The lever is a simple mechanism that has long been used by people to move heavy loads during the construction of various structures or when performing household tasks. This article gives the rule of equilibrium of the lever from the point of view of physics and considers an example of solving the problem of its application.
What is a lever?
Before giving the formula for the rule of equilibrium of the lever, we describe this simple mechanism. It is an object consisting of two parts: beams of a certain length and supports. The distances from the support to the ends of the lever are called the lengths of its shoulders. Devices can be equally-, multi- and single-armed. The shoulder ratio of this mechanism plays a key role in the application of the lever.
It allows you to make convenient the performance of a particular physical work. Note, he does not allow to win in the work, however, contributes to its division in such a way that it turns out to be much easier to accomplish. Depending on the ratio of the length of the shoulders, the lever gives a gain either in displacement or in the acting external force aimed at lifting loads.
Throughout history, this mechanism, together with other simple machines, has been used to lift and carry heavy loads. The lever was used by the Egyptians a few thousand years before our era. The rule of balance of the lever was experimentally established in the III century BC. e. Archimedes.
The moment of force and balance in statics
To understand what the rule of equilibrium of a lever consists of, which will be given below, one should study the concept of the moment of force. In physics, this term refers to the product of a shoulder of force and its module. The mathematical form of recording the moment of force has the form:
M = d * F.
This expression is written in scalar form. Here d is the shoulder of the force, which, as applied to the lever, is equal to the distance from the point of action F to the support.
The moment of force can be negative and positive. It will be negative when the force tends to rotate the lever in the direction of the clock.
By equilibrium in statics we mean the absence of translational and rotational motion of a system of bodies. In the first case, the sum of the forces acting on the system is zero, in the second case, the sum of the moments of these forces is zero.
Leverage balance rule in physics
Obviously, being on a support, the lever will not make translational motion, therefore, the sum of the forces acting on it is zero. However, it can make a rotation movement. The absence of the latter is called a state of equilibrium.
To understand when the lever will be in equilibrium, consider the following figure.
Here, leverage d F and d R are affected by forces F and R, respectively. In fact, there is a third force, which is the opposite of these two - this is the reaction of support. However, her shoulder is zero, so she will not be considered further.
In accordance with the condition of equilibrium in statics described above, the lever will not rotate if the sum of all the moment of forces is equal to zero. We write this rule of equilibrium of leverage in physics in mathematical form, we have:
R * d R - F * d F = 0.
We draw attention to the minus sign for the moment of force F, which tends to turn the lever clockwise. From this equality we obtain the relations:
R * d R = F * d F ;
d R / d F = F / R.
The formula for the lever balance rule in its final form was established by Archimedes as a result of experimental observations. Note that the Greek philosopher did not know about the magnitude of the moment of force.
The written equalities indicate that the smaller the ratio of shoulders d R / d F , the greater the gain in strength can be obtained (using a small F to lift a large weight of the load R). However, winning in strength, we will inevitably lose on the way. The last phrase is also true when it is formulated vice versa.
Example of a Leverage Balance
The figure above shows the single arm. It is known that a cargo weighing 100 kg is located at a distance of 1/3 of the length of the shoulder from the support. It is necessary to calculate the force F, which should be applied in order to balance the weight of the load.
In this problem, we consider a one-arm, not a two-arm lever. Nevertheless, the formula for its balance does not change, since it does not contain the lengths of the shoulders of the lever, but the shoulders of the application of forces. Denote the length of the entire shoulder by the letter d, then, applying the Archimedes equilibrium rule for the lever, we obtain:
d R / d F = F / R =>
1/3 * d / d = F / (m * g).
Where do we calculate the value of the force F:
F = 1/3 * m * g = 1/3 * 100 * 9.81 = 327 N.
The calculated force is three times less than the weight of the load, which clearly demonstrates the gain in strength with the help of the considered mechanism. This type of lever is used to move goods in a hand wheelbarrow.