Mechanical advantage

We can say that the biggest difficulty for drivers happens when changing the car tire. We know that loosening the car wheel bolts is not a very easy task, as it requires a little “force”. Even using the wheel wrench, we can still say that it is a little difficult, because sometimes the screws are too tight.

In some cases, it is even necessary to jump on top of the wheel wrench to be able to bend the screws. Thus, we perceive a certain ease when we step as far as possible from the key’s rotation axis, thus increasing the torque.

We can use another resource to loosen the screws: the use of a wheel wrench with a larger handle, which allows increasing the force arm and, consequently, the torque. To loosen the most difficult screws, the tire repairmen usually fit a pipe in the wheel wrench. After so much futile effort, they learn, through experience, to increase torque.

Many tools that we use in our daily lives reduce the force needed to loosen or tighten nuts, screws, cut wires, etc. In the wheel wrench shown in the figure below, for example, the force to tighten the bolt was applied away from the axis of rotation, which passes through point O.

To loosen or tighten the wheel bolt by hand, without the wheel wrench, we would need a lot of strength, much more than we have. But then, what is the advantage of using the wheel wrench?

When we use a wheel wrench to tighten a bolt, the magnitude of the force it exerts on the bolt, according to Newton’s third law, is the same force that the bolt exerts on the wheel wrench. We call the resistant force ( ) that which the screw applies to the tool, because it represents the reaction force of the screw to a force applied directly on it by the tool, sufficient to tighten it. The force we apply to the tool is called action force ( ) or powerful force .

It is not practical to use dynamometers to measure the intensity of these forces. For this reason, we will define the mechanical advantage of the wheel wrench, for example, based on the quantities associated with the structure of the tool or the screw. We will do this with the help of the torque concept.

The magnitude of the torque produced by the force applied to the end of the key, with respect to point O , is given by:

τ=F a .b

To exert the same torque with a force ( ) applied directly to the screw head (point P), we would have:

τ=F r .r

As the torques applied by the tool and the hand were assumed to be equal, we have:

Finally, we have:

This expression can be used to determine the mechanical advantage in spanners, pliers and screwdrivers, etc.

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