Curvilinear motion : A particle or body performs a curvilinear motion, when that particle describes a path that is not straight.
In nature, as well as in technique, it is very common to find movements whose trajectories are not straight lines, but curves. These movements are called curvilinear, and are found more often than rectilinear. Planets, satellites move along curved paths in cosmic space, and on Earth, all means of transport, parts of machines, water from rivers, air from the atmosphere move.
During this movement it cannot be said that only one coordinate of the body varies. For example, movement occurs in the plane, so as seen in Figure 1, two coordinates vary during movement: X and Y. The direction of movement, that is, the direction of the velocity vector varies throughout the duration the movement. In addition, the direction of the acceleration vector varies.
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- 1 Vector position at an instant t.
- 2 Speed
- 1 Instantaneous speed
- 2 Speed vector
- 3 Acceleration
- 1 Vector acceleration
- 4 Source
Vector position in an instant t.
How the position of the mobile changes over time. At time t the mobile is at point P, or in other words, its position vector is ṝ and at time t ‘it is at point P’, its position is given by vector ṝ ‘.
We will say that the mobile has moved Δṝ = ṝ’-ṝ in the time interval Dt = t’-t. Said vector has the direction of the secant that joins the points P and P ‘.
The velocity of the body at any point on a curvilinear path is tangentially directed at the path at this point and its direction coincides with that of the body’s movement at this point.
The speed at a point of a curvilinear movement is really tangential, if we observe the work on a grinding stone, when pressing on it the end of a steel knife, small particles will jump from it in the form of a spark.
These particles will fly with a tangential speed at the moment of jumping from the stone, it is very well seen that the direction of the sparks flight always agree with the tangent to the circumference at this point, and that the direction of their movement coincides with the from the periphery of the stone. Likewise the rubbers of a bicycle when rolling splash the mud in a direction tangent to the circumference (wheels), the brake guards prevent the rider from splashing.
The instantaneous velocity of the body at different points on the curvilinear path has different directions. The modulus of this speed can be the same at all points on the path or vary from point to point, from one instant to the next.
The mean velocity vector is defined as the quotient between the displacement vector Δṝ between the time Dt has traveled.
The mean velocity vector has the same direction as the displacement vector, the secant that joins points P and P ‘in the figure.
The speed vector in an instant is the limit of the average speed vector when the time interval tends to zero.
As we can see in the figure, as we make the time interval tend to zero, the direction of the mean velocity vector, the secant line that successively joins the points P, with the points P1, P2 ….., tends towards the tangent to the path at point P.
At time t, the mobile is at P and has a velocity whose direction is tangent to the trajectory at that point.
During the curvilinear movement of the point body, the direction of velocity varies constantly, although its modulus may or may not vary. But even if it does not change modularly, it can never be considered constant, since velocity is a vector quantity and for vector quantities, modulus, direction and direction are equally important. So curvilinear motion is accelerated motion.
During rectilinear motion with constant acceleration it is just the modulus of the speed that varies. Furthermore, it is known that in this case the acceleration vector is directed in the direction of the velocity vector or opposite to it and that the modulus of acceleration is determined by the variation of the modulus of speed in the unit of time. Thus also the curvilinear movement can be observed, in which the modulus of the velocity remains constant, but: How to only relate the acceleration with the variation of the direction of the velocity vector? How will it be directed since the acceleration will be the same? It is clear that the modulus, direction, and direction of acceleration are related to the shape of the curvilinear path, but each of the myriad shapes of curved paths cannot be observed.
Movement along any curvilinear path can be considered as movement by arcs of circles of different radii.
Therefore, the problem of finding the acceleration of a curvilinear motion lies in finding the acceleration during the uniform movement of the body by a circumference.
At time t the mobile is at P and has a speed → v whose direction is tangent to the trajectory at that point.
At time t ‘the mobile is at point P’ and has a speed → v ‘.
The mobile has changed, in general, its speed both in module and in direction, in the amount given by the difference vector Δ → v = → v ‘- → v.
Average acceleration is defined as the quotient between the velocity change vector and the time interval Dt = t’-t, in which said change takes place.
And acceleration in an instant