Part 2. Determination of the Acceleration of a Particle when its Motion is described by the Coordinate Method

The acceleration vector of a particle is . Hence, from the theorem of the projection of a derivative we obtain:

; ;

or

; ;

i.e. the projections of acceleration on the coordinate axes are equal to the first derivatives of the projections of the velocity or the second derivatives of the corresponding coordinates of the particle with respect to time. The magnitude and direction of the acceleration are given by the equations:

; ;

where a, b, c are the angles made by the acceleration vector with the coordinate axes.

Ex 3. Answer the following questions.

1. What are the projections of acceleration on the coordinate axes equal to?

2. What are the angles a, b, c made by?

Ex.4. Complete the following sentences with the information from the text (Parts 1 and 2).

1. Vector is always directed….

2. The ratio of the velocity increment vector to the corresponding time interval t defines….

3. The vector of instantaneous acceleration of a particle is equal…

4. The projections of acceleration on the coordinate axes are equal to the second derivative…..

Part 3.Determination of the Acceleration of a Particle when its Motion is described by the Natural Method

In the natural method of describing motion vector is determined from its projections on a set of coordinate axes Mrnb whose origin is at M and who move together with the body. These axes, called the axes of the natural trihedron (or velocity axes), are directed as follows: axis Mr along the tangent to the path in the direction of the positive displacement S, axis Mn along the normal in the osculating plane towards the inside of the path, and axis Mb perpendicular to the former two to form a right-hand set. The normal Mn, which lies in the osculating plane (or in plane of the curve itself if the curve is two-dimensional), is called the principal normal, and the normal Mb perpendicular to it is called the binormal.

The acceleration of a particle lies in the osculating plane Mrn, hence its projection on the binormal is zero (ab=0).

Let the particle occupy a position M and have a velocity at time t, and at time t1=t+t let it occupy a position M1 and have a velocity .

From the theorem of the projection of a vector sum (or difference) on an axis we obtain:

Noting that projections of vector on parallel axes are equal, draw through point M1 axes Mr’ and Mn’ parallel to Mr and Mn, respectively, and denote the angle between the direction of vector and the tangent Mr (this angle is called the angle of contiguity).

where k – the curvature of the curve at point M;

R – the radius of curvature at point M.

We see that

Vr = V; Vn = 0;

V1 r = V1 cos f; V1 n =V1 sin f.

Hence

When t tends to zero, f and S tends to zero too and V1 tends to V.

Then

Multiplying the numerator and denominator of the fraction under the limit sign an by fS, we find

,

since

Finally we obtain

We have thus proved that the projection of the acceleration of a particle on the tangent to the path is equal to the first derivative of the numerical value of the velocity, or the second derivative of the displacement 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 the curve, the projection of the acceleration on the binormal is zero (ab=0).

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 are given by the equation:

.

The relations obtained express that the tangential component of the acceleration is equal to the rate of change of the speed of the particle, while the normal component is equal to the square of the speed divided by the radius of curvature of the path at point P.

Normal acceleration characterizes the change in direction of the velocity depending upon whether the speed of the particle increases or decreases, a_r positive or negative, and the vector component points in the direction of motion or against the direction of motion. The vector component , on the other hand, is always directed towards the center of curvature of the path.

The absolute or numerical value of the velocity is called the speed: speed is thus essentially a positive quantity.

Ex 5. Answer the following questions.

1. What is the direction of a velocity axis?

2. What normal is called principal?

3. What is the binormal perpendicular to?

4. What is the projection of the acceleration on the tangent to the path equal to?

5. What is the projection of the acceleration on the principal normal equal to?

6. What is the projection of the acceleration on the binormal equal to?

7. Define the tangential and the normal components of the acceleration .

8. What is the difference between the velocity and the speed?

Ex.6 .Say whether the following statements are True or False. (Part 3)

1. Axis Mr is directed along the tangent to the path in the direction of the positive displacement S.

2. The normal Mn lying in the osculating plane is called the binormal.

3. The projection of the acceleration on the principal normal is equal to zero.

4. The tangential component of the acceleration is equal to the square of the speed divided by the radius of curvature of the path at point P.

5. Normal acceleration characterizes the change in direction of the velocity.

Ex.7 .Complete the phrases with the following prepositions:

As, towards, under, of, in, from, to, against, from, on, with, together.

Natural method ……describing motion; vector is determined ….its projections; move …. with the body; be directed …follows; the tangent …the path; in the direction …the positive displacement; the normal …the osculating plane; ….the inside of the path; be perpendicular …the axis; a projection ….the binormal; ….the theorem of the projection; be parallel …..; the fraction ….the limit sign; parallelogram constructed …the components and ….. its sides; point …..the direction of motion or …..the direction of motion.

Ex.8. Match the following nouns or noun phrases. Compose 5 sentences with any of these expressions.

	Nouns / noun phrases		Nouns / noun phrases
1	along the tangent	a	curvature of the path
2	angle between	b	the velocity increment vector
3	curvature of	c	average acceleration of the particle
4	direction of	d	the projection of a derivative
5	increase	e	the path
6	projection of	f	in velocity
7	radius of	g	the direction of vector and the tangent
8	ratio of	h	to the path
9	ratio to	i	the curve at point M
10	theorem of	j	the positive displacement
11	towards the inside of	k	acceleration on the coordinate axes
12	vector of	l	the corresponding time interval

Ex. 9. Translate the following phrases into English.

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.