Statics is one of the sections of modern physics that studies the conditions for bodies and systems to be in mechanical equilibrium. To solve balance problems, it is important to know what the support reaction force is. This article is devoted to a detailed consideration 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 balance is the action 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 equation:
F=ma
This entry is known as Newton's second law. Here the force F is the resultant of all forces acting on the body.
If one body acts with some force F1¯ on the second body, then the second one acts on the first one with exactly the same absolute force F2 ¯, but it points in the opposite direction than F1¯. That is, equality is true:
F1¯=-F2¯
This entry is a mathematical expression for Newton's third law.
When solving problems using this law, students often make a mistake comparing these forces. For example, a horse is pulling a cart, while the horse on the cart and the cart on the horse exert the same force modulo. Why then is the whole system moving? The answer to this question can be given correctly if we remember that both of these forces are applied to different bodies, so they do not balance each other.
Support reaction force
First, let's give a physical definition of this force, and then we'll explain with an example how it works. So, the force of the normal reaction of the support is the force that acts on the body from the side of 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 support reaction. It is usually denoted by the letter N.
Force N is a contact value. If there is contact between bodies, then it always appears. In the example above, the value of N is equal in absolute value to the weight of the body. However, this equality is only a special case. The support reaction and body weight are completely different forces of a different nature. Equality between them is always violated when the angle of inclination of the plane changes, additional acting forces appear, or when the system moves at an accelerated rate.
Force N is called normalbecause it always points 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, the weight of the body and the normal force N are not action and reaction, since they are both applied to the same body (glass of water).
Physical cause of N
As it was found out 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 body under the influence of a load is initially deformed elastically. The elastic force tends to restore the previous shape of the body, so it has 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 value N is the result of the Pauli principle. With a slight approach of atoms, their electron 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 force N?
It should be said right away that there is no definite formula for the support reaction force. Nevertheless, there is a technique that can be used to determine N for absolutely any system of interacting bodies.
The method for determining the value of N is as follows:
- first 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 action of the support reaction;
- solving the resulting Newton equation in the marked direction will lead to the desired value N.
When compiling a dynamic equation, one should carefully and correctly place the signs of the acting forces.
You can also find the support reaction if you use not the concept of forces, but the concept of their moments. The attraction of moments of forces is fair and convenient for systems that have points or axes of rotation.
Next, we will give two examples of solving problems in which we will show how to use Newton's second law and the concept of the moment of force to find the value of N.
Problem with a glass on the table
This example has already been given above. Assume that a 250 ml plastic beaker is filled with water. It was placed on the table, and a book weighing 300 grams was placed on top of the glass. What is the reaction force of the table support?
Let's write a dynamic equation. We have:
ma=P1+ P2- N
Here P1 and P2 are the weights of a glass of water and a book, respectively. Since the system is in equilibrium, then a=0. Considering that the weight of the body is equal to the force of gravity, and also neglecting the mass of the plastic cup, we get:
m1g + m2g - N=0=>
N=(m1+ m2)g
Given that the density of water is 1 g/cm3, and 1 ml is equal to 1cm3, we obtain according to the derived formula that the force N is 5.4 newtons.
Problem with a board, two supports and a load
A board whose mass can be neglected rests on two solid supports. The length of the board is 2 meters. What will be the reaction force of each support if a weight of 3 kg is placed on this board in the middle?
Before proceeding to the solution of the problem, it is necessary to introduce the concept of the moment of force. In physics, this value corresponds to the product of the force and the length of the lever (the distance from the point of application of the 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, let's calculate the total moment of forces relative to one of the supports (right). Let's denote the length of the board with the letter L. Then the moment of gravity of the load will be equal to:
M1=-mgL/2
Here L/2 is the lever of gravity. The minus sign appeared because the moment M1 rotates counterclockwise.
Moment of the reaction force of the support will be equal to:
M2=NL
Since the system is in equilibrium, the sum of the moments must be equal to zero. We get:
M1+ M2=0=>
NL + (-mgL/2)=0=>
N=mg/2=39, 81/2=14.7 H
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 forcethe left support will also be equal to 14.7 N.
