and
563:Simon Stevin
548:
535:
520:
492:
482:
480:
470:
462:
458:
443:
438:
435:Simon Stevin
419:
407:
336:
311:water slides
296:
208:normal force
194:and slanted
179:
177:
156:
141:
133:
112:
81:
67:
63:
61:
36:
5502:Sophie Roux
5492:Sophie Roux
4534:therefore,
616:Terminology
551:Renaissance
388: 1945
335:Earth ramp
214:. Aircraft
188:wheelchairs
58:, Florence.
5475:, 433–434.
4768:References
500:Stonehenge
315:ski slopes
192:Escalators
127:arctangent
5420:Vitruvius
5315:March 10,
5230:March 13,
4864:March 11,
4503:θ
4500:
4457:θ
4454:
4392:θ
4389:
4371:θ
4368:
4359:θ
4356:
4344:⋅
4318:⋅
4198:θ
4195:
4186:θ
4183:
4130:θ
4127:
4118:θ
4115:
4063:, and θ (
4051:Key: N =
4023:ϕ
4017:θ
3990:ϕ
3984:θ
3961:ϕ
3955:θ
3915:ϕ
3912:−
3909:θ
3903:
3895:ϕ
3892:
3805:ϕ
3802:
3796:θ
3793:
3787:−
3784:ϕ
3781:
3775:θ
3772:
3764:ϕ
3761:
3705:ϕ
3702:
3693:μ
3665:θ
3662:
3656:μ
3653:−
3650:θ
3647:
3581:θ
3578:
3562:−
3559:θ
3556:
3540:μ
3502:θ
3499:
3442:θ
3439:
3423:−
3410:μ
3362:μ
3318:θ
3315:
3299:−
3281:⊥
3273:∑
3245:θ
3242:
3226:−
3195:‖
3187:∑
3131:μ
3128:
3120:−
3109:ϕ
3078:ϕ
3072:θ
3066:
3058:ϕ
3055:
2968:ϕ
2965:
2959:θ
2956:
2947:ϕ
2944:
2938:θ
2935:
2927:ϕ
2924:
2908:θ
2905:
2895:ϕ
2892:
2884:ϕ
2881:
2868:θ
2865:
2846:θ
2843:
2837:ϕ
2834:
2825:θ
2822:
2764:ϕ
2761:
2752:μ
2744:Defining
2724:θ
2721:
2715:μ
2709:θ
2706:
2640:θ
2637:
2621:−
2618:θ
2615:
2599:μ
2596:−
2561:θ
2558:
2501:θ
2498:
2482:−
2469:μ
2466:−
2421:μ
2377:θ
2374:
2358:−
2340:⊥
2332:∑
2304:θ
2301:
2285:−
2272:−
2254:‖
2246:∑
2072:free body
1967:μ
1964:
1956:−
1945:ϕ
1923:μ
1903:ϕ
1875:ϕ
1855:θ
1831:μ
1766:μ
1562:θ
1559:
1471:θ
1468:
1438:θ
1336:⋅
1247:⋅
1081:. Due to
721:
713:−
702:θ
659:θ
495:causeways
371:railroad.
369:funicular
303:sidewalks
299:causeways
200:funicular
160:funicular
6005:Category
5568:(2010).
5504:(2008).
4729:See also
2014:Analysis
1621:friction
1067:friction
637:gradient
596:(1785).
582:friction
465:) equal.
343:, Israel
220:airliner
152:cylinder
123:friction
108:friction
4061:gravity
4003:: The '
2065:gravity
1893:tangent
1843:is the
1819:is the
1452:(above)
1079:casters
604:of the
602:tangent
512:Diolkos
439:A, B, C
412:History
337:(right)
129:of the
5975:Pulley
5894:
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3101:where
2212:Using
2170:normal
2063:, g =
1792:where
1482:Length
1409:Length
1340:Length
1075:barrel
341:Masada
5985:Wedge
5980:Screw
5970:Lever
5798:(PDF)
5185:, in
4701:37.50
4065:theta
2022:Key:
1454:that
632:slope
621:Slope
508:Siege
463:above
148:screw
144:wedge
84:force
5892:ISBN
5861:ISBN
5831:ISBN
5810:2012
5777:2012
5744:ISBN
5717:ISBN
5687:ISBN
5660:ISBN
5630:ISBN
5603:ISBN
5576:ISBN
5542:ISBN
5512:ISBN
5446:ISBN
5371:ISBN
5344:ISBN
5317:2012
5280:ISBN
5253:ISBN
5232:2012
5195:ISBN
5172:1943
5150:ISBN
5120:ISBN
5093:ISBN
5059:ISBN
5029:ISBN
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4964:2012
4924:ISBN
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4866:2012
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4707:48.1
4693:1804
4071:The
4057:mass
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2061:mass
1683:fric
1636:fric
1479:Rise
1412:Rise
1251:Rise
1087:work
908:The
766:The
733:Rise
688:Rise
645:rise
625:The
317:and
174:Uses
113:The
100:work
68:ramp
5498:in
5473:349
5436:,
5406:604
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4451:sin
4386:sin
4365:sin
4353:cos
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3900:sin
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3799:sin
3790:cos
3778:cos
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3758:cos
3699:tan
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3644:sin
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3116:tan
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2703:sin
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1952:tan
1696:out
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1500:So
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1102:out
736:Run
709:tan
684:Run
649:run
202:or
162:or
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