B1. Motion on prescribed trajectory

Inclined plane. Motion on an inclined plane. The inclined plane as a machine

When a mass is being raised, the direction of the motion (vertically upwards) and the direction of the load (weight, vertically downwards) are opposite, they form a stretched angle, a special one of all angles. In general, the angle between load and motion differs; then only a fractionof the load resists displacement, that is, only that fraction need be overcome.

Let the mass lie on an inclined plane CD (Fig. 28) and let it be moved upwards by a force which acts parallel to CD. Such a plane is inclined with respect to the horizontal, for example, a road leading up a mountain; its angle of inclination is a. The direction of the required motion is inclined to that of the force of gravity MP. What will the mass do when it is allowed to move on its own, that is, when it is only subject to gravity and no friction?

MP represents the force of gravity in magnitude as well as in direction; it causes vertical motion of the mass. The mass cannot move in this direction; it exerts therefore on the nclined plane, - that is, a resistance to its motion- a pressure . The driving force MP simultaneously moves the mass, but in a direction other than what it would be without the inclined plane. In fact, MP is replaced by the two, simultaneously acting forces MQ and MR. The force MQ acts as a pressure on the plane, which resists it due to its firmness and responds with an equally large pressure in the opposite direction and thus balances it. The mass can follow the force MR, since the plane does not resist motion along it (Fall along an inclined plane).

It must be balanced by some other force, if the motion is to be stopped. Since MR/MP = cos b, you have MR = MPcos b. Since MP is the weight of the mass, that is, MP = mg, and cosb=cosa, you find that MR = mgsina. On an inclined plane, the upwards directed force required to balance the load equals the load times the sine of the angle of inclination, that is, it is smaller than the actual load. The force, required to keep the mass at rest on the inclined plane (or, if its motion is uniform, in uniform motion) must only be as large as MR and in the opposite direction. However, if the force , which is to stop the mass from moving along the inclined plane downwards, acts parallel to the base of the inclined plane (K in Fig. 29), then it must be larger than mgsina, because only the component K cos a acts along the inclined plane; hence K cos a = mg sin a, whence K = mg tan a.

The component which acts as a pressure on the inclined plane is made ineffective by its firmness. Thus, the inclined plane becomes a device by which you can balance a force by a smaller force. Such devices are called machines.

Fall along an inclined plane

If you do not balance the component MR (mg sina), the mass will move downwards, that is, "fall" along the inclined plane. Its acceleration is given by force/mass = (mg sin a)/m = g sin a. If you replace here v by g sin a, you can answer all questions relating to the fall along an inclined plane. For example, the velocity at the end of the t-th second is: v1 = gt sin a (it is gt during free fall!). You can now make the velocity of fall as small as you please by making a sufficiently small, that is, let the inclined plane differ only very little from the horizontal plane. (Galilei used it to prove the laws of fall).

After it has fallen the distance s, the velocity of a freely falling mass is v=(2gs)1/2, whence it is
v
=(2gsin
a s)1/2 along the inclined plane with the angle of inclination a. Now let the mass drop from the point D (Fig. 28) to the horizontal plane - the base of the inclined plane - once freely along h and a second time along the inclined plane of length CD = l: What will be the velocities at which the mass arrives in these two experiments?

In the first case, you must replace s by h, whence v = (2gh)1/2. In the second case, you must replace s by l, whence v1= (2g sin a l)1/2; however, since h/l = sin a, then v1=(2gsinah/sina)1/2= (2gh)1/2 = v, that is, the mass arrives with the same velocity at the horizontal plane, irrespectively of whether it falls freely through the height of the inclined plane or along the inclined plane.

Tautochrone The fall along an inclined plane is an example of motion along a prescribed trajectory. It is very strange, if the prescribed trajectory is the concave side of the arc of a cycloid.

What is a cycloid? A point M of a circle which rolls along the straight line ab without slipping describes a cycloid, for example, every point on the periphery of a rolling tire or wheel. In order to reach the lowest point in a fall along an inclined plane, the mass requires the time interval t = (2s/gsin a)1/2, which depends on s, that is, it is longer, if it lies higher up on the inclined plane than if it were to start its fall lower down. However, if it falls along a vertical, upwards concave arc of a cycloid, it uses always the same time to reach the lowest point irrespectively from which point it starts to fall (Huygens 1673). The cycloid is therefore also called a Tautochrone (Greek: tautos = same, cronos = time).

Machines

The inclined plane is a device by which you can balance a force by means of a smaller force. For example, if you want to lift by hand a load from the ground on to a truck, you exert yourself less, if you push it along an inclined board on to the car than if you lift it vertically. The less inclined the board is, the smaller is the downwards force, which must be overcome. However, the distance over which the mass must be moved along the inclined plane is in the same proportion larger than the height by which it must be lifted. What you save in your application of force, you must give in the working distance.

If the height is h and the inclination a, then the length of the inclined plane is l = h/sina. While the (larger) force p must shift its point of application over the (smaller) distance h, that is, altogether perform the work ph, the smaller force psin a must shift its point of application over the larger distance l = h/sin a and perform the work psina h/sin a = ph, that is, the same as before and you do not save work. But with the aid of the inclined plane, you can undertake work for which your muscle strength is not sufficient; by suitable inclination of the inclined plane, you can make the force, required for a performance, as small as you please. The inclined plane is one of the devices, which are called machines - simple machines (in contrast to machines comprising several simple machines).

In general, a machine is defined as: A device, which is able to resist and makes it possible to balance a force of given magnitude by a smaller force. - The demand for a balancing capacity means: The machine must not be changed by the two groups of forces, that is, it must only transfer a force, but not use itself any of it. This problem is not strictly technically soluble, especially due to the deformability of solid bodies and friction.

Every inclined ladder, every upward road, every staircase is an inclined plane. In an unlimited area of applications, the inclined plane forms the base for two other simple machines: the screw and the wedge. You use the screw in a press (also the primitive copy press) and in devices which with the aid of screws raise loads. Every cutting instrument (knife, scissors, axe) employs a wedge.

Screw, wedge

Fig. 33 shows that the screw is an inclined plane: The hypotenuse AB of a right-angled triangle, placed around a circular cylinder tightly, describes on it a spiral. If its length CB equals the circumference of the cylinder, then, if the points B and C coincide, you have one turn of a spiral. Obviously, the line AB is an inclined plane with height AC, AB its length and BC its base. A flexible bar in place of AB (For example, of square cross-section), which covers the screw line, forms a protruding band, the thread of the screw.

Screws have several such turns, all of which can be imagined to have arisen in the same manner - The screw can transfer forces only after it is given a nut: You cut inside a hollow cylinder of circular cross-section with the diameter of the cylinder of Fig. 31 the same thread, which was on the cylinder a raised thread, that is the nut of the screw. If you insert the screw into the nut, that is, lay one inclined plane against the other, and let gravity act, it slides with its thread in the thread of the nut (assuming that there is no friction. In practice, friction cannot be avoided).

If you wish to eliminate the effect of gravity, it must be opposed by a force like on the inclined plane. You can apply this force at the circumference C of the screw, that is, in Fig. 29, parallel to the base of the inclined plane, from which the screw emerges. It is smaller in the same ratio as that, to be held in equilibrium, just as we found iy to be the case for the inclined plane.

The relations between the forces on a wedge are similarly related. A wedge is a three-sided prism (Fig. 31) in which one angle is very small compared with the other two. The two planes meeting in the sharp angle are the sides, the third the back and the edge opposite it the edge of the wedge. When the wedge has opened up a body, the separated parts press on it and drive it out, if there is no friction and the force driving it in ceases. Again friction is important!

An axe, which you drive into a block of wood, is not necessarily ejected when no force is applied to it. However, if friction could be almost eliminated, the wedge would be thrown out by the separated parts of the block. In order to keep the axe inside the block, you would have to apply an appropriate force to the back of the axe. The sharper the wedge, the smaller this force. All cutting gadgets like knives, chisels, planes, etc. work like wedges.

Figss. 34/ 35 show the relations between the forces which act on a wedge. A right-angled wedge has been driven under a beam, which is to support a wall, in order to stop it from falling to the right side of the figure. The wall presses against the beam, which presses against the wedge; it would eject it horizontally, if there were no friction, that is, you would then have to apply a horizontal force on its back, in order to keep it in place.

How large must be this force in relation to the pressure L of the beam? The pressure L acts at a right angle to the side AB of the wedge, but only the component at right angle to BC attempts to drive the wedge out. The component at right angle to the ground is of no interest (the resistance of the ground balances it). The force P, which you must apply against the back of the wedge, must therefore equal l. The figure shows that l/L = BC/BA, whence l is the smaller the sharper is the wedge, that is, the smaller is the ratio of the back to the wedge's hypotenuse.

Experience confirms this conclusion during the use of a knife, axe, needle, nail, etc. They intrude the easier, the more pointed they are.

Friction as impediment

When answering the question regarding the magnitude of the force required to balance a load of given size by means of a machine , we have neglected friction. However, it is ever present. A body on an inclined plane does not slide at all down it. As a rule, it stays put (unless the inclined plane is rather steep or the body has a special shape). Friction holds it in place. The more perfect the surface of the inclined plane as well as of the body, the less steep need be the inclined plane, on which the body would slide down. A method for the measurement of friction employs determination of the angle of inclination of a plane, at which the body starts to glide down.

If you attempt to pull the mass m by the spiral spring along the table T (Fig. 36), the spring will stretch a certain amount before the mass starts to move. However, the magnitude of this tension does not correspond to Newton's Second Law, but is larger. It can be measured like on a letter balance by a weight. For example, if m is 1000 g and you must apply a 600 g weight to stretch the spring, before m starts to move, this means: You must employ 3/5 times the force by which the mass presses on the table, in order to overcome the friction of m on T; the number 3/5 is the friction coefficient. You obtain the same number, if you place m on an inclined plane and find out at which angle of inclination m starts to glide down. (The scarp angle of a pile of sand, of grain, etc. also depends on this angle before the substance starts to slide).

In order to maintain the motion of the moving mass m, a much smaller tension in the spring is needed, may be 2/5 of its weight. This number, the friction coefficient, does not only differ for various pairs of material, but also changes for the same pair depending on the state of the surfaces, that is, the lubricant between the surfaces (oil. graphite, grease, etc.).

Rolling on well greased wheels encounters the least frictional resistance among all modes of motion. The friction coefficient is then much smaller than during sliding. A cartwheel has on its circumference rolling, on its axle sliding friction; ball-bearings are used, in order to convert also the latter into rolling motion (Fig. 37).

If you press surfaces, which slide along each other, ever more strongly together, the friction becomes substantially larger. Then you must apply a much larger force, in order to generate motion and ongoing motion becomes then much slower. This is the principle of the brake in cars.

You employ all the time friction intentionally as well as unintentionally. You could not move on foot or otherwise, if friction did not stop you from slipping; you would also not stay at rest while sitting or lying or standing up, unless friction stops you from sliding. Lighting a match on the friction surface of its box by frictional heat belongs to the conscious technical applications of friction. Devices like the brake dynamometer also employ friction.

Motions, which can be generated by mechanical means and remain visible, remain so also after the moving forces have stopped, but you see them slow down and eventually stop; for example, a carriage, when it is detached from a moving locomotive, keeps on rolling, a boat which is no longer rowed, keeps on drifing, etc.

Frictional effects: But a motion has only ceased to be visible; in reality, is has converted itself into another motion which, however, is not visible, but can be noted through its effect: Contacting surface have been heated. As a rule, one imagines that the end of the visible motion has destroyed it.

As a rule, the heating is not large enough to be seen without effort. But occasionally, it becomes so, for example, when very fast motion of a mass is suddenly interrupted or very down much slowed down. In the case of a railway carriage, which is being braked, the braking blocks on the wheels become so hot that you can feel it when you touch them, meteors, on entering the atmosphere from the airless interstellar space, become heated by the atmosphere on their surfaces so much that they light up (shooting stars), a flying bullet, impeded by a resisting material, can become so hot that it melts on the surface, etc.

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