Back to the learning aids

Example of the system of the operationally connected categories

Each important concept/term is indicated by a circle with a number inside (for example, the circle with number 33 inside it - - indicates the term «an angular speed»).

On the schematics the different circles are connected by lines. Each line represents a verbal statement or formula, which includes BOTH the terms linked by the line. For example, the line connecting circles 33 and 30,

represents the formula (or the statement), relating a radius of a circle (R, number 30) with an angular speed (w, number 33).

Each circle answers a question “what kind of physics quantity is used to describe the situation at the given problem. Each line answers questions “is there a direct connection between two given physical quantities”; or “dose the value of this quantity affects the value of that”; “how are the physical quantities related to each other, what kind of dependence is there between them”?

There are more then one lines getting the same answer.

The line 33 – 30 represents the equation v = wR, but the same formula is represented also by the lines 33 – 32 and 30 – 32 (the circle 32 represents the speed of a body under a circular motion).

The schematics below (Fig. 1) visualises the main connections, dependencies, relations connected between physical quantities in a school Kinematics.

This kind of schematics is called the system of operationally connected categories (SOCC), because each link represent a possible logical/algebraic operation which can be done on the categories connected by the link.

Fig. 1

Maybe this schematics dose not look beautiful, but it is quite useful to find the solution of a problem on Kinematics.

After the schematics usually a brief definition of every physical quantity is given to explain an each number and each line on the schematics (all the information can be found in any school text-book)

1. Object is any body in a nature (a car , a spacecraft, a stone etc.).

2. A point mass is a body, which sizes are not essential for the given problem. Usually it is fair for a small objects, particles, which is placed in the vicinity of a large objects (a stone falling to Earth, a train goes from one city to another etc.).

3. Mechanical motion is a transition in space with during time, one body is changing its position with respect to the other body. While there is a mechanical motion the distance between bodies or the parts of bodies is changing.

4. Translational motion is a motion when all points of a body have a like trajectory of motion, i.e. the trajectories of any points of the body can be put on each other and they will completely coincide. For a body under a translational motion it is possible to monitor only any one point of it.

5. Circular motion is a motion when all points of a body lie on circles, which centres belong to one axis (line), and the planes of the circles are mutually parallel.

6. The trajectory is a line along which a mass point is passing. For example, the jet plane leaves in the sky a white trace. This trace is a trajectory of the jet.

7. Distance is the length of a trajectory. To find a distance it is necessary to put a thread along a trajectory, then to stretch this thread into strait line and to apply to a ruler and read the number.

8. Displacement is a vector connecting an initial and final position of a body.

9. Vector is an arrow, a strait segment directed in some direction. As a matter of fact, the picture of an arrow departed from a bow is a sample of a vector.

10. Coordinate is numerical indicators assigned to a position of a point mass in space. In general case it is necessary to know three numbers to indicate precisely a position of a point mass.

11. Cartesian frame of reference – a special, convenient and most widespread frame of reference. It represents three axes (i.e. simply three bar rulers), which we have placed under an angle 90⁰ to each other. The point, from which the axes begin, is named an origin. At the axes this point is located at the zero position at each ruler. When the position of an point object is necessary to indicate the perpendiculars from the point onto each axes have to drown accurately. Numbers, which cuts off by the perpendiculars on the axes is named as a coordinate. Usually the letters x, y and z are used to notate the coordinates (see. Figure 2).

If the body is moving along a strait line one axis only is necessary to indicate the position of the body. In this case the displacement of the body is just the difference between final and initial coordinates х_f – х_i = S.

12. The projection of a vector on an axis is a segment between points on the axis, which will be received, if perpendiculars are drown from a tail and a head of the vector on the axis. The numerical value of the projection of a vector on an axis is equal to length of this segment, if the direction of a projection coincides a direction of an axis, and «the negative length» if a direction is opposite. In the Fig. 3 projections of a vector A are represented. Sometimes the word a component of a vector is used for the projection of a vector.

13. The absolute value of a vector is simply its length, i.e. the length of the arrow by what the vector is represented. If а_х, а_y, а_z are the projections of a vector, its length is calculating with the help of the Pythagorean theorem:

14. The sum of vectors is such a new vector, which is drawn by the following rule. We have two vectors, i.e. two arrows a and b. By the parallel transposition we shift one vector (anyone, we shall name it as a second vector) so that its beginning (tail) has coincided with a finish (a head) of the other (we shall name it as a first vector). Then we have to draw an arrow, which goes from a tail of the first vector to a head of the second one. This arrow is the sum of two vectors. It is very important to understand the sum of vectors is always a vector, not a number!

15. The initial time is an instant, at which we have started the stop watch (even if it is just an imaginary stop watch). The final time is the time at which we have toped the stop watch.

16. The interval of time of motion, or the time taken for the motion – a time passed from the initial time to the final time. Do not be confused with a instant of time and a time taken for the motion. When needed we can always represent a whole time interval as a sequence of smaller time intervals, which can be convenient way to describe a motion with the variable acceleration.

17. The average velocity is the displacement of a body, which the body makes in average per the time unit. The formula for an average velocity is: Here S is the displacement of the body during the time t. This formula is represented on the schematics by the lines 8 - 16, 8 - 17 and 16 - 17.

18. The average speed is a distance, which the body is passing in average during the time unit. The formula for an average speed is: , there L is the distance of a body passed during a time t. This formula is represented on the schematics by the lines 7 - 16, 7 - 18 and 16 - 18.

19. The instantaneous velocity is an average velocity taken for a very small interval of time (such small, that an initial and final position of a body almost the same). This velocity is always directed tangentially to a trajectory.

20. A uniform motion is a motion when the body is making an equal displacement for any equal time intervals. For uniform motion the average velocity coincides with the instantaneous one. A trajectory is the strait line, and it always convenient to chose the X axis along the trajectory.

21. Considering the motion with a uniform velocity the displacement coincides the distance and is related to velocity and time by the formula . This formula is represented on Figure 1 by the lines 16 - 21 and 19 - 21. The coordinate of a body in the motion with the constant velocity is described by the equation x = x_o + v t, where x is an initial coordinate of the body, and v is the speed of motion. This formula is not represented by any specific line on the schematics, but it can be derived from the quantities 16, 17 and 19.

22. The average acceleration characterizes rate of a changing of an instantaneous velocity of a body. The average acceleration by a on definition is equal to

here Dt = t_final – t_initial and

23. The instantaneous acceleration is an average acceleration taken for a very small interval of time (just as an instantaneous velocity).

24. The motion with a constant accelerated – is the motion when the velocity of a body makes equal changes during equal time intervals. For the motion with a constant accelerated the instantaneous acceleration coincides with an average acceleration and is called just "acceleration".

25. A velocity and an acceleration for a body which undertakes the motion with a constant acceleration are connected by formula . This formula is represented by lines 16 - 25 and 23 - 25.

26. The linear motion is the motion for which a trajectory is a strait line. To describe the position of the body which is under the linear motion to set one coordinate (for example, х) is enough. The coordinate of a body is connected to displacement by the formula х = х_о + S (х_о is the initial coordinate, S is the displacement).

27. The linear motion with a constant acceleration is a motion which is simultaneously is both a linear and with a constant acceleration. Such a motion is observed, if an acceleration and an initial velocity of a body are parallel to each other. Hence the body always will be on a strait line paralleled to these vectors.

28. The displacement of the body with a linear motion with a constant acceleration is calculated as . The given formula is represented by the lines 16 - 28 and 23 - 28.

29. The circular motion is the motion for of which a trajectory is a circle.

30. The radius of a circle is a segment connecting the centre of the circle to the point lying on the circle.

31. An angular displacement is an angle between two radiuses drown to an initial and final positions of a body. The angular displacement is usually notated by the letter j (“phy”).

32. A uniform circular motion is a circular motion with the constant speed (but the velocity is not a constant because the direction is constantly changing!).

33. An angular speed is a physical quantity describing the rate of changing of angular displacement during the time. The angular speed is defined by the formula w = j/t, which is represented by lines 16 - 33 and 31 - 33. The angular speed also is connected to the speed by the formula v = w R, which is represented by a line 19 - 33.

34. A frequency of rotation “n” is a number of revolutions per unit of time. The frequency is connected to an angular speed by a line 33 - 34 and formula w = 2 p n.

35. A period of rotation “T” is the time taken for one full revolution. The period is connected to frequency by a line 34 - 35 (formula Т = 1/n) and with an angular velocity by a line 33 - 35 (formula w T = 2p).

36. A centripetal acceleration is a physical quantity describing a rate of changing of a direction of a velocity. In a case of uniform circular motion the centripetal acceleration is directed to centre of a circle and is given by the formula a_c = v²/R. This formula is represented by lines 19 - 36 and 30 - 36.

The indicated list rather brief and partial, but is enough for solving of majority of typical school problems on a kinematics.