Mechanics I INTRODUCTION Mechanics, branch of physics concerning the motions of objects and their response to forces. Modern descriptions of such behavior 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 Greek philosopher and scientist Aristotle, scientists reasoned that a cannonball falls down 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 analyze motion in terms of distance traveled 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 Sir Isaac Newton improved this analysis by defining force and mass and relating these to acceleration. For objects traveling at speeds close to the speed of light, Newton's laws were superseded by Albert Einstein's theory of relativity. For atomic and subatomic particles, Newton's laws were superseded by quantum theory.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 Freefall Falling objects accelerate in response to the force exerted on them by Earth's gravity. Different objects accelerate at the same rate, regardless of their mass. This illustration shows the speed at which a ball and a cat would be moving and the distance each would have fallen at intervals of a tenth of a second during a short fall. © Microsoft Corporation. All Rights Reserved. Kinetics is the description of motion without regard to what causes the motion. Velocity (the time rate of change of position) is defined as the distance traveled divided by the time interval. Velocity may be measured in such units as kilometers per hour, miles per hour, or meters per second. Acceleration is defined as the time rate of change of velocity: the change of velocity divided by the time interval during the change. Acceleration may be measured in such units as meters per second per second or feet per second per second. 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 center 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 center of mass. To fully describe the motion of an object, the direction of the displacement must be given. Velocity, for example, has both magnitude (a scalar quantity measured, for example, in meters per second) and direction (measured, for example, in degrees of arc from a reference point). The magnitude of velocity is called speed. 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, traveled 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 = atThe distance traveled during this time is d = yat2An 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 notating 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/sec/sec (32 ft/sec/sec). 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/sec (32 ft/sec). At the end of the second second, the ball would have fallen 19.6 m (64 ft) and would have a speed of 19.6 m/sec (64 ft/sec). 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 required acceleration is directed toward the center of the circle and is called centripetal acceleration (see Centripetal Force). For an object traveling 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 analyzed separately. The resulting path of the ball is in the shape of a parabola. See Ballistics. III DYNAMICS Components of Velocity Neglecting air resistance, a ball thrown into the air at an angle will travel in a parabolic path. The velocity of the ball (V) has independent vertical (V) and horizontal (H) components; the horizontal component stays the same the entire time the ball is in the air, while the vertical component, the only component affected by gravity, changes continuously while the ball is aloft. © Microsoft Corporation. All Rights Reserved. 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 amplitude of a force: 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 = kxwhere k is a constant that depends on the nature of the spring material and its dimensions. IV VECTORS Vectors and Net Force Often, an object will have many forces acting on it simultaneously. Calculating the effect of each of the forces separately can be extremely complex and difficult. However, forces are vectors, and as such, any number of forces can be combined into a single net force vector (R) from which the object's behavior can be determined. © Microsoft Corporation. All Rights Reserved. If an object is motionless, the net force on it must be zero. A book lying on a table is being pulled down by the earth's gravitational attraction and is being 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. See Vector. V TORQUE For equilibrium, all the horizontal components of the force must cancel one another, and all the vertical components must cancel one another as well. This condition is necessary for equilibrium, but not sufficient. For example, if a person stands a book up on a table and pushes on 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 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 is the product of a force and the perpendicular distance to a turning axis. 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 canceled by the counterclockwise torques about that axis. Therefore, one could prove that if the torques cancel for any particular axis, they cancel for all axes. VI NEWTON'S THREE LAWS OF MOTION Acceleration and Newton's Laws Newton's second law states that the net force on an object is proportional to the acceleration that object undergoes. If there is no net force, then according to Newton's first law, there can be no acceleration. A book on a table experiences a downward force due to gravity, and an upward force due to the table pushing on the book (called the normal force). The two forces cancel each other out exactly; there is no net force, so the book does not accelerate off the table. © Microsoft Corporation. All Rights Reserved. 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 = maIn the International System of Units (also known as SI, after the initials of Système International), acceleration, a, is measured in meters per second per second. Mass is measured in kilograms; 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/sec/sec; this is equivalent to about 0.2248 lb. A massive object will require a greater force for a given acceleration than a small, light object. What is remarkable is that mass, which is a measure of the inertia of an object (inertia is 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. The implication of this phenomenon is that it is impossible to distinguish at a point whether the point is in a gravitational field or in an accelerated frame of reference. Einstein made this one of the cornerstones of his general theory of relativity, which is the currently accepted theory of gravitation. B Friction Friction Microscopic bumps on surfaces cause friction. When two surfaces contact each other, tiny bumps on each of the surfaces tend to run into each other, preventing the surfaces from moving past each other smoothly. An effective lubricant forms a layer between two surfaces that prevents the bumps on the surfaces from contacting each other; as a result the surfaces move past each other easily. © Microsoft Corporation. All Rights Reserved. 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. Also, the friction force 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, add to the weight of the object. The friction force is proportional to the total perpendicular force. Where friction is present, Newton's second law is expanded to The left side of the equation is simply the net effective force. (Acceleration will be constant in the direction of the effective force). When an object moves through a liquid, however, the magnitude of the friction depends on the velocity. For most human-size 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 sliding 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 Conservation of Momentum The principle of conservation of momentum is one of the lynchpins of mechanics. It asserts that in an isolated system, the product of the mass times the speed of all the bodies present remains constant. The principle is illustrated here using a pair of railroad cars that can slide on the rails without friction. © Microsoft Corporation. All Rights Reserved. Newton's third law of motion states that an object experiences a force because it is interacting with some other object. The force that object 1 exerts on object 2 must be of the same magnitude but in the opposite direction as the force that object 2 exerts on object 1. If, for example, a large adult gently shoves away a child on a skating rink, in addition to the force the adult imparts on the child, the child imparts 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, or 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 pushes 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 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 standing on a friction-free point 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 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 Energy of a Pendulum A moving pendulum changes potential energy into kinetic energy and back again. When the bob (weight on the end of string) is first released, it has potential energy due to its height, but no kinetic energy since it is not yet moving. As the bob accelerates downward, potential energy is traded for kinetic. At the bottom of its swing, the bob has no potential energy since it cannot fall any further. The bob is moving quickly at this point since all of its former potential energy has been transformed into kinetic energy. © Microsoft Corporation. All Rights Reserved. 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--ergs, 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: electric 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 thereby creates friction between the molecules of the ball material. This friction is transformed into heat, or thermal energy. Contributed By: Clifford E. Swartz Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.