Most measurements performed in the laboratory reduce essentially to the measurement of a length. By using this measurement and cer- tain conventions expressed by formulas , we obtain the desired quantity. When he measures something, the physicist must take great care to produce the miniTrmm possible disturbance of the system that is rmder observation.
For example, when we measure the temperature of a body, w'e place it in contact with a thermometer. In addition, all measurements are affected by some degree of experimental error because of the merutable imperfections hi the measuring device, or the limitations imposed bj our senses idsion and hearing which must record the information. There- ore, a physicist designs liis measuring technique so that the disturbance of the quantity measured is smaller than his experimental error.
In general, this is a najs possible when we are measuring quantities in the macroscopic range i e Ir,?
Thus wdiatever the dis turbaiice produced, it is netfgible compared udth the e. Equal arm balance for comparing the masses of two bodies. But in Chapter 7 we shall see a means for comparing masses dynamically. Mass ob- tamed dynamically is called inertial mass. No difference has been found between the two methods of measm-ing mass, as will be discussed in Chapter 13 With a few exceptions, all other quantities used thus far in physics can be re- lated to these four quantities by their definitions, expressed as mathematical rela- lons mvohung length, mass, time, and charge.
The units of all these derived quantities are in turn expressed in terms of the units of the four fundamental quantities by means of these defining relations. Therefore it is only necessary to agree on the nmts for the four fundamental quantities in order to W a con states of the krypton atom.
Definition of the tropical year Fig. The coulomb is thus defined as the amount of electric charge that passes through a section of a conductor during one second when the current is one ampere. Our decision to use the coulomb is based mainly on our wish to ex- press the more fundamental character of electric charge, without departing essentiaUy from the recommendations of the Eleventh Conference. The meter and the kilogram are units originally introduced during the French revolution, when the French government decided to establish a rational system of units, known since then as the metric system, to supplant the chaotic and varied units in use at that time.
Later measurements indicated that the standard bar was shorter by 1. Duplicates of the standard meter exist in many countries. However, the convenience of having a standard of more permanent character and easy availability at any laboratory was recognized. For that reason the red line of was chosen. Tins temperature was chosen because it is the temperature at which the density of water is a maximum. A platinum block, having a mass of one Idlogram, was built.
Later on it was decided to adopt this block as the standard kilogram without fur- ther reference to the water. The unit of length is the foot, abLeviated ft, the umt of mass is the pound, abbrerdated lb, and the unit of time is again the second. Another useful concept is relative density. If pi and pg are the densities of two different substances, their relative density is 2. It is customary to express relative densities Avith respect to water as a reference. It is the second that is more important in physics. Each degree is divided into 60 minutes ' and each minute into 60 seconds ".
To express a plane angle in radians, one draws, with an arbitrary radius R Fig. Then the measure of 0 in radians abbreviated rad is 2. In the world of meas urement, however, precision has the connotation of inaccuracy. What we mean is that when a physical property is described by some numerical quantity and some units, the numerical quantity is dependent on a number of different factors, in- cluding the particular piece of apparatus used to make the measurement, the type and number of measurements made, and the method employed by the experimenter to extract the number from the apparatus.
Unless the numerical quantity is ac- companied by another which describes the precision of the measurement, the num- ber quoted is as good as useless. A number may be extremely accurate that is, be exactly correct , but not be precise because the person quoting the number has failed to state at least something about his method of measurement. It is precise and accurate, since the number of units to be counted is small and integral. If there are two peo- ple, one slowly putting apples into the basket and another sloivly removing them, then one can make accurate and precise statements about the number of apples at any given time.
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Consider the number of people in a small village. Here the number is larger, but still fairly reasonable and definitely in- tegral. An observer standing in the center of the village's one street, by observing the commg and going of people after a census count, could make accurate state- ments about the number of people in the village. But his numerical quantity would not necessardy be precise, since it wnuld be difficult for him to discover the exact time of the budh and death of the townspeople. Make the village a city or a county, and the job becomes even more difficult.
S 3feasiirement in the laboratory With a relatively simple example, the period of a pendulum, we shall describe the methods used in obtaining the numerical quantity associated with a physical prop- erty. The period of a pendulum is the time between tvro consecutive passes of the bob through the same point, moving in the same direction. A particular pendulum was set to swing, and its period for a single oscillation was measured fifty separate times. Table contams the fifty measurements, in seconds.
From the table you may see that there is no one particular period for the pen- dulum. By adding all the periods and then dividing by the total number of measurements, we find that the mean or average value for the period of the pendulum is 3. Note that for the moment we have kept the entire number; we shall modify it at the proper time. By talcing the difference between this mean value and each measurement, we obtain the deviation of each measurement from the mean.
The sum of the absolute values of the deviations divided by the number of meas- urements is called the mean deviation, which gives an indication of the precision of the measurement. For our example, the mean deviation of the period is 0. Therefore, we should write the period of the pendulum, as measured in the laboratory, as 3.
Another way of expressing the precision of the measurement is by use of the rms deviation, defined as the square root of the quantity obtained by adding the squares of the deviations divided by the number of measurements. For our meas- urements, the rms root-mean-square deviation is 0. Assuming that the randomness that appears in the set of measurements is not due to any bias, but that these are just normal fliictuations, the rms deviation tells us that roughly two-thirds of all the measure- ments fall within this deviation from the mean value.
TABLE 3. One amu is equal to 1. Express, in kilograms and grams, the masses of one atom of a hydrogen and b oxygen. In 18 grams? In one cubic centimeter? Verify that this definition is compatible with the value of the amu given in Problem 2. Use the values of relative densities given in Table , Extend your calculation to other gases. What general conclusion can you draw from this result?
Estimate the number of molecules in one cubic centimeter of air at STP. Assuming that the gas is mainly hydrogen, estimate the number of hydro- gen atoms per cubic centimeter. Compare the result with air at STP Problem 2. In 2 hours the water level drops 1 mm. Estimate, in grams per hour, the rate at which water is evaporating. We suggest that the stu- dent perform this experiment and obtain his own data. Why do you get different results on different days?
VTaen we refer to a chemical element and not a compound, we use the atomic mass. Verify that the number of molecules or atoms in one mole of any substance is the same, and is equal to 6. The radius of the nu- cleus of uranium is 8. Using the atomic mass of uranium given in Table A-1, obtain the density of "nuclear matter. From your result, would you conclude that it is rea- sonable to treat nuclear matter in the same manner as matter in bulk, i.
When you compare these values with the data in Table , what do you conclude about the structure of these two bodies? Assuming that all atoms are distributed uniformly over all the uni- verse, how many atoms would there be in each cubic centimeter? Assume that all atoms are hydrogen. How many times could a light ray travel around the earth in one second?
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If it is discarded, the mean value of the nine remaining data points is Meas- ure the time it takes for the ball or pencil to go from rest, at the top, to the bottom when it hits the table. Repeat the experiment ten or more times. If you do not have a sweep-second hand, use your pulse as a timing source. Determine the height and weight of each member. Discriminate so that you cover onty one sex and have an age span of no more than three years. Calculate the mean height, mean weight, and the rms deviation.
Note that you cannot talk about the precision of your measurement in the same sense as above.
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Vector algebra is important because it enables the scientist to write in a convenient, terse, shorthand notation some very complicated expres- sions. It is also Figure possible to describe this same relation in yet another way; namely, by the shorthand notation of a graphical plot of this equation, as shown in Fig. Both these examples are readily xmderstandable to any student who has studied algebra and analytic geometry, because he understands the short- hand notation.
In the same manner, vector algebra is readily understandable, once the shorthand notation is understood.
By the end of this chapter it wiU be discovered that vector notation is not un- like the notation of algebra and analytic geometry. The major difference is in the interpretation of this notation. A thoughtful reading of this chapter accompanied by careful working of aU exercises will save the student many difficult moments in succeeding chapters. Once the posi- tive sense has been determined, we say that the line is oriented and caU it an axis. The coordinate axes X and Y are oriented lines in which the positive senses are as indicated in Fig. The positive sense is usually indicated by an arrow An onented line or axis defines a direction.
Two angles arc requited to define a direction m space the opposite direction is determined 3. Similarly, force and acceleration are vector quantities. When written donm, a symbol in boldface t 3 q e such as V or with an arrow, as V, indicates a vector i. A unit vector is a vector whose magnitude is one. If a particle is displaced first from A to B Fig. The procedure can be generalized to fit any kind of vectors. Displacement is a vector quantity. Vector difference is anticom- mutative. The difference between two vectors is obtained by adding to the first the negative or opposite of the second Fig.
Find: a the sum of the two vectors; b the difference between the two vectors. Solution: Before starting to apply the previous equations, the vectors on a set of coordinate axes Fig. We see from Fig. This may be done by moving either vector or both, just so long as the direction of the vector is not changed Fig.
Rectangular components of a vector in a plane. Components of a vector in a certain direction. Using Eq. TSTien we compare this result with Eq. Also from Fig. Rectangular components of a vector in three dimensions. Find the components of the vector that is 13 units long and makes an angle 6 of Find also the angles -with the X- and 7-axes. Solution: Using Fig. Now a simple application of Eq. In terms of Eq. Express the equation of a straight line parallel to UyiJ i- and passmg through a point Pq.
Solution: Appljdng Eq. The only physical assumption required is that we recognize that velocity is a vector quantity. If the water is still, Vb is also the velocity of the boat as measured bj' an observer on the shore. Thus the resultant velocity of the boat, as measured by an observer on the shore, is the vector sum of the velocity of the boat Vb relative to the water and the drift velocity Vc due to the water current.
Find the resultant velocity of the boat. Solution: This problem is solved graphically in Fig. Tn tliis case we know that F must have the direction 00'. This has been done in Fig. Thus, using Eq. Therefore the direction of Fa must be N We leave the answer to the student. Solution: Let P Fig. The plane AB is inclined an angle 0. The component a gives the acceleration of the body along the plane. Obviously A. Thus from Eq. Vector equation of a plane. Fig, , Vector relations m Fig. If two vectors are 3. Obviously A y. This can be seen as follows Fig.
The vector product is equivalent to the area of the paral- lelogram defined by the two vectors. Y Fig. The vector product is distributive. In this case Fig. Therefore Eq. The proof in the general c ase of three vectors in space is similar, but somewhat more complex. Solution: First we compute the vector product of A and B, using Eq. Find the distance from point P 4, —1, 5 to the straight line passing through points Pi — 1, 2, 0 and P 2 1, 1, 4.
Z Solution: The geometry of the problem has been illustrated in Fig. We shall adopt the convention of representing it by a vector S, whose mag- nitude IS equal to the area of the surface and whose direc- tion is perpendicular to the surface. The sense of the vector is the direction in which a right-handed screw advances when its head is rotated in the same sense as the periphery IS onenled.
The components of S have a simple geometric meaning Suppose that the plane of surface S makes an angle 6 with the XF-plane Fig. Z Fig. Vector addition of surfaces For Ktample, let us consider a plot of land, of which part is horizontal and part on the slope of a hill, as indicated in Fig. Finally, consider a closed surface, as shown in Fig. Divide this surface into small plane surfaces, each one represented by a vector Si in the outward direction.
We can always associate the small areas in pairs such that their combined pro- jection is zero. For example, in Fig. References 1. New York: Appleton-Centuryr Crofts, 2. Elementary Vectors, by E. New York; Pergamon Press, 3. Mechanics second edition , by K.
Reading, Mass. Physical Mechanics third edition , by R. Princeton, N. Vector Mechanics, by D. New York: McGraw-Hill, 8. Introduction to Engineenng Mechanics, by J. Leighton, and M Sands. Readmg, hlass. Obtain the magnitude of the resultant and the angles it makes with the X-, Y-, and Z- axes. Prove that the value of the triple product is equal to the volume of the parallelepiped made from the three vectors. Also write the equation of the straight hue passing through them.
Also find the distance from point P to the plane through Q perpendicu- lar to F. It is composed of three tri- angular surfaces each having one side coincident with the sides of the triangle and one common vertex at point a, b, c. Show that the vector representing the com- plete surface is independent of a, b, c. Was this result to be expected in view of Problem 3. Consider the tetrahedron with vertices at points 0, 0, 0 , 2,0,0 , 0,2,0 , and 1,1,2. Find: a the vector representing each face; b the vector representing the whole tetra- hedron; c the magnitude of the surface of the tetrahedron.
Were you expecting the result obtained in b? Find, using vector methods, the angle of each side with the opposite face and the dis- tance from one vertex to the opposite face. Equilibrium of a Rigid Body Composilion of concurrent forces 4J 4,1 Introduction An important usage of vector algebra is its application to the composition of forces. The precise definition of force will be analyzed in Chapter 7, where we shall discuss the dynamics of motion. However, to gain more skill in the manipu- lation of vectors, we shall now discuss the composition of forces, and in particular the equilibrium of forces, a problem of wide apphcation in engineering.
We shall assume at present an intuitive notion of force, derived from our every- day experience, such as the force needed to push or pull a given weight, the force exerted by certain tools, etc. Experience confirms that forces are combined according to the rules of vector algebra. In this chapter we shall consider forces applied only to mass points or particles and rigid bodies.
In this chapter, however, we shall also express force in other irnits, such as kilogram-force kgf , pound-force Ibf , poundal pdl , and ton T. Therefore, the resultant R of several concurrent forces Fj, F2, F3, The directions are as mdicated in the figure. According to the properties of the vector product, the torque is represented by a vector perpendicular to both r and F; that is, perpendicular to the plane that may be drawn through both r and F, and directed according to the sense of advance of a right-handed screw rotated in the same sense as the rotation produced by F around 0.
This is indicated in Fig. Determine the torque applied to the body in Fig. Find also the equation of the line of action of the force. If all the forces are coplanar, and 0 is also in the same plane, all torques appearing in Eq. Consider three forces applied at point A of Fig. Using 0 as the reference point, find the resultant torque due to these forces. The resultant torque can also be found by applying Eq. The force is chosen equal to R for translational equivalence and is applied at the point about which the torques were evaluated, so that its torque is zero.
The couple with a torque equal to t is then chosen for rotational equivalence. Find the resultant force and the re- sultant torque of the system illustrated in Fig. Therefore 2 X F 2 Ux — 4. T Composition o! Parallel forces Let us consider a system of forces parallel to a unit vector u Then T. We conclude that a system of parallel forces can be reduced to a single force, parallel to each of the forces, given by Eq. The vector equation 4. Find the resultant of the forces acting on the bar of Fig. Equilibrium of a particle 4. EqaiUhrium of a Particle Statics is the branch of mechanics that deals with the equilibrium of bodies.
Discuss the equilibrium of three forces acting on a particle. Solution: We shall consider the three forces illustrated in Fig. This indicates that the three concurrent forces in equilibrium must be in one plane. Also, applying the Law of Sines hl. Figure Figure Fig. Equilibrium on an inclined plane. Discuss the equilibrium ot a particle on a smooth inclined plane. We may proceed in two different ways. Figure 70 Forces 4. The student must decide, in each particular problem, which method is more direct or convenient.
JEqttilibrinm of a Jttiffid MIotly When forces are acting on a rigid body, it is necessary to consider equilibrium relative to both translation and rotation Therefore the two following conditions are required I. We now illustrate the technique of solving some typical problems of plane statics. The bar of Fig. Find the forces exerted on the bar at points A and B. The bar weighs 40 kgf and its length is 8 m.
Solution: Applying first the condition 4. It is more convenient to mpute the torques relative to A, because in this way the torque of force F is zero. If the tension in the cable is Ibf, what are the horizontal and verti- cal forces exerted on the pole by the cable' 4. There is a stone block weight 10 kgf on the plane, held in place by a fixed obstacle. Find the force exerted by the block a on the plane and b on the obstacle. Use the resultant force to deter- mine the resultant torque. Prove that the resultant torque is perpendicular to the resultant force.
The angles between the m Figure 4. Determine the equation of the line of action of the force. Each square is 1 ft on a side. There are forces of , , and djmes acting down- ward at 0, 50, and cm from one end, and forces of and 13, dynes acting upward at 20 and cm from the same end.
Determine the magnitude and line of action of the resultant. Each segment of the beam AB is 1 decimeter. Also find the force needed at A and B to balance the other forces. It is resting on its ends A and B and is supporting the masses, as shown in Fig. Calculate the reac- tions at the supports.
Calculate the reactions of the two planes on the sphere. Calculate the reaction of the wall and the plane on the sphere. If a is the angle between the rope and the wall, determine the tension in the rope and the reaction of the wall on the sphere. CE and DC are cables. Neglect the weight of the boom. There is a fixed point C around which the beam can rotate.
The beam is resting on point A. A man weighing 75 kgf is walking along the beam, starting at A. Calculate the maximum distance the man can go from A and still maintain equilibrium. Plot the reaction at A as a function of the dis- tance X. Determine the magnitude and position of the resultant. The spheres C and D respectively 40 kg and 20 kg , linked by the beam CD, are rest- ing on it. The distance between the cen- ters of the spheres is 0.
Figure 4 41 A bridge m long and neighing 10, kgf IS held up by two columns at its ends. Extend your plot until all cars are off the bridge. Plot the re- action at each end of the plank as a func- tion of the distance of the man from the end. The pulleys marked C are movable. The plane and the pulleys are all smooth. Rope AC is horizontal and rope AB is parallel to the plane. Also calculate the reaction of the plane on the weight A. Find the position of equilibrium of the stick. Calculate the reactions of the hemisphere on the stick. Determine the position of equilibrium and the reaction forces as a function of the angle a.
Calculate the reactions of the surfaces on the spheres. Show that each sphere is independently in equilibrium. Everything else in the example remains the same. Was this result to be expected? Use the data listed in Table However, if the observers know their relative motion, they can easily reconcile their respective observations.
In Chapter 6 we shall dis- cuss in more detail this important matter of comparing data gathered by ob- servers who are in relative motion.
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For the time being we shall assume that we have a well-defined frame of reference. Let us take the OZ-axis of Fig. The position of the object is defined by its displacement x from an arbitrary point 0, or origin. Obviously, x may be positive or negative. Thus the average velocity during a certain time interval is equal to the are age displacement per unit time during the time interval.
To determine the in- stantaneous velocity at a point, such as A, we must make the time interval At as small as possible, so that essentiallj'- no changes in the state of motion occur during that small interval. In mathematical' language this is equivalent to computing the limiting value of the fraction appearing in Eq. Thus Ax 2. Thus , Ax 0. If the velocity remains con- stant, the motion is said to be uniform. Again referring to Fig. Thus the average acceleration during a certain time interval u the change ni velocity per unit Ume diirmg the time interval.
In general, the acceleration varies during the motion. If the rectilinear motion has constant acceleration, the motion is said to be uniformly accelerated. From Eq. The acceleration is also related to the position by combining Eqs. When we multiply the left-hand side of this equation by the left-hand side of Eq 5. Vector relation between velocity and acceleration in rectilinear motion. Graph of velocity and displacement in uniformly accelerated motion.
We may plot both c and x against time. Both equations have been plotted in Fig. Graphs of this kind are very useful in analyzing all types of motion. Solution: a Using Eqs. Also the student should solve the problem by placing the origin of coordinates at A. Eq- 6. Displacement and average ve- locitj" in curvilinear motion. The velocity is tangent to the path in curvilinear motion. Or, if vre take into account Eq. Let Oq Fig. As in the rectilinear case, s may be positive or negative, depending on which side of Oq the particle is.
When the particle moves from A to B, the dis- placement As along the cuiwe is given by the length of the arc AB. Multiplying and dividing Eq. Now from Fig. Acceleration m curvilinear motion Acceleration is a vector that has the same direction as the instantaneous change m velocity Since velocity changes in the direction m which the curve bends, acceleration is always pointing toward the concavity of the curve, and m general 5. Vector relation between velocity and ac- celeration in curvilinear motion. Similarly, Eq. The time required for the projectile to reach the highest point A is obtained by setting Vy 0 in Eq.
The time is obviously twice the value given by Eq. The equation of the path is ob- tained by eliminating the time t between the two equations in 5. The multiflash photograph in b shows that the mass falls with uniformly- accelerated motion. Verify this by taking actual measurements on the photograph.
But we know that i? Therefore the velocity of A is V UX. This equation also gives the velocity of any point B on the rim of the disk. The tan- gential acceleration of B is thus the same as the acceleration of A. Vector relation between angular velocity, linear velocity, and position vector in circular motion Circular motion: angular velocity 5. The unit was named hertz after the German physicist H.
Hertz , who was the first to prove experimentally the existence of electromagnetic waves. For example, the motion of the earth around the sun is neither cucular nor uniform, but periodic. It is a motion that repeats itself every time the earth completes one orbit.
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The period is the time required to complete one cycle, and the frequency is the number of C 3 '-cles per second, one hertz cor- responding to one c 3 ''cle per second. Find the angular velocity of the earth about its axis. In this case of uniform circular motion we may compute the acceleration directly by using Eq. Then, smce co is constant. This can be verified very easily. Referring to Fig. This value coincides with om- previous result 5. The latitude of point. That is, dr dB 5.
Mteferenees 1. Mechanics, Keith R. Physical Mechanics, Robert B, Lindsay. Introduction to Engineering Mechanics, John V. Vector Mechanics, D. Eeymnan, R. Source Book in Physics, W. Cambridge, Mass. RoUer Reading Mass. Also find its acceleration. Find the velocity iJk and the displacement as functions of time. Also find a; as a function of i and a as a function of x. Indi- cate a where the motion is in the positive or negative X-direction, b when the mo- tion is accelerated or retarded, c when the body passes through the origin, and d when the velocity is zero.
Also make a sketch of the velocity and the acceleration as functions of time. X m Fig. Calculate the velocity and the distance traveled by the stone after 10 s. Solve the same problem for the case of a balloon rising at the given velocity. Cal- culate the angular velocity and angular acceleration after 4 s. After it has been rotating for some time at this speed, the brakes are applied, and it takes 5 min to stop the wheel.
If the total number of revolutions of the wheel is , calculate the total time of rotation. The point d is attached to the rim of a flvnvheel whose diameter is 9 in. Obtain a relation between the angular velocities and the radii of the two wheels. Find the time necessary for wheel B to reach an angular velocity of rpm. Calculate the distance cov- ered by the train in the time it takes for the lamp to fall to the ground.
It strikes the ground at a horizontal distance of 4 km from the gun. Calculate the range horizontal distance from the base of the cliff of the gun. Repeat the problem for a firing angle below the horizontal. Repeat the problem for a car moving away from the cliff. At the in- stant the plane is directly over an antiair- craft gun, the gun fires at the plane. Cal- culate the minimum velocity vq and the aiming angle a which the projectile would need in order to hit the plane. At the moment he fires his gun, the squirrel drops off the branch. Prove that the bomb should be dropped when the hori- zontal distance between the plane and the ship is m.
Solve the same problem for the case in which the ship is moving in the opposite direction. Also find the components of the velocity and the acceleration of the point. A reiative to B are defined by 6. Taking the derivative of Eq. Equations 6. For situation c we use Eq. There- fore, using Eq. For simplicity we shall assume that both 0 and O' are in the same region of space and that each uses a frame of reference attached to itself but with a common origin.
For example, observer 0, who uses the frame XYZ Fig. That is, by substituting the appropriate quantities into Eq. Also from Eqs. The second term, 2co X V, is called the Coriolis acceleration. The third term is similar to Eq. Both the Coriolis and cen- tripetal accelerations are the result of the relative rotational motion of the ob- semrs. In the next section we shall illustrate the use of these relations. As indicated in Example 5. Let us call the acceleration of gravity as measured by a nonrotating observer at A.
Then corre- sponds to a in Eq. Solving Eq, 6. Radial and horizontal components of the centrifugal acceleration. Remembering Example 5. Its maximum value, at the equator, is about 0. In Fig. As indicated before, the acceleration of gravity go points dovmward along AB. The two components are illustrated in Fig. According to the definition of g given by Eq. The component as decreases as one moves from the poles toward the equator, where it is zero.
Thus at the equator the Coriolis acceleration produces no hori- zontal effect on the horizontal motion. The vertical effect is small compared with the acceleration of gravity, and in most instances may be neglected. The horizontal effect may be seen in two common phenomena. One is the whirl- ing of nund in a hurricane. If a low-pressure center develops in the atmosphere, the wind will flow radially toward the center Fig. However, the Coriolis acceleration deviates the air molecules toward the right of their paths in the north- ern latitudes, resulting in a counterclockwise or wlurhng motion.
As a second example, let us consider the oscillations of a pendulum. When the amplitude of the oscillations is small, we can assume that the motion of the bob is along a horizontal path. If the pendulunr were initially set to oscillate in the east- west direction and were released at A see Fig. Therefore, at the end of the first oscillation, it reaches B' instead of B. On its return, it goes to A' and not to A. Therefore, in successive complete oscilla- tions. Ihe end result is the cyclonic motion illustrated in Fig. Physicists in earlier days had assumed that vibrations of this hypothetical ether were related to light in the same way that vibrations in air are related to sound.
Remember Example 6. In the American physicists Michelson and Morley started a memorable series of experiments for measuring the velocity of light in different directions relative to the earth. To their great surprise they found that the velocity of light was the same in ah directions. Une possible alternative explanation would be that the earth drags the ether with it, as it drags the atmosphere, and therefore close to the earth's surface the ether should be at rest relative to the earth.
This is a rather improbable explanation, since the ether drag w'ould manifest itself in other phenomena connected with light propagation. Such phenomena have never been observed. For the above reasons, the idea of an ether has been discarded by phy sicists. This principle states that all laws of nature must he the same i. Again this reduces to Eq. Therefore observer O' also measures a velocity c. Solving Eq. Note that if 7' and v are both smaller than c, then 7 is also smaller than c. Obtain the relation between the acceleration of a particle as measured obfervers in relative motion.
For simpUcity, supose that, at the instant of the comparison, the particle is at rest relative to observer 0. This result differs from Eq. In other words, the requirement that the velocitj' of light be invariant in all frames of reference which are in uniform motion relative to each other destroys the invariance of the acceleration. It is important to know the relation between the magnitudes of the accelerations ob- served by 0 and O'. That is, observer 0 saw the events occur at two different positions in space. Let us consider two observers 0 and 0' in relative motion along the X-axis with velocity v.
This is the same distance as measured by 0, since the mirror is at a posi- tion perpendicular to the direction of motion. Suppose that, when 0 and O' are coincident, a light signal is flashed from their common origin toward the mirror. For the system that sees the mirror in motion, the light signal must be sent out at an angle dependent on the velocity of the mirror and the distance L.
Let T and T' be the times recorded by 0 and O' for the light signal to return to 0' after it has been reflected from the mirror. In the O'-system, the light will return to the origin, but in the 0-system, the light Avill cross the X-axis at a distance vT from the origin. Analysis of the Michelson- Morley experiment. At the beginning of Sec- tion 6. Note pig. Basic components of the that the light path drawn in Fig. Let us call v the velocity of the earth relative to the ether, and orient the interferometer so that the line PMi is parallel to the motion of the earth.
Fig- There is, however, a fundamental difference between the two under- lying hjqiotheses used for obtaining these two apparently identical results: 1 The con- traction 6. The v appearing in the formula is the velocity of the object relative to the observer, and thus the contrac- tion is different for different observers. It was the genius of Einstein that led him to realize that the idea of an ether was artificial and unnecessary, and that the logical explanation was the second one.
This was' the basic post'ulate which Einstein used to formulate the principle of relativity, as we shall see in Chapter Rush, Set. Bronowski, Sci. Shankland, Am. Frisch and J. Smith, Am. Scott and M. Viner, Am. Weisskopf, Physics Today, September , page 24 9. Holton, Am.
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Van Nostrand Co. The Special Theory of Relativity, D. New York: W. IW Mechanics, D. Find, in each case, the velocity of the bail relative to an observer standing on the. Find the e. Find the Coriolis acceleration. Show that in the Northern Southern hemi- sphere it pushes the water against the right left bank. This effect produces a some- what larger erosion on the right left bank that has been noticed in some cases.
YTiat is your Coriolis acceleration? Ydiat is the maximum devi- ation of the plumb line from the radial direction on the surface of Jupiter? Consider this problem when the origins coincide and then when they are displaced. How long is the stick according to O'? The orbital velocity of the earth relative to the sun IS 30 km and the radius of the earth is given in Table 6 27 A rocket ship heading toward the moon passes the earth with a relative velocity of 0 8c a How long does the trip from the earth to the moon take, ac- cording to an observer on the earth? A hght placed at 0 is turned on for one year How long IS the light on, according to 0?
Under what conditions are events that ap- pear simultaneous to 0 also simultaneous to tion? Why does a spring oscillate when it is stretched? We assume that the motion of the particle is relative to an observer Avho is himself a free particle or system; i. Such an observer is called an inertial observer, and the frame of reference he uses is called an inertial frame of reference. Nor is the sun an inertial frame of reference. Because of its inter- actions with the other bodies in the galaxy, it describes a curved orbit about the center of 'the galaxy Fig.
Let us illustrate some experiments performed in our terrestrial laboratories that support the law of inertia. A spherical ball resting on a smooth horizontal surface vill remain at rest imless acted upon. That is, its velocity remains constant, with value equal to zero. We assume that the surface on which the ball is resting bal- ances the interaction between the baU and the earth, and hence that the ball is essentially free of interactions.
When the ball is hit, as in billiards, it momentarily suffers an interaction and gains velocity, but afterward is free again, moving in a straight line with the velocity it acquired Avhen it was struck. Interaction between two particles. In writing this equation we have maintained our assumption that the masses of the particles are independent of their states of motion ; thus we have used the same masses as in Eq. In other words. This result constitutes the principle of the conservation of momentum, one of the most fimdamental and imivemal principles of physics.
For example, consider a hydrogen atom, composed of an electron revolving around a proton, and let us assume that it is isolated so that only the interaction between the electron and the proton has to be considered. Then the sum of the momenta of the electron and the proton relative to an inertial frame of reference is constant. Similarly, con- sider the system composed of the earth and the moon. If it were possible to neglect the mteractions due to the sun and the other bodies of the planetary sys- tem, then the sum of the momenta of the earth and the moon, relative to an in- ertial frame of reference, would be constant.
Although the above-stated principle of the conservation of momentum con- ers only two particles, this principle holds also for any number of particles omung an isolated system; i. There- of, the principle of conservation of momentum in its general form says that the total moxnentum of an isolated sxjstem of particles is constant. The lav- of inertia stated in Section 7. Because if we have only one isolated particle instead of several, Eq. We continually find arormd us examples of the principle of conservation of momentum. The recoil of a firearm is one.
Initially the system of gun plus bullet is at rest, and the total momentum is zero. When the gun is fired, it recoils to com- pensate for the forward momentum gained by the bullet. A gun whose mass is 0. Solution; Initially both the gun and the bullet are at rest and their total momentum is zero. Momentum is conserved in the explosion of a grenade. In this situation it is rather difficult to use the prmciple of conservation of momentum.
However, there is a practical way of circumvent- ing this difficulty, by introducing the concept of force. Sign In Register Help Cart. Cart items. Toggle navigation. Search Results Results 1 -2 of 2. Very Good. Pearson Higher Education. Spine creases, wear to binding and pages from reading. May contain limited notes, underlining or highlighting that does affect the text.
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