Mechanics

I

Introduction

Mechanics, branch of physics concerned with the motions of objects and their response to forces. Modern descriptions of such behaviour begin with a careful definition of such quantities as displacement (distance moved), time, velocity, acceleration, mass, and force. Until about 400 years ago, however, motion was explained from a very different point of view. For example, following the ideas of the Greek philosopher and scientist Aristotle, scientists reasoned that a cannonball falls because its natural position is in the Earth; the Sun, the Moon, and the stars travel in circles around the Earth because it is the nature of heavenly objects to travel in perfect circles.

The Italian physicist and astronomer Galileo brought together the ideas of other great thinkers of his time and began to analyse motion in terms of distance travelled from some starting position and the time that it took. He showed that the speed of falling objects increases steadily during the time of their fall. This acceleration is the same for heavy objects as for light ones, provided air friction (air resistance) is discounted. The English mathematician and physicist Isaac Newton improved this analysis by defining force and mass and relating these to acceleration. For objects travelling at speeds close to the speed of light, Newton's laws were superseded by the theory of relativity of Albert Einstein. For atomic and subatomic particles, Newton's laws were superseded by quantum theory. For everyday phenomena, however, Newton's three laws of motion remain the cornerstone of dynamics, which is the study of what causes motion.

II

Kinetics

Kinetics is the description of motion without regard to what causes the motion. Velocity (the rate of change of position) is defined as the distance travelled in a specific direction divided by the time interval. The magnitude of velocity is called speed. Speed may be measured in such units as kilometres per hour, miles per hour, or metres per second. Acceleration is defined as the rate of change of velocity: the change of velocity divided by the time taken by the change. Acceleration therefore has both magnitude and direction, and may be measured in such units as metres per second per second (m/s²), or feet per second per second (ft/s²).

Regarding the size or weight of the moving object, no mathematical problems are presented if the object is very small compared with the distances involved. If the object is large, it contains one point, called the centre of mass, the motion of which can be described as characteristic of the whole object. If the object is rotating, it is frequently convenient to describe its rotation about an axis that goes through the centre of mass.

Several special types of motion are easily described. First, velocity may be constant. In the simplest case, the velocity might be zero; position would not change during the time interval. With constant velocity, the average velocity is equal to the velocity at any particular time. If time, t, is measured with a clock starting at t = 0, then the distance d travelled at constant velocity v is equal to the product of velocity and time:

d = vt

In the second special type of motion, acceleration is constant. Because the velocity is changing, instantaneous velocity, or the velocity at a given instant, must be defined. For constant acceleration a, starting with zero velocity (v = 0) at t = 0, the instantaneous velocity at time t is

v = at

The distance travelled during this time is

d = yat2

An important feature revealed in this equation is the dependence of distance on the square of the time (t², or “t squared”, is the short way of writing t × t). A heavy object falling freely (uninfluenced by air friction) near the surface of the Earth undergoes constant acceleration. In this case the acceleration is 9.8 m/s² (32 ft/s²). At the end of the first second, a ball would have fallen 4.9 m (16 ft) and would have a speed of 9.8 m/s (32 ft/s). At the end of the next second, the ball would have fallen 19.6 m (64 ft) and would have a speed of 19.6 m/s (64 ft/s).

Circular motion is another simple type of motion. If an object has constant speed but an acceleration always at right angles to its velocity, it will travel in a circle. The acceleration is directed towards the centre of the circle and is called centripetal acceleration (see Centripetal Force). For an object travelling at speed v in a circle of radius r, the centripetal acceleration is

Another simple type of motion that is frequently observed occurs when a ball is thrown at an angle into the air. Because of gravitation, the ball undergoes a constant downward acceleration that first slows its original upward speed and then increases its downward speed as it falls back to Earth. Meanwhile the horizontal component of the original velocity remains constant (ignoring air resistance), making the ball travel at a constant speed in the horizontal direction until it hits the Earth. The vertical and horizontal components of the motion are independent, and they can be analysed separately. The resulting path of the ball is in the shape of a parabola. See Ballistics.

III

Dynamics

To understand why and how objects accelerate, force and mass must be defined. At the intuitive level, a force is just a push or a pull. It can be measured in terms of either of two effects. A force can either distort something, such as a spring, or accelerate an object. The first effect can be used in the calibration of a spring scale, which can in turn be used to measure the strength of other forces: the greater the force F, the greater the stretch x. For many springs, over a limited range, the stretch is proportional to the force

F = kx

where k is a constant that depends on the material and dimensions of the spring.

IV

Vectors

If an object is motionless, the net force on it must be zero. A book lying on a table is pulled down by the Earth's gravitational attraction and is pushed up by the molecular repulsion of the tabletop. The net force is zero; the book is in equilibrium. When calculating the net force, it is necessary to add the forces as vectors.