
#1
Jul1809, 10:43 AM

P: 37

The simple pulley :
A simple pulley consists essentially of a wheel with a rope round it which it contained in a suspended framework . suppose W is the load or weight in lb.wt. attached to one end of the rope , and P is the effort in lb.wt. applied downwards at the other end which can just raise W . if the rope is very light and there is no friction round the wheel , the tension in the rope is W lb.wt. , and P must hence equal W . however , there is considerably less strain in pulling the rope of a pulley downward to lift heavy loads than in raising them to the same height by a direct upward force , and the hauler can also use his own weight in this case . The mechanical advantage (M.A.) of a machine is defined as the ratio (load / effort) = W/P . we shall see later that the load raised in a practical system of pulleys is much greater than the effort , so that the mechanical advantage is much greater than one . in the case one of the simple pulley , however , W = P , neglecting friction ; hence the mechanical advantage , W/P , is one . The velocity ration(V.R.) of a machine is defined as the ratio r/s , where r,s are the distances moved by the effort and load respectively in the same time . in general , the effort (applied force) moves a much greater distance than the load when the latter is raised , and the velocity ratio is thus much greater than one . in the simple pulley , the distance (r) moved by the effort is equal to the distance (s) moved by the load , since the rope which raises the weight is also used to apply the effort . thus the velocity ratio , r/s , is one in this case . it should be carefully noted that the magnitude of the velocity ratio , unlike the mechanical advantage , is not affected by friction in the pulley system . The block and tackle : In practice , pulleys are designed to provide a large mechanical advantage , so that a large load or weight can be raised with a small effort . this figure : http://www.lhup.edu/~dsimanek/scenar...1/pulleyss.gif illustrates a useful system of pulleys , known as a block and tackle system , which contains tow sets of pulleys with one continuous rope round them . the load of weight W is attached to the lower set of pulleys , which is movable , while the upper set is supported from a beam and is fixed in position . assuming that the pulleys and rope are light , and that no friction is present , the tension in every part of the rope is equal to P , the applied effort . hence , since there are four portions of rope round the lower set of pulleys , which support W , it follows that 4P = W . thus the mechanical advantage , W/P , is 4 in this case . in practice the mechanical advantage is less than this figure , since there is friction between the rope and the pulleys , and , moreover , the pulleys have weight . for example , if the two lower pulleys in the figure above have a total weight of 50 lb.wt. , then 4P = W+50 , considering the equilibrium of the two lower pulleys . consequently 4P is greater in magnitude than W , so that W/P is less than 4 . to find the velocity ratio (V.R.) of the system of pulleys , suppose that the weight W is raised a distance x when the effort P is applied . if we imagine the lowest pulley A raised this distance from QR to PS (in the figure below :) , the length of rope made available = PQ + SR = x + x = 2x . the upper pulley , also rises a distance x , so that the movement of this pulley also makes a length 2x of rope available . thus a total length , 4x , of rope slips round the pulleys when the load is raised a distance x , and hence 4x is the distance moved by the effort . then V.R. = distance moved by effort / distance moved by load in same time = 4x/x = 4 . this is the magnitude of the (V.R.) obtained even when friction and the weights of the pulleys are taken into account , since the distance moved by the effort must always be 4x when the load is raised a distance x . it should again be noted that the magnitude of the mechanical advantage is affected by friction and the weights of the pulleys . in general , the V.R. of this system of pulleys is n , where n is the total number of pulleys in the system . the mechanical advantage is also equal to n when friction and the weights of the pulleys are neglected . in the case of an odd number of pulleys , the upper fixed block has one more pulley in it than the lower movable . thus , if there were 5 pulleys , the upper fixed block in the first figure would have 3 pulleys , and the string would be connected to the lower limb at the lower block . this system of pulleys is used at railway stations and engineering works for hauling heavy loads . archimedean system of pulleys : the basic form of another pulley system , sometimes known as the Archimedean or first system of pulleys , is shown in the figure(a) : the load W is attached to a pulley a A1 , which has a rope passing round it . one end of the rope is attached to a fixed point on a beam , while the downward effort P is applied to the other end of the rope with the aid of a small fixed pulley F . suppose that the pulleys are light and smooth , and that the rope is light . the tension in the rope is then equal to P lb.wt. the upward tensions in the two portions of the rope on either side of the pulley A1 support the load W , and hence W = 2P . the mechanical advantage is thus ideally 2 , although in practice it is less . the V.R. can be deduced by imagining W raised a distance y , when a length 2y of rope moves round the pulley A . the effort , P , thus moves a distance 2y , and hence the V.R. is 2y/y , or 2 . it will be noted that the fixed pulley F offers a convenient means of applying the force P in a downward direction ; as it is fixed , F is not a part of the pulley system , which comprises only A1 in this case . a more practical form of the same pulley system , often used by builders , is shown in figure (b) . in this case there are three movable pulleys , A1 , A2 , A3 , with a load of weight W attached to the lowest pulley . we shall assume the pulleys are very light and that friction is absent . the relation between the effort P and W can be found by noting that the tension in the rope round A1 is W/2 , since the tension in the two parts of the rope round A1 support W . the two parts of the rope round A2 must each supply a tension of ½ of W/2 , or W/4 , by considering the pulley A2 ; and , similarly , the tension in the rope supporting the pulley A3 must be ½ of W/4 , or W/8 . but the tension in the latter rope is P , the effort applied . thus P = W/8 . consequently a theoretical mechanical advantage of 8 is obtained with this pulley system . by considering the lowest pulley A1 and the attached weight W to be raised a vertical distance x , and then following the lengths of rope moving round A1 , A2 , A3 , it can be shown that the effort P moves a distance 8x . in general , the V.R. is 2^n , where n is the number of pulleys ; thus if n = 3 , as in the figure above , the V.R. is 2^3 , or 8 . the mechanical advantage is 2^n only if friction and the weights of the pulleys are neglected . sources : 1 principles of physics . 



#2
Jul1809, 10:53 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,882

There are a lot of words here. Is there a question?




#3
Jul1909, 07:18 AM

P: 37

no there is not a question.



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