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This emphasizes the need for a clear distinction between weight and mass since the weight of an object is expressed in units of force. Briefly, the weight of a body is the force with which that body is attracted toward the earth. Its mass is the quantity of matter in the body, irrespective of its location in space. For example, if a given body is moved from a valley to the top of an adjacent mountain, its mass is constant; but its weight is less on the mountain top because the magnitude of the gravitational attraction decreases with increasing elevation above sea level.
Although the interpretation of Eq. First, the acceleration of gravity is measured with respect to a reference frame fixed in the earth. This is not an inertial frame in the strict sense because the earth is rotating in space.
Therefore, as we shall show more clearly in Chapter 2, the so-called inertial forces must be included in a calculation of the acceleration of gravity. Thus it turns out that centrifugal as well as gravitational forces enter the probleln. The effect of the centrifugal force is to cause a slight change in the magnitude and direction of the acceleration of gravity, the amount of the change depending upon the latitude.
A second reason for the slight variation In the acceleration of gravity SEC. Furthennore, in any discussion of orbits about the earth, we shall use the symbol go which includes gravitational effects only and refers to a spherical earth. In particular, let us consider the concepts of space, time, mass, and force from the viewpoint of Newtonian mechanics and indicate experimental procedures for defining the corresponding fundan1entaI units.
In the latter process, we shall not be concerned so much with the practical utility of the operational definitions as with their theoretical validity. Newton conceived of space as being infinite, homogeneous, isotropic, and absolute.
The infinite nature of space follows from the implicit assumption that ordinary, Euclidean geometry applies to it. Therefore the angular velocity Q is a free, vector. N ote 1hat QllL different Also, recall again that angular velocity refers to the motion of a rigid body, or, in essence, to the motion of three rigidly connected points that are not collinear.
The term has no unique meaning for the motion of a point or a vector or a straight line in three-dimensional space. I 38 CHAP. The velocity and acceleration of a point P as it moves on a curved path in space may be expressed in terms of tangential and normal components. Let us assume that the position of P is specified by its distance s along the curve from a given reference point. This is known as the normal direction at P and is designated by the unit vector en.
The plane of et and en at any x point P is called the osculating plane. Tangential, norn1al, and Therefore, any Inotion of et Inust binormal unit vectors for a curve in take place in the osculating plane at space.
Its magnitude is equal to its rotation rate i b about an axis perpendicular to the osculating plane. Thus where rob is the rotation rate of et about an axis in the binonnal direction ebo The binormal unit vector eb is perpendicular to the osculating plane and is given by thus completing the orthogonal triad of unit vectors at P.
The angular rate i b can be expressed in terms of the radius of curvature p and the speed s of the point P as it moves alorig the curve. Of course, the position of the center of curvature changes, in general, as P moves along the curve.
The time derivatives of the unit vectors can also be obtained in terms of the angular rotation rate OJ of the unit vector triad. We have Note that the normal component of OJ is zero. This can be explained by the fact that en was defined to lie in the dIrection of Ct.
Any normal component of OJ would result in a binormal component of Ct, in conflict with the original assumption. Of course, f t does not influence Ct. It can be seen that j tis also the rotation rate of the osculating plane.
Consider the motion of a point P in a circular path of radius a Fig. Circular motion of a point. Note that the same result is obtained by using the tangential and normal unit vectors of Eq.
Helical Motion. Helical motion of a point. In this case, the locus of the centers of curvature is a helix with the same radius and helix angle. Harmonic Motion. A common type of particle motion is that in which the particle is attracted toward a fixed point by a force that is directly proportional to the distance of the particle from the point.
From Newton's law of motion as given by Eq. But we' have assumed that the applied force is proportional to x and is directed toward the origin. So we can write the differential equation where 02 is a constant.
Simple harmonic motion. Thus the extreme values of the displacement and acceleration occur at the moment when the velocity is zero, and vice versa. Now consider the case of two-dimensional harmonic motion. Suppose the particle P moves in the x direction in accordance with Eq. Differentiating Eq.
Ct 44 CHAP. Harmonic motion in two dimensions. We saw that the x and y motions have the same frequency but are independent of each other. The relative phase angle 3 between these motions is determined from the initial conditions and the value of 3 will influence the shape and sense of the motion along the elliptical path. In any event, however, the path is inscribed within the dashed rectangle shown in Fig.
In this case, we again have simple rectilinear harmonic motion. Also, SEC. The acceleration is also constant in magnitude and, from Eq. For the case of simple harmonic motion, the two planes are orthogonal and the path reduces to a line.
The velocity and acceleration vectors in the case of elliptical nlotion are found by projecting the corresponding vectors for the case of circular motion. Hence it can be seen that the maxilnum velocity occurs at the ends of the Ininor axis and the maximunl acceleration occurs at the ends of the major axis.
In each of these two cases, the vectors at the given points are parallel to the plane onto which they are projected. In case the motions in the x and y directions are of different frequencies, then a different class of curves known as Lissajous figures is generated. Again, the motion relnains within the dashed rectangle of Fig. For the case of rational frequency ratios, the Inotion is periodic and retraces itself with a period equal to the least common multiple of the periods of the two component vibrations.
We shall not, however, pursue this topic further. Now consider the case shown in Fig. Also, the body is rotating with an angular velocity co relative to this system. Suppose we define the velocity ry where v and vA are the absolute velocities of the points P and A, respectively. Motion of a point in a rigid body.
Now suppose the observer is translating in an arbitrary fashion but is not rotating relative to X YZ. The velocities of points P and A as viewed by the translating observer would not be the same as in the previous case.
The velocity difference would, however, be the same as before because the apparent velocities of the points P and A would each change by the same amount, namely, by the negative of the observer's translational velocity.
Thus the velocity of point P relative to point A is identical for any nonrotating observer. On the other hand, it is important to realize that the velocity of P relative to A will be different, in general, when the motion is viewed from various reference frames in relative rotational motion, as we shall demonstrate. Hence a statement of the relative velocity of two points should also specify the reference frame.
An inertial or a nonrotating frame is assumed if none is stated explicitly. Returning now to a consideration of the relative velocity vPA as viewed by a nonrotating observer, we note that it can also be considered to be the velocity of P as viewed by an observer on a non rotating system that is translating with A. In this case, the base point A would have no velocity relative to the observer, and therefore we could use the results of Sec. Proceeding in this fashion, we obtain from Eq. So, from Eqs.
Since p is of constant magnitude, we can again use Eq. Finally, denoting the acceleration vA by aA, we obtain from Eqs. Since A is considerz ed to be a free vector during the differentiation process, no generality is lost by taking the point 0 as the common origin of the unit vector triad and also the vector A.
Now, as each observer views the vector A, he might choose to express it in terms of the unit vectors of his -,p::;y own system. Thus each observer would give a different set of components. Nevertheless, they would be viewing the same vector and a simple x coordinate conversion based upon the relative orientation of the coFig. The vector A relative to ordinate systems would provide a fixed and rotating reference frames. On the other hand, if each observer were to calculate the time rate of change of A, the results would, in general, not agree, even after performing the coordinate conversion used previously.
To clarify this point, recall from Eq. Now let us consider the last three terms on the right side of Eq. Using Eq. It is important to note that A can be any vector whatever.
A relatively simple application of Eq. On the other hand, a more complicated situation would arise if, for example, A were the velocity of P relative to another point P', as viewed from a third system that is rotating separately.
Thus Eq. Although we have referred to the coordinate systems in the preceding discussion as fixed or rotating, the derivation of Eq.
Therefore we need not consider either system as being more fundamental than the other. If we call them system A and system B, respectively, we can write , where' W BA is the rotation rate of system B as viewed from system A. Since the result must be symmetrical with respect to the two systems, we could also write where we note that either system. A turntable rotates with a constant angular velocity W about a perpendicular axis through o Fig. Motion of a point that is moving on a turntable.
Solve for the absolute velocity and acceleration of P in terms of its motion relative to the turntable. To find the acceleration, we again apply Eq. Rather, it is the time rate of change of the absolute velocity vector as viewed from the rotating system. In genera], the subscript after a differentiated dotted vector refers to the coordinate system from which the rate of change is viewed.
If the subscript is omitted, a nonrotating reference frame is assumed. Next, using the result given in Eq. Now we shall use the general result of Eq. In Fig. Suppose that r is the position vector of P and R is the position vector of 0', both relative to point 0 in the fixed XYZ system.
The position vectors of a point P relative to a fixed system and a moving system. Next we express p in terms of its value relative to the rotating xyz system, using Eq. Then from Eqs. We see that Ii is the absolute velocity of 0' and that j X P is the velocity of P' relative to 0' as viewed by a nonrotating observer. The remaining term P r is the velocity of P relati. It lSlrlteresting to note that P r can also be interpreted as the velocity of P relative to P', as viewed by a nonrotating observer.
In other words, at this pa rticular moment, the velocity of P relative to P' is the same, whether viewed from a rotating or nonrotating system. Also, of course, the velocity of P relative to any point fixed in the xyz system, whether it be P' or 0', is the same when viewed by an observer moving with the system. To obtain the absolute acceleration of P, we find the rate- of change of each of the terms in Eq.
R is the absolute acceleration of 0'. Thus the first three terms of Eq. The fourth term P r is the acceleration of the point P relative to the xyz system, that is, as viewed by an observer moving with the xyz system. The fifth term 2 1 X. Note that the Coriolis acceleration arises from two sources, namely, Eqs.
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