Newton's second law is perhaps the most famous of the three laws of classical mechanics that an English scientist postulated in the middle of the 17th century. Indeed, when solving problems in physics for the movement and balance of bodies, everyone knows what the product of mass and acceleration means. Let's take a closer look at the features of this law in this article.
The place of Newton's second law in classical mechanics
Classical mechanics is based on three pillars - three laws of Isaac Newton. The first of them describes the behavior of the body if external forces do not act on it, the second describes this behavior when such forces arise, and finally, the third law is the law of the interaction of bodies. The second law occupies a central place for good reason, since it links the first and third postulates into a single and harmonious theory - classical mechanics.
Another important feature of the second law is that it offersa mathematical tool to quantify the interaction is the product of mass and acceleration. The first and third laws use the second law to obtain quantitative information about the process of forces.
Impulse of power
Further in the article, the formula of Newton's second law, which appears in all modern textbooks on physics, will be presented. Nevertheless, initially the creator of this formula himself gave it in a slightly different form.
When postulating the second law, Newton started from the first. It can be mathematically written in terms of the amount of momentum p¯. It is equal to:
p¯=mv¯.
The amount of motion is a vector quantity, which is related to the inertial properties of the body. The latter are determined by the mass m, which in the above formula is the coefficient relating the speed v¯ and momentum p¯. Note that the last two characteristics are vector quantities. They point in the same direction.
What will happen if some external force F¯ starts acting on a body with momentum p¯? That's right, the momentum will change by the amount dp¯. Moreover, this value will be the greater in absolute value, the longer the force F¯ acts on the body. This experimentally established fact allows us to write the following equality:
F¯dt=dp¯.
This formula is Newton's 2nd law, presented by the scientist himself in his works. An important conclusion follows from it: the vectorchanges in momentum are always directed in the same direction as the vector of the force that caused this change. In this expression, the left side is called the impulse of the force. This name has led to the fact that the amount of momentum itself is often called momentum.
Force, mass and acceleration
Now we get the generally accepted formula of the considered law of classical mechanics. To do this, we substitute the value dp¯ into the expression in the previous paragraph and divide both sides of the equation by the time dt. We have:
F¯dt=mdv¯=>
F¯=mdv¯/dt.
The time derivative of velocity is the linear acceleration a¯. Therefore, the last equality can be rewritten as:
F¯=ma¯.
Thus, the external force F¯ acting on the considered body leads to the linear acceleration a¯. In this case, the vectors of these physical quantities are directed in one direction. This equality can be read in reverse: the mass per acceleration is equal to the force acting on the body.
Problem Solving
Let's show on the example of a physical problem how to use the considered law.
Falling down, the stone increased its speed by 1.62 m/s every second. It is necessary to determine the force acting on the stone if its mass is 0.3 kg.
According to the definition, acceleration is the rate at which speed changes. In this case, its modulus is:
a=v/t=1.62/1=1.62 m/s2.
Because the product of mass byacceleration will give us the desired force, then we get:
F=ma=0.31.62=0.486 N.
Note that all the bodies that fall on the Moon near its surface have the considered acceleration. This means that the force we found corresponds to the force of the moon's gravity.
