Newton's Laws

Newton's Laws of Motion

Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces. They have been expressed in several different ways over nearly three centuries, and can be summarized as follows.

1. First law. If an object experiences no net force, then its velocity is constant: the object is either at rest (if its velocity is zero), or it moves in a straight line with constant speed (if its velocity is nonzero).
2. Second law. For a body with constant mass, the acceleration a of the body is parallel and directly proportional to the net force $F$ acting on the body, is in the direction of the net force, and is inversely proportional to the mass $m$ of the body, i.e., $F = m a$. In general, the force is directly proportional to the rate of change of momentum of the body, i.e., $F=\frac{d}{d t}(mv)$.
3. Third law. When a first body exerts a force $F_1$ on a second body, the second body simultaneously exerts a force $F_2 = -F_1$ on the first body. This means that $F_1$ and $F_2$ are equal in magnitude and opposite in direction.

The three laws of motion were first compiled by Sir Isaac Newton in his work Philosophiae Naturalis Principia Mathematica, first published in 1687. Newton used them to explain and investigate the motion of many physical objects and systems. For example, in the third volume of the text, Newton showed that these laws of motion, combined with his law of universal gravitation, explained Kepler's laws of planetary motion.

Newton's Laws of Gravitation

Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. (Separately it was shown that large spherically symmetrical masses attract and are attracted as if all their mass were concentrated at their centers.) This is a general physical law derived from empirical observations by what Newton called induction. It is a part of classical mechanics and was formulated in Newton's work Philosophiae Naturalis Principia Mathematica ("the Principia"), first published on 5 July 1687. (When Newton's book was presented in 1686 to the Royal Society, Robert Hooke made a claim that Newton had obtained the inverse square law from him; see the History section below.) In modern language, the law states the following.

A Python program

Here is a Python program to plot the gravitational force between two point masses.

    import numpy as np
    import matplotlib.pyplot as plt

    G = 6.67430e-11

    print("Calculating the gravitational force between two 
        point masses\r\n")

    m_1 = float(input("Mass 1 (kg): "))
    m_2 = float(input("Mass 2 (kg): "))
    r_max = float(input("Maximum distance (m): "))
    r_min = -1

    while (r_min != r_min) or  (r_min <= 0) or (r_min >= r_max):
        try:
            r_min = float(input("Minimum distance, > 0 (m): "))
        except ValueError:
            print("This needs to be a positive number.")

    r = np.linspace(r_min, r_max, 300)
    F = G * m_1 * m_2 / r**2

    plt.plot(r, F)
    plt.title("Gravitational force")
    plt.xlabel("distance, r")
    plt.ylabel("force, F")
    plt.show()

                

A live Python demonstration

Here a live Python demonstration to plot the gravitational force between two point masses. Click the triangle to run it.

Appendix

The table below shows the SI units for the quantities involved in the above discussion.