The most simple case of curvilinear movement is the uniform circular motion, in which a point P is moving with constant speed on a trajectory given by a circle of radius R and center O . In this case, the time interval T , used by point P to make a complete turn on the circumference, is called the period of motion. If in the unit of time (1 second), the point P completes f turns of circumference, the period used for each single turn will be equal to 1 / f seconds; the number fit is called the frequency of motion and is measured in revolutions per second (rev / s) (the unit of measurement of frequency in the International System is the hertz, symbol Hz, where 1 Hz = 1 s -1 ). The period and the frequency in a uniform circular motion are linked by the relationship:
The speed in uniform circular motion
Known the period T of a uniform circular motion, its speed can be easily deduced by remembering that, by definition, the point P completes, in a time interval equal to a period, exactly one turn of circumference, thus covering a space equal to 2R . The relationship between space traveled and time spent therefore leads to a constant value given by:
Or, substituting the frequency f for period T :
Once the intensity value has been established, the velocity as a vector quantity is fully defined by assigning the direction of the tangent to the circumference at point P and the direction of motion as direction (see fig. 4.1). The velocity in uniform circular motion can also be expressed in terms of angular velocity, which represents the displacement of the anglefollowing the motion of the point P on the circumference. The relationship between the angular velocityand the velocity of the point P is given by:
Acceleration in uniform circular motion
In uniform circular motion, while the modulus of the velocity vector remains constant, its direction changes continuously, since, as the point moves along the circumference, the position of the tangent to the curve changes continuously. It is therefore possible to express this variation by introducing an acceleration vector. In this case the tangential component of the acceleration is zero, while the centripetal component can be determined. To calculate the acceleration, it may be useful to use a graphic method, illustrated starting from figure 4.2; on a circumference of radius R , the velocity vectors v are displayed1, v2, v3and v4, relating to the various positions P1, P2, P3and P4of the point P in four instants of time t1, t2, t3and t4. Now imagine to transport all the speed vectors parallel to themselves until their origins coincide in a single point (see fig. 4.2 B); their opposite ends will therefore draw a circumference of radius equal to the modulus v of the speed (this circumference does not coincide with the original trajectory of the motion). The velocity vector of the arrow moves on this new circumference so as to make a full turn in a period T equal to that of the motion P . A whole turn of the new circumference thus equates to a space:
(being the modulus of v equal to the radius).
By applying the usual relationship between space and time in a uniform circular motion, we obtain the velocity of the velocity vector of P , i.e. the value of its acceleration, at :
By substituting its value v = (2R ) / T at speed , we obtain:
The acceleration vector (or velocity of the velocity) will then, again according to what has been said for uniform circular motions, direction tangent to the velocity trajectory curve, i.e. direction perpendicular to the radius of the circumference built with the velocity vectors and, therefore, to the vector speed v . But, returning to the original circumference (see fig. 4.2 A), v has a direction perpendicular to the radius R , and therefore the modulus vector a will have the direction of the perpendicular to the perpendicular to the radius, i.e. the direction of the radius itself.
The direction of a , as shown in figure 4.3, will point to the center of the circumference; for this reason, the acceleration thus constructed is called centripetal.