The world that surrounds us is in constant motion. Nevertheless, there are systems that can be in a relative state of rest and equilibrium. One of them is the lever. In this article, we consider what it is from the point of view of physics, and also solve a couple of problems on the condition of equilibrium of the lever.
What is a lever?
In physics, a lever is a simple mechanism consisting of a weightless beam (board) and one support. The location of the support is not fixed, so it can be closer to one end of the beam.
Being a simple mechanism, the lever serves to convert force into a path, and vice versa. Despite the fact that force and path are completely different physical quantities, they are related to each other by the formula of work. To lift a load, you need to do some work. You can do this in two different ways: apply great force and move the load a short distance or act with a slight force, but at the same time increase the travel path. Actually, the lever serves for this. In short, this mechanism allows you to win along the way and lose in strength or, conversely, win in strength, but lose in transit.
Leverage acting
This article is devoted to the equilibrium conditions of the lever. Any balance in statics (a branch of physics that studies bodies at rest) implies the presence or absence of forces. If we consider the lever in its free form (weightless beam and support), then no forces act on it, and it will be in balance.
When work is performed using a lever of any type, three forces always act on it. We list them:
- Cargo weight. Since the mechanism in question serves to lift loads, it is obvious that their weight will have to be overcome.
- External reaction force. This is the force applied by a person or another machine to counteract the weight of the load on the lever beam.
- Support reaction. The direction of this force is always perpendicular to the plane of the lever beam. The reaction force of the support is directed upward.
The condition for the balance of the lever involves consideration not so much of the marked active forces as the moments of forces created by them.
What is a moment of power
In physics, the moment of force, or torque, is called the value equal to the product of external force on the shoulder. The shoulder of force is the distance from the point of application of force to the axis of rotation. The presence of the latter is important in calculating the moment of force. Without the presence of an axis of rotation, it makes no sense to speak of a moment of force. Given the above definition, we can write the following expression for the torque M:
M = F * d
In fairness, we note that the moment of force is actually a vector quantity, nevertheless, to understand the topic of this article, it is enough to know how the modulus of the moment of force is calculated.
In addition to the formula above, it should be remembered that if the force F tends to rotate the system so that it begins to move counterclockwise, then the created moment is considered positive. On the contrary, the desire to rotate the system along the clock indicates a negative torque.
Formula of Leverage Equilibrium Condition
The figure below shows a typical lever, as well as the values of its right and left shoulders. External force is indicated by the letter F, and the weight of the load to be lifted is indicated by the letter R.
In statics, in order for the system to rest, two conditions must be met:
- The sum of the external forces that affect the system must be equal to zero.
- The sum of all the moments of the mentioned forces relative to any axis should be zero.
The first of these conditions means the lack of translational movement of the system. It is obvious to the lever, since its support stands firmly on the floor or ground. Therefore, checking the condition of the equilibrium of the lever involves only checking the justice of the following expression:
∑ i = 1 n M i = 0
Since in our case only three forces act, we rewrite this formula as follows:
R * d R - F * d F + N * 0 = 0
The reaction force does not create moment support. We rewrite the last expression in the form:
R * d R = F * d F
This is the condition for the balance of leverage (in the 7th grade of secondary schools in the physics course it is studied). The formula shows: if the value of the force F is greater than the weight of the load R, then the shoulder d F should be less than the shoulder d R. The latter means that by applying great force over a short distance, we can move the load over a long distance. The reverse situation is also true when F <R and, accordingly, d F > d R. In this case, the gain is observed in force.
Challenge with elephant and ant
Many people know the famous statement of Archimedes about the possibility of using the lever to move the whole globe. This bold statement makes physical sense, given the leverage balance formula described above. Leave Archimedes and the Earth alone and solve a slightly different problem, which is no less interesting.
The elephant and ant were placed on different shoulders of the lever. Suppose that the elephant’s center of mass is one meter from the support. At what distance should the ant be located to balance the elephant?
To answer the question of the problem, we turn to tabular data on the masses of the animals in question. We take the ant mass 5 mg (5 * 10 -6 kg), we will consider the elephant mass equal to 5000 kg. Using the leverage equilibrium formula, we obtain:
5000 * 1 = 5 * 10 -6 * x =>
x = 5000 / (5 * 10 -6 ) = 10 9 m.
An ant can really balance an elephant, but for this it must be located at a distance of 1 million kilometers from the lever support, which corresponds to 1/150 of the distance from the Earth to the Sun!
Problem with support at the end of the beam
As noted above, at the lever, the support under the beam can be located anywhere. Suppose that it is near one of the ends of the beam. Such a lever has a single shoulder, shown below in the figure.
Assume that the load (red arrow) has a mass of 50 kg and is located exactly in the middle of the lever arm. What size should the external force F (blue arrow) be, which is applied to the end of the shoulder to balance this load?
Denote the length of the lever arm by the letter d. Then we can write the equilibrium condition in the following form:
F * d = R * d / 2 =>
F = m * g / 2 = 50 * 9.81 / 2 = 245.25 N
Thus, the magnitude of the applied force should be half the weight of the load.
This type of lever is used in inventions such as a hand wheelbarrow for moving loads or a nut cracker.