MECHANICS' INSTITUTE

a voluntary organization common in Britain and the United States between 1820 and 1860 for educating manual workers. Ideally such an institute was to have a library, a museum, a laboratory, public lectures about applied science, and courses in various skills, but few had all of these. Mechanics of different trades were to learn from each othera denial of guild exclusivenessand to add to human knowledge. A forerunner of such institutes was the Birmingham Brotherly Society founded in England in 1796. In Glasgow, George Birkbeck collected information about different trades and offered lectures at the Andersonian University (also called Anderson's University) from 1800 to 1804. He then moved to London, where in 1809 he helped to found the London Institute for the Diffusion of Science, Medicine, and the Arts, while Andrew Ure continued his work in Glasgow. Timothy Claxton founded the Mechanical Institution in London in 1817; it offered lecture-discussions for three years, until Claxton left London in 1820. The New York Mechanic and Scientific Institution, founded in 1822, was the first of many short-lived efforts in New York. The Glasgow Mechanics' Instituteconsidered a model because of its library, museum, and lecture programwas founded in 1823. The same year, Birkbeck helped organize the London Mechanics' Institute. The Franklin Institute of the State of Pennsylvania for the Promotion of the Mechanic Arts was founded in Philadelphia in 1824, and the Maryland Institute for the Promotion of the Mechanic Arts in Baltimore in 1825. Timothy Claxton, who had moved to Boston, founded the Boston Mechanics' Institute in 1826, but its reliance on lectures doomed it. Claxton tried again, founding the Boston Mechanics' Lyceum in 1831. In Cincinnati the Ohio Mechanics' Institute opened in 1829. In France, Baron Charles Dupin founded several institutes before 1826, beginning at La Rochelle and Nevers. From 1830 to 1860 hundreds of institutes were founded in the United States and Britain. Britain's Society for the Diffusion of Useful Knowledge (founded 1825) provided a central organization unknown in the United States. But many institutes were short-lived, and some of the more successful were taken over by non-mechanics with money, leisure, and the desire to hear lectures. Rules requiring mechanic majorities on governing boards were disregarded. The Franklin Institute early became a centre for advanced research in applied science, publishing reports which few mechanics could understand. The Ohio Mechanics' Institute became a school, offering courses and certificates in skills. The Maryland Institute fell dormant after its building burned in 1835 but was revived in 1847. Some institutes became lyceums; others, public libraries; others, exhibiting agencies. After 1860 mechanics' institutes largely disappeared. But the Franklin Institute has remained an important research centre; the Ohio Mechanics' Institute was an independent school until 1969, when it became part of the University of Cincinnati (O.M.I. College of Applied Science); and the Manchester Mechanics' Institute offered courses until it became a municipal trade school in 1892. The Maryland Institute opened a School of Design in 1850; that school gradually became the focus of the Institute, which became the Maryland Institute, College of Art. Motion of a particle in one dimension Uniform motion According to Newton's first law (also known as the principle of inertia), a body with no net force acting on it will either remain at rest or continue to move with uniform speed in a straight line, according to its initial condition of motion. In fact, in classical Newtonian mechanics, there is no important distinction between rest and uniform motion in a straight line; they may be regarded as the same state of motion seen by different observers, one moving at the same velocity as the particle, the other moving at constant velocity with respect to the particle. Although the principle of inertia is the starting point and the fundamental assumption of classical mechanics, it is less than intuitively obvious to the untrained eye. In Aristotelian mechanics, and in ordinary experience, objects that are not being pushed tend to come to rest. The law of inertia was deduced by Galileo from his experiments with balls rolling down inclined planes, described above. For Galileo, the principle of inertia was fundamental to his central scientific task: he had to explain how it is possible that if the Earth is really spinning on its axis and orbiting the Sun we do not sense that motion. The principle of inertia helps to provide the answer: Since we are in motion together with the Earth, and our natural tendency is to retain that motion, the Earth appears to us to be at rest. Thus, the principle of inertia, far from being a statement of the obvious, was once a central issue of scientific contention. By the time Newton had sorted out all the details, it was possible to account accurately for the small deviations from this picture caused by the fact that the motion of the Earth's surface is not uniform motion in a straight line (the effects of rotational motion are discussed below). In the Newtonian formulation, the common observation that bodies that are not pushed tend to come to rest is attributed to the fact that they have unbalanced forces acting on them, such as friction and air resistance. As has already been stated, a body in motion may be said to have momentum equal to the product of its mass and its velocity. It also has a kind of energy that is due entirely to its motion, called kinetic energy. The kinetic energy of a body of mass m in motion with velocity v is given by Falling bodies and uniformly accelerated motion During the 14th century, the French scholar Nicole Oresme studied the mathematical properties of uniformly accelerated motion. He had little interest in whether that kind of motion could be observed in the realm of actual human existence, but he did discover that, if a particle is uniformly accelerated, its speed increases in direct proportion to time, and the distance it traverses is proportional to the square of the time spent accelerating. Two centuries later, Galileo repeated these same mathematical discoveries (perhaps independently) and, just as important, determined that this kind of motion is actually executed by balls rolling down inclined planes. As the incline of the plane increases, the acceleration increases, but the motion continues to be uniformly accelerated. From this observation, Galileo deduced that a body falling freely in the vertical direction would also have uniform acceleration. Even more remarkably, he demonstrated that, in the absence of air resistance, all bodies would fall with the same constant acceleration regardless of their mass. If the constant acceleration of any body dropped near the surface of the Earth is expressed as g, the behaviour of a body dropped from rest at height z0 and time t = 0 may be summarized by the following equations: where z is the height of the body above the surface, v is its speed, and a is its acceleration. These equations of motion hold true until the body actually strikes the surface. The value of g is approximately 9.8 metres per second squared (m/s2). A body of mass m at a height z0 above the surface may be said to possess a kind of energy purely by virtue of its position. This kind of energy (energy of position) is called potential energy. The gravitational potential energy is given by Technically, it is more correct to say that this potential energy is a property of the Earth-body system rather than a property of the body itself, but this pedantic distinction can be ignored. As the body falls to height z less than z0, its potential energy U converts to kinetic energy K = 1/2mv2. Thus, the speed v of the body at any height z is given by solving the equation Equation (8) is an expression of the law of conservation of energy. It says that the sum of kinetic energy, 1/2mv2, and potential energy, mgz, at any point during the fall, is equal to the total initial energy, mgz0, before the fall began. Exactly the same dependence of speed on height could be deduced from the kinematic equations (4), (5), and (6) above. In order to reach the initial height z0, the body had to be given its initial potential energy by some external agency, such as a person lifting it. The process by which a body or a system obtains mechanical energy from outside of itself is called work. The increase of the energy of the body is equal to the work done on it. Work is equal to force times distance. The force exerted by the Earth's gravity on a body of mass m may be deduced from the observation that the body, if released, will fall with acceleration g. Since force is equal to mass times acceleration, the force of gravity is given by F = mg. To lift the body to height z0, an equal and opposite (i.e., upward) force must be exerted through a distance z0. Thus, the work done is which is equal to the potential energy that results. If work is done by applying a force to a body that is not being acted upon by an opposing force, the body is accelerated. In this case, the work endows the body with kinetic energy rather than potential energy. The energy that the body gains is equal to the work done on it in either case. It should be noted that work, potential energy, and kinetic energy, all being aspects of the same quantity, must all have the dimensions ml2/t2. Rigid bodies Statics Figure 17: (A) A body in equilibrium under equal and opposite forces. (B) A body not in equilibrium Figure 17: (A) A body in equilibrium under equal and opposite forces. (B) A body not in equilibrium Statics is the study of bodies and structures that are in equilibrium. For a body to be in equilibrium, there must be no net force acting on it. In addition, there must be no net torque acting on it. Figure 17A shows a body in equilibrium under the action of equal and opposite forces. Figure 17B shows a body acted on by equal and opposite forces that produce a net torque, tending to start it rotating. It is therefore not in equilibrium. Figure 18: The resultant force (FR) produces the same net force When a body has a net force and a net torque acting on it owing to a combination of forces, all the forces acting on the body may be replaced by a single (imaginary) force called the resultant, which acts at a single point on the body, producing the same net force and the same net torque. The body can be brought into equilibrium by applying to it a real force at the same point, equal and opposite to the resultant. This force is called the equilibrant. An example is shown in Figure 18. The torque on a body due to a given force depends on the reference point chosen, since the torque t by definition equals r F, where r is a vector from some chosen reference point to the point of application of the force. Thus, for a body to be at equilibrium, not only must the net force on it be equal to zero but the net torque with respect to any point must also be zero. Fortunately, it is easily shown for a rigid body that, if the net force is zero and the net torque is zero with respect to any one point, then the net torque is also zero with respect to any other point in the frame of reference. A body is formally regarded as rigid if the distance between any set of two points in it is always constant. In reality no body is perfectly rigid. When equal and opposite forces are applied to a body, it is always deformed slightly. The body's own tendency to restore the deformation has the effect of applying counterforces to whatever is applying the forces, thus obeying Newton's third law. Calling a body rigid means that the changes in the dimensions of the body are small enough to be neglected, even though the force produced by the deformation may not be neglected. Figure 19: (A) Compression produced by equal and opposite forces. (B) Tension produced by equal and Figure 19: (A) Compression produced by equal and opposite forces. (B) Tension produced by equal and Equal and opposite forces acting on a rigid body may act so as to compress the body (Figure 19A) or to stretch it (Figure 19B). The bodies are then said to be under compression or under tension, respectively. Strings, chains, and cables are rigid under tension but may collapse under compression. On the other hand, certain building materials, such as brick and mortar, stone, or concrete, tend to be strong under compression but very weak under tension. The most important application of statics is to study the stability of structures, such as edifices and bridges. In these cases, gravity applies a force to each component of the structure as well as to any bodies the structure may need to support. The force of gravity acts on each bit of mass of which each component is made, but for each rigid component it may be thought of as acting at a single point, the centre of gravity, which is in these cases the same as the centre of mass. Figure 20: (A) A body supported by two rigid members under tension. (B) A body supported by two Figure 20: (A) A body supported by two rigid members under tension. (B) A body supported by two Figure 20: (A) A body supported by two rigid members under tension. (B) A body supported by two To give a simple but important example of the application of statics, consider the two situations shown in Figure 20. In each case, a mass m is supported by two symmetric members, each making an angle q with respect to the horizontal. In Figure 20A the members are under tension; in Figure 20B they are under compression. In either case, the force acting along each of the members is shown to be The force in either case thus becomes intolerably large if the angle q is allowed to be very small. In other words, the mass cannot be hung from a perfectly horizontal member. Figure 20: (A) A body supported by two rigid members under tension. (B) A body supported by two The ancient Greeks built magnificent stone temples; however, the horizontal stone slabs that constituted the roofs of the temples could not support even their own weight over more than a very small span. For this reason, one characteristic that identifies a Greek temple is the many closely spaced pillars needed to hold up the flat roof. The problem posed by equation (71) was solved by the ancient Romans, who incorporated into their architecture the arch, a structure that supports its weight by compression, corresponding to Figure 20B. Figure 20: (A) A body supported by two rigid members under tension. (B) A body supported by two A suspension bridge illustrates the use of tension. The weight of the span and any traffic on it is supported by cables, which are placed under tension by the weight. Corresponding to Figure 20A, the cables are not stretched to be horizontal, but rather they are always hung so as to have substantial curvature. It should be mentioned in passing that equilibrium under static forces is not sufficient to guarantee the stability of a structure. It must also be stable against perturbations such as the additional forces that might be imposed, for example, by winds or by earthquakes. Analysis of the stability of structures under such perturbations is an important part of the job of an engineer or architect. Rotation about a fixed axis Consider a rigid body that is free to rotate about an axis fixed in space. Because of the body's inertia, it resists being set into rotational motion, and equally important, once rotating, it resists being brought to rest. Exactly how that inertial resistance depends on the mass and geometry of the body is discussed here. Take the axis of rotation to be the z-axis. A vector in the x-y plane from the axis to a bit of mass fixed in the body makes an angle q with respect to the x-axis. If the body is rotating, q changes with time, and the body's angular frequency is w is also known as the angular velocity. If w is changing in time, there is also an angular acceleration a, such that Because linear momentum p is related to linear speed v by p = mv, where m is the mass, and because force F is related to acceleration a by F = ma, it is reasonable to assume that there exists a quantity I that expresses the rotational inertia of the rigid body in analogy to the way m expresses the inertial resistance to changes in linear motion. One would expect to find that the angular momentum is given by and that the torque (twisting force) is given by Figure 21: Rotation around a fixed axis. One can imagine dividing the rigid body into bits of mass labeled m1, m2, m3, and so on. Let the bit of mass at the tip of the vector be called mi, as indicated in Figure 21. If the length of the vector from the axis to this bit of mass is Ri, then mi's linear velocity vi equals wRi (see equation ), and its angular momentum Li equals miviRi (see equation ), or miRi2w. The angular momentum of the rigid body is found by summing all the contributions from all the bits of mass labeled i = 1, 2, 3 . . . : In a rigid body, the quantity in parentheses in equation (76) is always constant (each bit of mass mi always remains the same distance Ri from the axis). Thus if the motion is accelerated, then Recalling that t = dL/dt, one may write (These equations may be written in scalar form, since L and t are always directed along the axis of rotation in this discussion.) Comparing equations (76) and (78) with (74) and (75), one finds that The quantity I is called the moment of inertia. According to equation (79), the effect of a bit of mass on the moment of inertia depends on its distance from the axis. Because of the factor Ri2, mass far from the axis makes a bigger contribution than mass close to the axis. It is important to note that Ri is the distance from the axis, not from a point. Thus, if xi and yi are the x and y coordinates of the mass mi, then Ri2 = xi2 + yi2, regardless of the value of the z coordinate. The moments of inertia of some simple uniform bodies are given in the table. The moment of inertia of any body depends on the axis of rotation. Depending on the symmetry of the body, there may be as many as three different moments of inertia about mutually perpendicular axes passing through the centre of mass. If the axis does not pass through the centre of mass, the moment of inertia may be related to that about a parallel axis that does so. Let Ic be the moment of inertia about the parallel axis through the centre of mass, r the distance between the two axes, and M the total mass of the body. Then In other words, the moment of inertia about an axis that does not pass through the centre of mass is equal to the moment of inertia for rotation about an axis through the centre of mass (Ic) plus a contribution that acts as if the mass were concentrated at the centre of mass, which then rotates about the axis of rotation. The dynamics of rigid bodies rotating about fixed axes may be summarized in three equations. The angular momentum is L = Iw, the torque is t = Ia, and the kinetic energy is K = 1/2Iw2.