Pulley

It is a wheel or a flat disc which has a grooved edge or a rim. A pulley rotates about a fixed axis which passes through the centre. The axle is fixed to a frame or a block. If the block is clamped, the pulley is called fixed pulley and if the block is not clamped, the pulley is called moveable pulley.

Single fixed pulley

It is a single pulley which is usually used to draw water from well. The frame of the pulley is clamped.

A rope is passed through the groove. To one end of the rope, a load is tied and [cm_tooltip_parse] effort [/cm_tooltip_parse] is applied at the other end. The effort is applied in the downward direction. The tension 'T' of the rope acts on both the sides of the rope upward.

If the pulley is not rotating and the load due to friction and moveable parts is neglected, then both the load and the effort are equal to the tension in the string.

Thus, W = P = T

[cm_tooltip_parse] Mechanical advantage [/cm_tooltip_parse] $=\dfrac{Load}{Efforts} = \dfrac{W}{T}=1$

In practice, the effort applied is greater than load due to friction in the moveable parts of the pulley.

If the effort moves through a distance 'x', load is lifted through the same distance also. Hence,\(\text{velocity ratio} =\dfrac{\text{ Distance moved by the effort}}{\text{ Distance moved by the load}}=1\) Thus, there is no gain in mechanical advantage or in velocity ratio of the pulley. It is used only to change the direction of effort. It is easier to exert effort in the downward direction, i.e., in the direction of gravity which makes lifting of loads convenient.

Efficiency of an ideal pulley

Efficiency(η) = [cm_tooltip_parse] Ideal mechanical advantage [/cm_tooltip_parse]/Velocity ratio (V.R.) If Ideal mechanical advantage (I.M.A.)=1 and Velocity ratio (V.R.) = 1 Then Efficiency (η) = 1 or 100% Thus, the [glossary_exclude]efficiency[/glossary_exclude]of an ideal pulley is 100%. In practice, the effort applied is used to left useful loads, to overcome loads due to friction and load due to Moveable parts. But the total loads is equal to effort. If the useful load is \(W_1\) and R is load due to friction and Moveable parts, the total load = \(W_1\)+R $⇒W_1+R = P$ $⇒\dfrac{W_1+R}{P} = 1⇒\dfrac{W_1}{P}+\dfrac{R}{P}=1$ $⇒\dfrac{W_1}{P}=1-\dfrac{R}{P}$ But$ \dfrac{W_1}{P}=\dfrac{\text{Useful load}}{Effort}$ = Actual Mechanical Advantage (A.M.A) $∴ A.M.A. = 1-\dfrac{R}{P}$

Single Moveable Pulley

If the block of the pulley is not clamped, it is called moveable pulley. In this case, the pulley moves as the effort pulls the rope. One end of a rope is fixed to a hook. The rope passes through the rim of the pulley and the effort is applied in the upward direction at the other end of rope. The moveable pulley divides the rope into two segments. Tension 'T' acts in an upward direction along the two segments of the rope. If P is the effort applied to overcome total load, then P = T.

Sheath Pulley System or Block and Tackle System

It consists of two sets of pulleys. One set of pulleys is attached to a rigid support at the top and is called block. The lower set of pulleys carries a load and is called tackle. One end of the string or rope is fixed to a hook of the tackle. The rope passes round the rim of all the pulleys and the effort is applied at the free end of the rope in the downward direction, as shown in the Fig. 3. Tackle is supported by 'n' segments of the rope, where 'n' is the number of pulleys in the system. Tension on each segment of the rope is T and it acts vertically upwards. If P is the effort used to overcome load W, then W = nT and P = T. Thus, the ideal W nT = P T mechanical advantage of the pulley system, = \(\dfrac{W}{P}\) = \(\dfrac{nT}{T}\) = n Thus, the I.M.A. of a pulley system is equal to number of pulleys in the system. The effort required to lift the load is (W/n). Thus, the system of pulleys acts as a force multiplier and it also changes the direction of force. Thus, the larger the number of pulleys, the lesser is the effort. If the effort moves through a distance x, then length of each segment of the rope is decreased by (W/n). Thus, the velocity ratio of the pulley system (V.R.) $=\dfrac{\text{Displacement of the effort}}{\text{Displacement of the load}}⇒ ^{V.R.} = \dfrac{x}{\dfrac{x}{n}}=n$ The efficiency of system of Pulleys $η=\dfrac{M.A.}{V.R.}=\dfrac{n}{n}$ = 1 or 100% η is 100% only in an ideal system of Pulleys. If \(W_1\) is the useful load to overcome by the effort, then M.A. = (W/P), where W is the total load. But W = \(W_1\)+R where R is the load due to friction and Moveable parts of the system. Then,  $ M.A. = \frac{W_1+R}{P} = n$ Since     M.A. = n We get  $\frac{W_1+R}{P} = n$ $\frac{W_1}{P}+\frac{R}{P} = n$ But\(\frac{W_1}{P}\) is [glossary_exclude]actual mechanical advantage[/glossary_exclude] of the pulley system A.M.A. $⇒    A.M.A+ \frac{R}{P} = n$ Hence, $A.M.A = n-\frac{R}{P}$ Since (R/P) is positive, A.M.A. of the pulley system is always less than n.

Effect of Weight of Pulleys in Tackle

If \(W_2\) is the Weight of the Pulleys in Tackle, then the load and Weight is balanced by nT, where n is the number of Pulleys and T is the Tension on each part of the rope, thus, $W+W_2 = nT = nP$      (∴ T = P) Where, P is the effort $⇒W+W_2 = nP$ $⇒\dfrac{W}{P}+\dfrac{W_2}{P} = n⇒\dfrac{W}{P} = n-\dfrac{W_2}{P}$ Hence, $\text{Mechanical advantage (M.A.)} = n-\dfrac{W_2}{P}$ $(∴\frac{W}{P} = \text{Mechanical advantage (M.A.)}$ The efficiency of the system of Pulleys, $η= \dfrac{\text{Mechanical advantage(M.A.)}}{\text{Velocity ratio(V.R.)}}$ ⇒    $η = \dfrac{n-\frac{W_2}{P}}{n}$ ⇒    $η = 1-\dfrac{W_2}{nP}$ Therefore, to achieve higher efficiency in system of Pulleys, the Weight of the Pulleys in the tackle should be lesser.