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Consider the bottle opener in Figure 10-9 When you use the opener, you lift the handle, thereby doing work on the opener The opener lifts the cap, doing work on it The work that you do is called the input work, Wi The work that the machine does is called the output work, Wo Recall that work is the transfer of energy by mechanical means You put work into a machine, such as the bottle opener That is, you transfer energy to the opener The opener, in turn, does work on the cap, thereby transferring energy to it The opener is not a source of energy, and therefore, the cap cannot receive more energy than the amount of energy that you put into the opener Thus, the output work can never be greater than the input work The machine simply aids in the transfer of energy from you to the bottle cap Mechanical advantage The force exerted by a person on a machine is called the effort force, Fe The force exerted by the machine is called the resistance force, Fr As shown in Figure 10-9a, Fe is the upward force exerted by the person using the bottle opener and Fr is the upward force exerted by the bottle opener The ratio of resistance force to effort force, Fr /Fe , is called the mechanical advantage, MA, of the machine Mechanical Advantage MA

machine effort force resistance force mechanical advantage ideal mechanical advantage efficiency compound machine

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Figure 10-9 A bottle opener is an example of a simple machine It makes opening a bottle easier, but it does not lessen the work required to do so

Figure 10-10 A fixed pulley has a mechanical advantage equal to 1 (a) A pulley system with a movable pulley has a mechanical advantage greater than 1 (b)

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In a fixed pulley, such as the one shown in Figure 10-10a, the forces, Fe and Fr , are equal, and consequently MA is 1 What is the advantage of this machine The fixed pulley is useful, not because the effort force is lessened, but because the direction of the effort force is changed Many machines, such as the bottle opener shown in Figure 10-9 and the pulley system shown in Figure 10-10b, have a mechanical advantage greater than 1 When the mechanical advantage is greater than 1, the machine increases the force applied by a person You can write the mechanical advantage of a machine in another way using the definition of work The input work is the product of the effort force that a person exerts, Fe , and the distance his or her hand moved, de In the same way, the output work is the product of the resistance force, Fr, and the displacement of the load, dr A machine can increase force, but it cannot increase energy An ideal machine transfers all the energy, so the output work equals the input work: Wo Wi or Fr dr Fe de This equation can be rewritten Fr /Fe de /dr Recall that mechanical advantage is given by MA Fr /Fe Therefore, for an ideal machine, ideal mechanical advantage, IMA, is equal to the displacement of the effort force, divided by the displacement of the load The ideal mechanical advantage can be represented by the following equation Ideal Mechanical Advantage IMA

The ideal mechanical advantage of an ideal machine is equal to the displacement of the effort force, divided by the displacement of the load

And the second result, like this (and so on):

Note that you measure the distances moved to calculate the ideal mechanical advantage, but you measure the forces exerted to find the actual mechanical advantage

Efficiency In a real machine, not all of the input work is available as output work Energy removed from the system means that there is less output work from the machine Consequently, the machine is less efficient at accomplishing the task The efficiency of a machine, e, is defined as the ratio of output work to input work Efficiency e

The efficiency of a machine (in %) is equal to the output work, divided by the input work, multiplied by 100

An ideal machine has equal output and input work, Wo /Wi 1, and its efficiency is 100 percent All real machines have efficiencies of less than 100 percent Efficiency can be expressed in terms of the mechanical advantage and ideal mechanical advantage Efficiency, e Wo/Wi, can be rewritten as follows:

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