Mechanical Efficiency and Mechanical Advantage

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Two factors are important in selecting an appropriate machine for performing a given task: mechanical efficiency and mechanical advantage.

Mechanical efficiency gives an indication of the losses that may be incurred while performing a certain job. It is the fraction of the input work W_i that is available for carrying out the desired output work W_o:

${{\eta}_m}=\frac{{W_c}}{{W_i}} \qquad \qquad(1)$

In a real machine, some of the input work is inevitably lost as heat, so all real machines have efficiencies that are less than 1.

Question: Suppose you hook two machines together, one with efficiency of 40% and the other with an efficiency of 60%. What is the efficiency of the two machines operating together?

Answer: If you are tempted to say 100%, try to resist. The first machine will feed only 40% of the original input energy into the second machine, which then puts out 60% of the 40%. In other words, each machine loses some energy and the net efficiency is 40%x60% = 24%.

Question: A transmission gearbox transmits power from the engine to the axle through a number of interlocking gears. To get more power we need to shift the transmission to a lower gear, which will cause the speed to decrease. Can we design transmissions that increase power and at the same time increase velocity?

Answer: No. A machine can increase the magnitude of the effort (input) force or increase the velocity of the object to be moved, but not both. If a machine did both at the same time, the output work would exceed the input work, which is impossible.

Mechanical advantage (MA) is the amount by which the machine multiplies the force. It is the ratio of the force exerted by a machine (or the resistance force F_r) to the force exerted up on a machine (or the effort force F_e). In an ideal machine no energy is wasted so the output work equals the input work $(W o = W i)$ , i.e; $F r . d r = F e . d e$ . Therefore,

$MA=\frac{{F_r}}{{F_e}}=\frac{{d_e}}{{d_r}} \qquad \qquad(2)$

where d_r and d_e are corresponding displacements by F_r and F_e (lever arms). Forces are not, however, the only things that are magnified by levers. Since the longer arm of the lever has to travel a greater distance than the smaller arm of the lever in the same amount of time, it travels faster.

$\frac{{V_e}}{{V_r}}=\frac{{d_e}}{{d_r}} \qquad \qquad(3)$

You can infer from equations 2 and 3 that the gain in velocity or displacement is inversely proportional to reduction in effort, and is therefore inversely proportional to mechanical advantage. Hence you may use a lever to gain either mechanical advantage or speed, but not both.

Question: What is the mechanical advantage of a hammer driving a nail through a block of wood?

Answer: The energy used to carry the hammer through its flight (from the time of raising your hand until the time that hammer hits the nail) is used to push the nail only a few millimeters into the wood. For example, if we lower our hand by 50 cm, but the nail is pushed down only by 0.5 cm, then we have reduced the displacement by100 times, and at the same time multiplied the force by a factor of 100.

Question: Using the pulley arrangement shown in Figure 1, how much force is required to lift a 200-pound package? What is the mechanical advantage?

Answer: The load is divided between two cables holding the pulley at the right. The force is one half (100 pounds) and the mechanical advantage is two.

Unlike mechanical efficiency, mechanical advantage may or may not be greater than 1. Machines are designed to make a task simpler by increasing the force applied; therefore, for most machines mechanical advantage is greater than one. The muscles in our bodies also act as levers. Unlike machines, however, in many instances they do not make a particular task (such as lifting a weight) simpler. In fact, the human body is not designed for strength, but for agility, speed and wide ranges of motion.

Figure 1: A human elbow acts as a third class lever during the lifting of a weight

Question: Our muscles are much stronger than they seem. For instance, the bicep is attached to the forearm about 8 times closer to the fulcrum (the elbow) than one’s hand is when lifting a weight. What is the advantage of this seemingly inefficient arrangement?

Answer: The advantage is speed (also a factor of 8), which is frequently more important in the animal world.

Example: What is the mechanical advantage of the system of pulleys?

Solution: The mechanical advantage of a pulley system is approximately equal to the number of supporting ropes or strands. A single pulley can only change the direction of the pull to lift a load. Therefore, pulley shown in (a) has a mechanical advantage of 1 (MA = 1). The pulley system in (b) has two supporting pulleys and therefore has a MA = 2.

Example: What is the force required for moving a 200 kilogram stone? What is the mechanical advantage? If we want to lift the stone at a rate of 5 cm a second, what is the speed at which we need to push down on the lever?

Solution: From equation (2) we have $(200 x 9.8) x 0.2 = F e x 1.0$ , or Fe = 392 N. The mechanical advantage is:

$MA=\frac{{F_r}}{{F_e}}=\frac{(200)(9.8)}{392}=\frac{1.0}{0.2}=5.0$

Using equation 3, we have

${V_e}={V_r}. \frac{{d_e}}{{d_r}}=5. \frac{1.0}{0.2}=25 cm/s$

References

(1) Toossi Reza, "Energy and the Environment:Sources, technologies, and impacts", Verve Publishers, 2005

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Mechanical Efficiency and Mechanical Advantage

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