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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 For objects travelling at speeds close to the speed of light, Newton's laws were superseded by the theory of relativity of Albert Einstein. For atomic and subatomic particles, Newton's laws were superseded by quantum theory.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 (. 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.

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.

V TORQUE  
For equilibrium, the horizontal components of the forces acting on an object must cancel one another, and so must the vertical components. This condition is necessary for equilibrium, but not sufficient. For example, if a person stands a book on a table and pushes the book equally hard with one hand in one direction and with the other hand in the other direction, the book will remain motionless if the person's hands are opposite each other. (The net result is that the book is being squeezed.) If, however, one hand is near the top of the book and the other hand near the bottom, a torque, or turning force, is produced, and the book will fall on its side. For equilibrium to exist it is also necessary that the sum of the torques about any axis be zero.

A torque, or moment of a force, is the product of the force and the perpendicular distance to an axis of rotation. When a force is applied to a heavy door to open it, the force is exerted perpendicularly to the door and at the greatest distance from the hinges. Thus, a maximum torque is created. If the door were shoved with the same force at a point halfway between handle and hinge, the torque would be only half of its previous magnitude. If the force were applied parallel to the door (that is, edge on), the torque would be zero. For an object to be in equilibrium, the clockwise torques about any axis must be cancelled by the anti-clockwise torques about that axis. It can be proved that if the torques cancel for any particular axis, they cancel for all axes.

VI NEWTON'S THREE LAWS OF MOTION  
Newton's first law of motion states that if the vector sum of the forces acting on an object is zero, then the object will remain at rest or remain moving at constant velocity. If the force exerted on an object is zero, the object does not necessarily have zero velocity. Without any forces acting on it, including friction, an object in motion will continue to travel at constant velocity.

A The Second Law  Newton's second law relates net force and acceleration. A net force on an object will accelerate it—that is, change its velocity. The acceleration will be proportional to the magnitude of the force and in the same direction as the force. The proportionality constant is the mass, m, of the object

F = ma

In the International System of Units (also known as SI, the acronym for “Système International”), acceleration, a, is measured in m/s2. Mass is measured in kg; force, F, in newtons. A newton is defined as the force necessary to impart to a mass of 1 kg an acceleration of 1 m/s2; this force is roughly equal to the weight of a 100-g (3.5-oz) object at sea level.

A more massive object will require a greater force for a given acceleration than a less massive one. What is remarkable is that mass, which is a measure of the inertia of an object (its reluctance to change velocity), is also a measure of the gravitational attraction that the object exerts on other objects. It is surprising and profound that the inertial property and the gravitational property are determined by the same thing. Einstein made this one of the cornerstones of his general theory of relativity, which is the currently accepted theory of gravitation

B Friction  
Friction acts like a force applied in the direction opposite to an object's velocity. For dry sliding friction, where no lubrication is present, the friction force is almost independent of velocity. The friction force also does not depend on the apparent area of contact between an object and the surface upon which it slides. The actual contact area—that is, the area where the microscopic bumps on the object and sliding surface are actually touching each other—is relatively small. As the object moves across the sliding surface, the tiny bumps on the object and sliding surface collide, and force is required to move the bumps past each other. The actual contact area depends on the perpendicular force between the object and sliding surface. Frequently this force is just the weight of the sliding object. If the object is pushed at an angle to the horizontal, however, the downward vertical component of the force will, in effect, be added to the weight of the object. The friction force is proportional to the total perpendicular force.

Where friction is present, Newton's second law can be expanded to

When an object moves through a liquid, however, the magnitude of the friction depends on the velocity. For most human-sized objects moving in water or air (at subsonic speeds), the resulting friction is proportional to the square of the speed. Newton's second law then becomes

The proportionality constant, k, is characteristic of the two materials that are moving past each other, and depends on the area of contact between the two surfaces and the degree of streamlining of the moving object.

C The Third Law  
Newton's third law of motion states that when an object exerts a force on another object, it experiences a force in return. The force that object one exerts on object two must be of the same magnitude as the force that object two exerts on object one but in the opposite direction. On a skating rink, for example, if a large adult gently pushes away a child, then in addition to the force the adult exerts on the child, the child exerts an equal but oppositely directed force on the adult. Because the mass of the adult is larger, however, the acceleration of the adult will be smaller.

Newton's third law also requires the conservation of momentum, the product of mass and velocity. For an isolated system, with no external forces acting on it, the momentum must remain constant. In the example of the adult and child on the skating rink, their initial velocities are zero, and thus the initial momentum of the system is zero. During the interaction, internal forces are at work between adult and child, but net external forces equal zero. Therefore, the momentum of the system must remain zero. After the adult has pushed the child away, the product of the large mass and small velocity of the adult must equal the product of the small mass and large velocity of the child. The momenta are equal in magnitude but opposite in direction, thus adding up to zero.

Another conserved quantity of great importance is angular (rotational) momentum. The angular momentum of a rotating object depends on its speed of rotation, its mass, and the distance of the mass from the axis. When a skater on (almost) frictionless ice spins faster and faster, angular momentum is conserved despite the increasing speed. At the start of the spin, the skater's arms are outstretched. Part of the skater's mass is therefore at a large radius. As the skater's arms are lowered, thus decreasing their distance from the axis of rotation, the rotational speed must increase in order to maintain constant angular momentum.

VII ENERGY  
The quantity called energy ties together all branches of physics. In the field of mechanics, energy must be provided to do work; work is defined as the product of force and the distance an object moves in the direction of the force. When a force is exerted on an object but the force does not cause the object to move, no work is done. Energy and work are both measured in the same units—joules or foot-pounds, for example.

If work is done lifting an object to a greater height, energy has been stored in the form of gravitational potential energy. Many other forms of energy exist: electrical and magnetic potential energy; kinetic energy; energy stored in stretched springs, compressed gases, or molecular bonds; thermal energy; and mass itself. In all transformations from one kind of energy to another, the total energy is conserved. For instance, if work is done on a rubber ball to raise it, its gravitational potential energy is increased. If the ball is then dropped, the gravitational potential energy is transformed to kinetic energy. When the ball hits the ground, it becomes distorted and there is friction between the molecules of the ball material. This friction is transformed into heat, or thermal energy.

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