Statics is one of the branches of modern physics that studies the conditions for finding bodies and systems in mechanical equilibrium. To solve equilibrium problems, it is important to know what the strength of the support reaction is. This article is devoted to a detailed discussion of this issue.
Newton's second and third laws
Before considering the definition of the reaction force of the support, we should remember what causes the movement of bodies.
The reason for the violation of mechanical equilibrium is the effect on the body of external or internal forces. As a result of this action, the body acquires a certain acceleration, which is calculated using the following equality:
F = m * a
This entry is known as Newton’s second law. Here the force F is the result of all the forces acting on the body.
If one body acts with a certain force F 1 ¯ on the second body, then the second acts on the first with exactly the same force F 2 ¯ in absolute value, but it is directed in the opposite direction than F 1 ¯. That is, the equality is true:
F 1 ¯ = -F 2 ¯
This entry is a mathematical expression for the third Newtonian law.
When solving problems using this law, students often make a mistake by comparing these forces. For example, a horse carries a cart, while a horse on a cart and a cart on a horse exert equal modulo forces. Why, then, is the whole system moving? The answer to this question can be correctly given if we recall that both of these forces are applied to different bodies, so they do not balance each other.
Support reaction force
First, we give a physical definition of this force, and then explain with an example how it acts. So, the force of the normal reaction of the support is the force that acts on the body from the surface. For example, we put a glass of water on the table. To prevent the glass from moving with the acceleration of free fall down, the table acts on it with a force that balances the force of gravity. This is the reaction of support. It is usually denoted by the letter N.
Force N is a contact quantity. If there is contact between the bodies, then it always appears. In the example above, the value of N is modulo body weight. Nevertheless, this equality is only a special case. The support reaction and body weight are completely different forces that have a different nature. The equality between them is always violated when the angle of inclination of the plane changes, additional acting forces appear, or when the system moves accelerated.
The force N is called normal because it is always directed perpendicular to the plane of the surface.
If we talk about Newton’s third law, then in the example above with a glass of water on the table, body weight and normal force N are not an action and a reaction, since both of them are applied to the same body (a glass of water).
Physical reason for the appearance of force N
As it was found above, the reaction force of the support prevents the penetration of some solids into others. Why does this power appear? The reason is the deformation. Any solid bodies under the influence of load are first deformed elastically. The force of elasticity tends to restore the previous shape of the body, so it exerts a buoyant effect, which manifests itself in the form of a support reaction.
If we consider the issue at the atomic level, then the appearance of the quantity N is the result of the Pauli principle. With a small approach of atoms, their electronic shells begin to overlap, which leads to the appearance of a repulsive force.
It may seem strange to many that a glass of water can deform a table, but it is. The deformation is so small that it cannot be observed with the naked eye.
How to calculate the force N?
It should immediately be said that there is no specific formula for the reaction force of the support. Nevertheless, there is a technique, applying which, it is possible to determine N for absolutely any system of interacting bodies.
The method for determining the value of N is as follows:
- First, they write down Newton’s second law for the given system, taking into account all the forces acting in it;
- find the resulting projection of all forces on the direction of the reaction of the support;
- solving the obtained Newton equation in the indicated direction will lead to the desired value of N.
When compiling a dynamic equation, the signs of the acting forces should be carefully and correctly placed.
You can also find the support reaction if you use not the concept of forces, but the concept of their moments. The attraction of the moments of force is fair and convenient for systems that have points or axes of rotation.
Next, we give two examples of solving problems in which we show how to use the second Newtonian law and the concept of the moment of force to find the value N.
Task with a glass on the table
This example has already been given. Suppose a 250 ml plastic cup is filled with water. He was put on a table, and a book weighing 300 grams was placed on top of a glass. What is the reaction force of the table support?
We write the dynamic equation. We have:
m * a = P 1 + P 2 - N
Here P 1 and P 2 - the weight of a glass of water and books, respectively. Since the system is in equilibrium, a = 0. Given that the body weight is equal to gravity, and also neglecting the mass of the plastic glass, we get:
m 1 * g + m 2 * g - N = 0 =>
N = (m 1 + m 2 ) * g
Given that the density of water is 1 g / cm 3 and 1 ml is 1 cm 3 , we obtain according to the derived formula that the force N is equal to 5.4 Newton.
Task with a board, two supports and a load
The board, the mass of which can be neglected, lies on two solid supports. The length of the board is 2 meters. What will be the reaction force of each support if a load weighing 3 kg is placed on the middle of this board?
Before proceeding to the solution of the problem, the concept of the moment of force should be introduced. In physics, this quantity corresponds to the product of the force and the length of the lever (the distance from the point of application of force to the axis of rotation). A system with an axis of rotation will be in equilibrium if the total moment of forces is zero.
Returning to our task, we calculate the total moment of forces with respect to one of the supports (right). Denote the length of the board by the letter L. Then the moment of gravity will be equal to:
M 1 = -m * g * L / 2
Here L / 2 is the lever of gravity. A minus sign appeared because the moment M 1 rotates counterclockwise.
The moment of reaction force of the support will be equal to:
M 2 = N * L
Since the system is in equilibrium, the sum of the moments must be equal to zero. We get:
M 1 + M 2 = 0 =>
N * L + (-m * g * L / 2) = 0 =>
N = m * g / 2 = 3 * 9.81 / 2 = 14.7 N
Note that the force N does not depend on the length of the board.
Given the symmetry of the location of the load on the board relative to the supports, the reaction force of the left support will also be equal to 14.7 N.