PULLEYS

shows a single fixed pulley arrangement which consists of a wheel with a groove around its circumference to receive a rope or chain. It is mounted on an axle about which it can rotate freely. The ends of the axle are supported by a frame called block. Block may be fixed or movable, if it is fixed the pulley is called fixed pulley and if it is movable the pulley is called movable pulley.

Figure 13.7 (a) Single Fixed Pulley Arrangement, (b) Single Movable Pulley, and (c) One Fixed and One Movable Pulley Arrangement

There are following assumptions made for pulley arrangement:

• Pulley weight is neglected since the pulley weight is considered very small compared to the weight to be tied.
• Pulley is smooth hence tensions in both side of the string tied on pulley is same.

Direction of effort P in Figure 13.7 is vertically downward; it may be upward or inclined which does not change the value of tension in the string. In all cases, P = W

Figure 13.7 (b) shows a single movable pulley arrangement. The block is movable. In this case, the effort may be applied in any convenient direction. The weight W will be equally shared by the two portions of the string. In this case, mechanical advantage is more than one. But, the force cannot be applied easily. For easy application of force P, a fixed pulley is introduced as shown in Figure 13.7 (c).

It is clear from Figure 13.7 (c) that to raise the load W through a certain distance, the effort will have to move double the distance.

Hence,

13.6.1  Order of Pulley

First Order Pulley: VR = 2ⁿ where n is number of movable pulleys, only one pulley is fixed.

Second Order Pulley: Three fixed and three movable pulleys are in the system. Single string passes over all the pulleys. VR = n where n is total number of pulleys.

Third Order Pulley: One pulley is fixed; one end of each string is connected to load. VR = 2ⁿ – 1 where n is total number of pulleys.

All the three orders of pulleys are shown in Figure 13.8.

Figure 13.8 (a) First Order Pulley, (b) Second Order Pulley, and (c) Third Order Pulley

Example 13.6: Four pulleys are arranged in first order in which one pulley is fixed. If the efficiency of the system is 95%, find the effort required to raise a load of 5,000 N. Also, find the friction of the machine.

Solution:

Number of movable pulleys, n = 3

VR for first order pulleys = 2ⁿ = 2³ = 8

Example 13.7: Six pulleys are arranged in second order, three pulleys in the upper block and three pulleys in lower block. It was found that a load of 1,200 N was raised by effort of 300 N. Calculate the efficiency and effort lost by friction in the machine.

Solution:

Ideal effort required to raise a load of 1,200 N, 

     Effort lost in friction = 300 – 200 = 100 N

Example 13.8: Four pulleys arranged in third order. A load of 1,500 N was raised by an effort of 300 N. Calculate the efficiency and effort lost in friction of the machine.

Solution:

VR for second order pulleys = 2ⁿ – 1 = 2⁴ – 1 = 15

Ideal effort required to raise a load of 1,200 N, 

    Effort lost in friction = 300 – 100 = 200 N

13.6.2  Weston’s Differential Pulley Block

Figure 13.9 shows Weston’s differential pulley block arrangement consisting of two blocks A and B. Upper block A is provided with two pulleys 1 and 2, having differential diameters. Both pulleys behave as one pulley. Lower block B carries a movable to which load W is attached. One string passes over all the pulleys. The effort is applied to the string passing over the bigger pulley of block A, which causes unwinding of rope from pulley 1 and winding on the pulley 2. The string on the effort side lengthens while the string on load side shortens, thereby raising the load.

Figure 13.9 Weston’s Differential Pulley Block

Let     W = load lifted by lower block B

          P = Effort applied to upper block A

          D = Diameter of pulley 1

          d = Diameter of pulley 2

When upper pulley rotates one by revolution, the effort will move by π D. Length of string will be reduced by the smaller pulley by π d. Net shortening of the string length = π (D – d).

Shortening of string is divided equally between the two portions of the string. Distance moved by the load 

Example 13.9: A differential pulley block consists of a lower block and an upper block. The upper block has two grooves one of which has a radius of 200 mm and the other a radius of 120 mm. If the efficiency of the machine is 45%, calculate the effort required to raise a load of 1,200 N.

Solution: