Newton studied motion at its very basis. For such cases, being characterized by low speeds, his could choose to ignore friction (such as wind drag) and still obtain reasonable numerical answers. Another advantage of the idealization, "little or no friction," is that any general motion of a body in space can be considered as composed of two components. One component is the projection of the motion on a plane parallel to earth. The rules of uniform (or unforced) motion adequately describe this component. The second component, the projection of the motion of the body on a vertical "Z-axis," was termed natural motion. Since this component must included the constant force of nature, the force of gravity, accelerated motion is exhibited.
Newton postulated that all matter exerts mutual attractive forces and that the forces are directed from the one to the other (and vice versa).. One mass, Earth, is immensely large and widely distributed. The collective attractive effect of all of Earth's mass was called gravity. Because of Earth, all other matter was subject, at all times, to the constant force of gravity force, |F|gravity,Earth. Newton and others agreed upon the equation form to represent the magnitude of the gravity force of Earth.
In Newton's time, system mass had been made quantitative and the radius of Earth had been measured by Eratosthanes. And since initial usages of the equation would be at the Earth surface, the radius, r, would equal ro,Earth. With these considerations the unknowns of the above equation are reduced to the product, the proportionality constant times the mass of Earth. To proceed to make gravity force quantitative, Newton and others realized the need to obtain a number for the constant, μME by experimentation.
Direct Measurement of μMEarth:
The determination of this constant was one of the very early scientific endeavors. Galileo (who died about the time Newton was born) sought the constant by use of block-sliding-down-inclined-plane experiments, He was unsuccessful. This example shows why the direct "free-fall" method was prohibitive in Newton's time. (This method is used with electronic timers in today's physics labs). The history here is quite interesting. Atwood (~1740) developed the first Atwood's Machine which obtained the constant in a quite ingenious way. Later Cavendish measured the gravational constant, μ. So read this example as a beginning explanation of this famous constant.
The magnitude of μME has been determined [9.81m/s2 (or 32.2ft/s2)] and the group has been given a new symbol, go,E and the name: "Earth surface acceleration of gravity." When this result is substituted into Newton's postulated equation the magnitude of the force of gravity is made quantitative as:
BLUE OCEAN TOWING: Our company has contracted to protect oil production structures in the Davis Straight (northeast of Canada) from ice hazzards. A massive slab (~ 800 Gkg) has cleaved and is moving with the sea current toward a production rig. Our biggest ocean tug will apply a steady pull to deflect the slab from its path. Use the geometry of the situation with the initial conditions. Solve Newton's Second Law to determine the least constant Towing Force required of our ocean tug. HINT: This solves like "free-fall" but the slab moves horizontally.
GOD Lifted Earth:
In Massachusetts (1710) an accused witch was hanged in the public square. At the moment of her death, when the rope snapped taut, it seemed to all that an earthquake had occurred. But later the rumor spread that she was innocent and "God Lifted Earth" to receive her. The next Sunday, a Bishop explained, "She was guilty. It was gravity, not God. She fell down, Earth fell up." Can this be explained?
Hi! we are now quite a ways into this writing but we have not encountered the famous high-school explanation of Newton's Laws of Motion?
Where F = ma? We have used Newton's Laws of Motion repeatedly. The "f = ma" form is a watered-down algebraic form. Newton was obliged to understand vectors and calculus (not grade school topics).
WHERE is f = ma ?
This writing does not use the algebraic, high-school-physics equation, f = ma. This understanding is that Newton's focus was momentum, not force. The equation, f = ma, represents Newton's ideas algebraically, stripped of the mathematics he used.
Newton began with two axioms and the powerful mathematics of vectors and calculus. To understand what Newton understood requires better mathematics than is taught in high school.
Newton's perspective, what he was driving at, is made clear upon understanding the axioms he set down as basis for his Laws of Motion. In "quantity of matter," the first axiom of two, he established that there is mass, as a measurable scalar property of obvious importance in any motion of a body. His second axiom addressed "quantity of motion," something less easy as mass to define. Newton was obliged to use vectors to quantify position. Position was a prerequisite idea (with its space and vector basis). Change of position in time begat velocity. which required vector calculus to specify. The focus of his Second Law was the product, mass of a body times its vector velocity. Momentum is the principle, second idea. We will show the few steps beginning with f = ma required to return to Newton's formulation, the vector, first-order differential momentum equation.
The form of Newton's Laws of Motion selected for this writing is arranged in accord with Newton's axiomatic approach. Calculus is made explicit; the derivative of body momentum is placed prominently, left of equality. Subscripts of the derivative to identify the system (body for now). Superscripts to the derivative to identify the vector space. Finally ΣF replaces F right of equality (where the nonhomogeneous term belongs). Vectors... all properly notated. Phew... also we need the initial conditions near the the differential equation
Mathematical notation says this better than words.
In its use, Newton's equation is accompanied by a sketch of the physical situation. The sketch below might seem over-complete but it puts a picture to all of the ideas together.
There is a highly compelling reason students should use the differential equation form of Newton's Second Law of Motion. Later in studying engineering you will find the mass equation, momentum equation and energy equation for a body have the same form. All three physical statements are first order differential equations (with their respective initial conditions). The skills and understanding gained in solving any of them are the same skills needed to solve and understand the others. All three equations take the same perspective - system.
PROPERTY EQUATIONS (at their simplest levels) are written above. Each of them is a first-order differential equation with the property as the independent variable. To learn how to solve any of them is to learn how to solve the others. The mathematics is the same. The calculus involved is no more than the concept of derivative.