No law or ordinance is mightier than understanding. --Plato
Forces
Those intimidated by the difficult mathematical aspects of physics unfortunately lose sight of the fact that the questions asked are very simple and basic. It is not too much of a simplification to characterize physics as the study of how one body pushes or pulls on another to change its motion. Such pushes or pulls are forces. A Force causes an acceleration, which is, in essence, a change in a body's motion. Familiar forces are gravity, which causes a dropped object to accelerate towards the floor, or the electromagnetic force, which accelerates a metal nail towards a magnet. Forces are not restricted to objects we can hold in our hand, however. The force of gravity keeps the Earth orbiting the Sun and the Sun orbiting the center of our Galaxy. The electromagnetic force keeps electrons orbiting their nuclei in atoms.
Newton's Laws
How exactly do forces change the motion of bodies?
Sir Isaac Newton
provided the answer for
macroscopic
objects moving at speeds much less than the speed
of light. In his
Principia, first published in 1687,
he laid out his three famous
laws of
motion:
- Unless acted upon by an unbalanced force, an object will maintain a constant velocity.
- An applied force is equal to the rate of change of momentum.
- All forces occur in pairs, and these two forces are equal in magnitude and opposite in direction.
The First Law informs us that a body will remain at rest or will continue traveling in a straight line unless some unbalanced force acts on the body. A skater on the ice will slide in a straight line unless she pushes sideways on the ice to make herself turn or down on the ice to make herself jump. Only the slight frictional force between her skate and the ice causes her to slow. The second law is the fundamental law of classical dynamics. It tells us that the acceleration a of an object is proportional to the force F applied to it. The proportionality constant is the inertial mass m. Students know the Second Law as F = ma. The quantities a and F are bold-faced because they are vectors, which are quantities with both a magnitude and a direction. The Third Law is perhaps more familiarly known as the law that says, "For every action there is an equal and opposite reaction."
The Second Law
In order to calculate how a body responds to a force, we use the Second Law. Since the acceleration is the first time derivative of the velocity, which we denote by v, we may write
The velocity v itself is the first time derivative of the vector x that locates the position of the particle relative to the origin of the coordinate system used:
In some cases we may directly solve these differential equations with ordinary methods of calculus. In general, however, the differential equations do not have closed-form solutions, and we must solve the problem numerically.
The Newton Tool
The purpose of the Newton Tool is to allow an internet user to solve Newton's Second Law numerically and graph and download the results. The tool requires the user to enter an acceleration expression and initial conditions on the position and velocity. Upon submission of the data, the server at Clemson University integrates the appropriate differential equation and then allows the user to graph and download the results. In this way, the Newton Tool is intended to be an easy-to-use means to calculate the motion of a single particle in classical mechanics.
The best way to start using the tool is to try out the tutorials. Help links in the tool also aid the user in determining the correct input. More background and other related information is available from the Papers & Links sidebar link. We hope you enjoy the tool, and, as always, we welcome feedback.
