por What is centripetal acceleration?
The centripetal acceleration ac, also called radial or normal, is the acceleration that a moving object has when it describes a circular path. Its magnitude is v2/rwhere r It is the radius of the circle, it is directed towards the center of the circle and it is responsible for keeping the mobile in its path.
The dimensions of centripetal acceleration are length per unit time squared. In the International System they are m/s2. If for some reason the centripetal acceleration disappears, so does the force that forces the mobile to maintain the circular path.
This is what happens to a car that tries to corner on a flat and icy track, where the friction between the ground and the wheels is insufficient for the car to take the curve. Therefore, the only possibility left to him is to move in a straight line and that is why he goes out of the curve.
circular movements
When an object moves in a circle, at all times the centripetal acceleration is directed radially towards the center of the circle, a direction that is perpendicular to the trajectory followed.
Since velocity is always tangent to the path, then velocity and centripetal acceleration turn out to be perpendicular. Therefore, velocity and acceleration do not always have the same direction.
Under these circumstances, the mobile has the possibility of describing the circumference with constant or variable speed. The first case is known as Uniform Circular Movement, or MCU for its acronym, the second case will be a Variable Circular Movement.
In both cases, the centripetal acceleration is in charge of keeping the mobile spinning, taking care that the speed varies only in direction and direction.
However, to have a Variable Circular Movement, another acceleration component would be needed in the same direction as the speed, which is responsible for increasing or decreasing the speed. This component of acceleration is known as tangential acceleration.
The variable circular movement and the curvilinear movement, in general, have both components of acceleration, because the curvilinear movement can be imagined as the route through innumerable arcs of circumference that make up the curved trajectory.
centripetal force
Now, a force is responsible for providing acceleration. For a satellite orbiting the Earth, it is the force of gravity. And since gravity always acts perpendicular to the trajectory, it does not change the speed of the satellite.
In such a case, gravity acts as a force. centripetal forcewhich is not a special or separate kind of force, but one which, in the case of the satellite, is directed radially towards the center of the Earth.
In other types of circular motion, for example a car taking a turn, the role of the centripetal force is played by static friction, and for a stone attached to a string that is spun in a circle, the tension in the string It is the force that forces the mobile to turn.
Formulas for Centripetal Acceleration
The centripetal acceleration is calculated by the expression:
ac = v2/r
This expression will be derived below. By definition, acceleration is the change in velocity over time:
The diagram above shows in the left figure, two points through which a mobile passes on a circle of radius r counterclockwise. Note that the magnitude of the velocity is the same in both cases, but not the direction or sense.
The mobile spends a time Δyou on the tour, which is small, since the points are very close.
The figure also shows two position vectors r1 and r2whose module is the same: the radius r of the circumference. The angle between both points is Δφ. In green highlights the bow traversed by the mobile, denoted as Δl.
In the figure on the right it is seen that the magnitude of Δv, the change in velocity, is approximately proportional to Δl, since the angle Δφ is small. But the change in velocity is precisely related to acceleration. From the triangle it can be seen, by addition of vectors, that:
v1 + Δv = v2 → Δv = v2 – v1
Δv is interesting, because it is proportional to the centripetal acceleration. From the figure it can be seen that the angle Δφ being small, the vector Δv is essentially perpendicular to both v1 like a v2 and points to the center of the circle.
Although up to here the vectors are highlighted in bold, for the effects of a geometric nature that follow, we work with the modules or magnitudes of these vectors, disregarding the vector notation.
Something else: you need to make use of the definition of central angle, which is:
Δφ= Δl/r
Now both figures are compared, which are proportional, since the angle Δφ it is common:
Dividing by Δt:
Because by the definition of speed it is v = Δl/ Δt. But Δv/ Δyou it is precisely the magnitude of the centripetal acceleration, which is what is sought. In this way, we arrive at the expression described at the beginning:
ac=v2/r
solved exercise
A particle moves in a circle of radius 2.70 m. At a certain moment its acceleration is 1.05 m/s2 in a direction that makes an angle of 32.0º with the direction of motion. Calculate your speed:
a) At that time
b) 2.00 seconds later, assuming constant tangential acceleration.
Answer
It is a varied circular movement, since the statement indicates that the acceleration has a given angle with the direction of the movement, which is neither 0º (it could not be a circular movement) nor 90º (it would be a uniform circular movement).
Therefore, the two components—radial and tangential—coexist. They will be denoted as ac and at and are drawn in the following figure. The vector in green is the net acceleration vector or simply acceleration to.
a) Calculation of the acceleration components
ac = a.cos θ = 1.05 m/s2 . cos 32.0º = 0.89 m/s2 (in red)
at = a.sin θ = 1.05 m/s2 . sin 32.0º = 0.57 m/s2 (in orange)
Calculation of the speed of the mobile
Since ac = v2/rso:
b) 2 seconds later the speed v will be increased by the tangential component of acceleration
v = vo + at. t = 1.6 m/s + (0.57 x 2) m/s = 2.74 m/s
References
Giancoli, D. Physics. Principles with Applications. Sixth Edition. Prentice Hall.
Hewitt, Paul. Conceptual Physical Science. fifth edition. pearson.