This document describes the concept of a force and its relationship to acceleration.
Acceleration is the rate of change of velocity. In vectors, we defined velocity as the rate of change of position. In essence, we have acceleration -> velocity -> postion because acceleration affects velocity, which affects the position. The motion of objects that accelerate has continuity and fluidity because their movement becomes dependent on a previous (accelerated) state.
Acceleration doesn't merely refer to speeding up or slowing down; instead, it refers to any change in velocity—magnitude or direction.
Our definitions of force come from Sir Isaac Newton's three laws of motion:
- A body remains at rest or in motion at a constant speed in a straight line, except insofar as a force acts upon it
- At any instant of time, the net force on a body is equal to the body's acceleration multiplied by its mass or, equivalently, the rate at which the body's momentum is changing with time
- If two bodies exert forces on each other, these forces have the same magnitude but opposite directions
The first law is commonly stated as, "An object at rest stays at rest, and an object in motion stays in motion" (we could add, "...at a constant speed and direction unless acted upon by an unbalanced force"). As Newton established, in the absence of any forces, no force is required to keep an object moving. When a ball is tossed into the air, its velocity remains constant only in the absence of any forces or if the forces that act on it cancel each other out, meaning the net force adds up to zero. That is often referred to as equilibrium. In reality, an object's velocity changes because of unseen forces such as air resistance and gravity; hence, the falling ball will reach a terminal velocity (which stays constant) once the force of air resistance equals the force of gravity. In other words, the first law could be restated as "An object's velocity vector will remain constant if it's in a state of equilibrium".
Newton's second law is commonly stated as, "Force equals mass times acceleration", or, mathematically:
F = ma
In other words, acceleration is directly proportional to force and inversely proportional to mass. Putting this into base terms - the harder you're pushed, the faster you fall. That is, unless you're large, in which case the push will be less effective.
The third law is more commonly described as, "For every action, there is an equal and opposite reaction". That could be refactored as "Forces always occur in pairs. The two forces are of equal strength but in opposite directions". These forces do not cancel each other out because the forces act on different objects. And just because the two forces are equal doesn't mean that the objects' movements are equal (or that the objects will stop moving).
We can summarise Newton's three laws as:
A force is a vector that causes an object with mass to accelerate.
Furthermore, forces must accumulate. That is stated in the full definition of Newton's second law, F = ma, which implicitly suggests that "Net force equals mass times acceleration"; in other words, acceleration is equal to the sum of all forces divided by mass. Hence, if we ignore mass (set it to 1), as long as all the forces can be added together (accumulated), it doesn't matter how many forces there are, the sum total will give you the object's acceleration.
Mass is a measure of the amount of matter in an object (measured in kilograms in the metric system). An object that has a mass of 1 kilogram on Earth would also have a mass of 1 kilogram on the Moon (or anywhere else, for that matter).
Weight is the force of gravity on an object. From Newton's second law, you can calculate weight as mass times the acceleration of gravity. Weight is measured in newtons (N), a unit that indicates the magnitude of a gravitational force (on Earth, this is 9.8 N/kg). Because weight is tied to gravity, an object on the Moon weighs one-sixth as much as it does on Earth because the gravitational force there is weaker. Related to mass is density, which is defined as the amount of mass per unit of volume (grams per cubic centimetre, for example).
If you were to climb a ladder and drop two balls of different masses, they would hit the ground at the same time. Galileo made this discovery of simultaneous acceleration when he performed a similar test in 1589. The reason is that everything cancels out; gravity is calculated relative to an object's mass—so that the bigger the object, the stronger the force—but then you divide by the mass to determine the acceleration. Therefore, the acceleration of gravity for different objects is equal.
The best-known force is gravitational attraction. Every object with mass exerts a gravitational force on every other object, and the gravitational force between two bodies is proportional to the mass of those bodies and inversely proportional to the square of the distance between them. The formula for calculating the strengths of those forces is given for reference below:
Where the numerator (the top half of the right-hand side of the equation) shows G, which is the universal gravitational constant, m1 and m2 are the masses of two objects and r is the unit vector pointing from object 1 to object 2. Because r is a unit vector, magnitude is ignored, so this shows direction alone (i.e. it's a direction vector), and it can be computed by subtracting the position of one object from the other. Finally, the denominator (the bottom half of the right-hand side of the equation) is the distance between the two objects squared.
The formula shows that the larger the numerator, the bigger the force due to mass. However, the opposite is true for the denominator: the bigger the value (the farther away the object), the weaker the force. Mathematically, the strength of the gravitational force is inversely proportional to the distance squared. That is known as the inverse square law.
Friction is the force that resists motion when two surfaces come into contact. If the two surfaces do not move with respect to one another, it is called static friction. When the two surfaces move and slide against each other, it is called kinetic friction. Hence, friction is a dissipative force because it causes the kinetic energy of an object to be converted into another form, giving the impression of loss or dissipation. Here's the friction formula:
f = μN, where:
- f is the friction force
- μ is the coefficient of friction, which establishes the strength of a friction force for a particular surface
- N is the normal force in newtons, where N = mg, i.e. mass m times gravity g. It is the force perpendicular to the object's motion along a surface
The n-body problem involves solving an equation for the motion of a group of objects that interact via gravitational forces. For two objects, the motions can be precisely computed. Adding one more body turns the problem into a three-body problem for which no formal solution exists. However, that does not mean that it is impossible to model gravity applied to n-bodies; for example, the paper Classification of Symmetry Groups for Planar n-Body Choreographies by James Montaldi and Katrina Steckles, explores choreographic solutions to the n-body problem (defined as periodic motions of bodies following one another at regular intervals).
Springs and pendulums are examples of simple harmonic motion. A spring connects a movable bob (or mass) and a fixed anchor point. Hooke's law states that the force (F) needed to extend or compress a spring by some distance x scales linearly with respect to that distance—that is, F = -kx, where k is the spring constant that defines the spring's stiffness. The law is named after 17th-century British physicist Robert Hooke, who first stated the law in 1676 as a Latin anagram, "Ut tensio, sic vis" or "As the extension, so the force", or "The extension is proportional to the force".
At the maximum displacement −x, the spring is under its greatest tension, which forces the bob upward. At the maximum displacement +x, the spring reaches its greatest compression, which forces the bob back down. At either position of maximum displacement, the force is greatest and is directed toward the equilibrium position, the velocity (v) of the bob is zero, its acceleration is at a maximum, and the bob changes direction. At the equilibrium position, the velocity is at its maximum, and the acceleration (a) has fallen to zero.
Newton's second law, F = ma, can express how the displacement of the bob changes with time by setting ma = −kx. The acceleration a is the second derivative of x with respect to time t, and the resulting differential equation can be solved with x = A cos ωt, where A is the maximum displacement and ω is the angular frequency in radians per second (calculus is beyond the scope of this document). The time it takes the bob to move from A to −A and back again is the time it takes for ωt to advance by 2π. Therefore, the period T it takes for the bob to move from A to −A and back again is ωT = 2π, or T = 2π/ω. The frequency of the vibration in cycles per second is 1/T or ω/2π.
A pendulum is similar to a spring as it is a body suspended by a thread fixed to a pivot point so that it can swing back and forth under the influence of gravity. Pendulums are used to regulate the movement of clocks because the period for each complete oscillation remains constant. This constancy property was first noted in c.1583 by the Italian scientist Galileo, who compared the movement of a swinging lamp in a Pisa cathedral with his pulse rate. However, The Dutch mathematician and scientist Christiaan Huygens (c. 1656) solved the essential problem of making the period of a pendulum truly constant.
The period of a pendulum can be increased by using a longer thread. The formula for the period T of a pendulum is T = 2π √(L/g), where L is the length of the pendulum and g is the acceleration due to gravity. The force responsible for the motion is always directed toward the equilibrium position and is directly proportional to the distance from it. That is, F = −kx, Hooke's law. Indeed, a pendulum is a specialised form of spring because it is a bob hanging from an anchor connected by a spring with a fully rigid connection that can be neither compressed nor extended.
Newton's second law states that forces are accumulated and applied at some point in time. 3D physics engines incorporate time as a variable called delta time, where delta refers to the change in time; indeed, in Unity, Time.deltaTime is the interval in seconds from the last frame to the current one.