Friction Reduces Pulleys’ Mechanical Advantage

 The presence of friction in mechanical designs is as guaranteed as conflict in a good movie, and engineers inevitably must deal with the conflicts friction produces within their mechanical designs.   But unlike a good movie, where conflict presents a positive, engaging force, friction’s presence in pulleys results only in impediment, wasting energy and reducing mechanical advantage.  We’ll investigate the math behind this phenomenon now.

Friction Reduces Pulleys’ Mechanical Advantage

   A few topics back we performed a work input-output analysis of an idealized situation in which no friction is present in a compound pulley.   The analysis yielded this equation for mechanical advantage,

                                                MA = d₂ ÷ d₁                       (1)

where d₂ is the is the length of rope Mr. Toga extracts from the pulley in order to lift his urn a distance d₁ above the ground.   Engineers refer to this idealized frictionless scenario as an ideal mechanical advantage, IMA, so equation (1) becomes,

                                                IMA = d₂ ÷ d₁                       (2)

   We also learned that in the idealized situation mechanical advantage is the ratio of the urn’s weight force, W, to the force exerted by Mr. Toga, F, as shown in the following equation.   See our past blog for a refresher on how this ratio is developed.

                                                IMA = W ÷ F                          (3)

   In reality, friction exists between a pulley’s moving parts, namely, its wheels and the rope threaded through them.   In fact, the more pulleys we add, the more friction increases.

   The actual amount of lifting force required to lift an object is a combination of F_F , the friction-filled force, and F, the idealized friction-free force.  The result is F_Actual as shown here,

F_Actual = F + F_F                       (4)

   The real world scenario in which friction is present is known within the engineering profession as actual mechanical advantage, AMA, which is equal to,

                                                AMA = W ÷ F_Actual                  (5)

   To see how AMA is affected by friction force F_F, let’s substitute equation (4) into equation (5),

                                                AMA = W ÷ (F + F_F)                (6)

   With the presence of F_F in equation (6), W gets divided by the sum of F and F_F .   This results in a smaller number than IMA, which was computed in equation (3).   In other words, friction reduces the actual mechanical advantage of the compound pulley.

   Next topic we’ll see how the presence of F_F translates into lost work effort in the compound pulley, thus creating an inequality between the work input, WI and work output WO.