Ex. 9 . Translate the following phrases into English.

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     Пусть точка  занимает положение;  в данный момент времени; по отношению ко времени; соответствующие координаты точки; положительное перемещение; провести ось через точку; обозначить угол между; радиус кривой;  время стремится к нулю; полученное отношение; абсолютное и числовое значение скорости.

Ex. 10. Translate into English.

1. Эти составляющие взаимно перпендикулярны.

2. Скорость точки  может увеличиваться или уменьшаться с течением времени.

3. Этот угол называется смежным.

4. Эта ось перпендикулярна к двум другим осям.

5. Вектор ускорения - это  диагональ параллелограмма, сторонами которого являются составляющие и   .

6. Вектор  может быть положительно или отрицательно направленным в зависимости от того, увеличивается или уменьшается скорость частицы.

Unit 5.

Tangential and Normal Accelerations of a Particle.

Learn the following words and word combinations by heart:

Ex.1. Look at Appendix 1 and read the following mathematical symbols  and Greek letters.

τ, w,     Δφ, ., sin Dj, wn>0,

We can already compute the acceleration vector  according to its projections on stationary coordinate axes Oxyz. In the natural method of describing motion, vector is determined from its projections on a set of coordinate axes M τ nb whose origin is at M and who move together with the body (Fig. 6).

The acceleration of a particle lies in the osculating plane, i.e., plane M τ n, hence the projection of vector  on the binormal is zero (wb = 0).

Fig. 6

Let us calculate the projection of on the other two axes. Let the particle occupy a position M and have a velocity at any time t, and at time t₁=t+Δt let it occupy a position M₁ and have a velocity . Then, by virtue of the definition,

Let us now express this equation in terms of the projections of the vectors on the axes M τand Mn through point M (see Fig. 6). From the theorem of the projection of a vector sum (or difference) on an axis we obtain:

,            .

Noting that projections of a vector on parallel axes are equal, draw through point M₁ axes Mτ' and Mn' parallelto Mτ and Mn respectively, and denote the angle between the direction of vector  and the tangent Mτ by the symbol Δφ. This angle between the tangents to the curve at points M and M1 is called the angle of contiguity.

It will be recalled that the limit of the ratio of the angle of contiguity Δφ to the arc  = D s defines the curvature k of the curve at the point M. As the curvature is the inverse of the radius of curvature r at M, we have:

.

From the diagram in Fig. 6, we see that the projections of vectors  and  on the axes M t and Mn are

              v t = v,                       vn = 0,

              v1 t = v1 cos D j, v1n = sin D j,

where v and v₁ are the numerical values of the velocity of the particle at instants t and t₁. Hence,

,                    .

It will be noted that when , point M1 approaches M indefinitely, simultaneously ,  and .

Hence, taking into account that , we obtain for w t the expression

                                           .

We shall transform the right-hand side of the equation for wn in such a way so that it includes ratios with known limits. For this purpose, multiplying the numerator and denominator of fraction under the limit sign by D j D s, we find:

,                        (23)

since, when , the limits of each of the cofactors inside the brackets are as follows:

                        .

Finally we obtain

                        ,        .                             (24)

We have thus proved that the projection of acceleration of a particle on tangent to the path is equal to the first derivative of numerical value of the velocity, or the second derivative of the displacement (the arc coordinate) s, with respect to time; the projection of the acceleration on the principal normal is equal to the second power of the velocity divided by the radius of curvature of the path at the given point of curve, the projection of acceleration on the binormal is zero (wb = 0). This is an important theorem of particle kinematics.

When particle M is moving in one plane, the tangent M t sweepsaround the binormal Mb with an angular velocity . By introducing this quantity into Eq. (23) we can obtain one more equation for calculating wn that is frequently used in practice:

                                           Wn = v w                              (24.1)

i.e. normal acceleration equals the product of a particle’s velocity by angular velocity of the sweep of tangent to the path.

Fig. 7

Lay off vectors  and , i.e. the normal and tangential components of the acceleration, along the tangent M t and the principal  normal Mn, respectively (Fig. 7). The component  is always directed along the inward normal, as wn>0, while the component  can be directed either in the positive or in the negative direction of axis M t, depending on the sign of the projection w t (see Figs. 7 a   and  b).

The acceleration vector  is the diagonal of a parallelogram constructed with the components and  as its sides. As the components are mutually perpendicular, the magnitude of vector  and its angle m to the normal Mn  are given by the equations:

           (25)

Thus, if the motion of a particle is described by the natural method and the path (and, consequently, the radius of curvature at any point) and the equations of motion (20) are known, from Eqs. (22), (24), and (25) we can determine the magnitude and direction of the velocity and acceleration vectors of the particle for any instant.

Comprehension  check.

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