CENTRIPETAL ACCELERATION: DEFINITION, FORMULAS, CALCULATION, EXERCISES - PHYSICAL - 2024

The centripetal acceleration a _c, also called radial or normal, is the acceleration that a moving object carries when it describes a circular path. Its magnitude is v ² / r, where r is the radius of the circle, it is directed towards the center of it and it is responsible for keeping the mobile on its way.

The dimensions of the centripetal acceleration are length per unit time squared. In the International System they are m / s ². If for some reason the centripetal acceleration disappears, so does the force that forces the mobile to maintain the circular path.

Rotating objects have centripetal acceleration, which is directed toward the center of the path. Source: Pixabay

This is what happens to a car attempting to corner on a flat, icy track, where the friction between the ground and the wheels is insufficient for the car to corner. Therefore the only possibility that remains is to move in a straight line and that is why it leaves 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 circumference, a direction that is perpendicular to the path 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, centripetal acceleration is responsible for keeping the mobile spinning, ensuring that the speed varies only in direction and in direction.

However, to have a Variable Circular Motion, another component of the acceleration in the same direction as the speed would be needed, which is responsible for increasing or decreasing the speed. This component of acceleration is known as tangential acceleration.

Variable circular motion and curvilinear motion in general have both components of acceleration, because curvilinear motion can be thought of as the path through innumerable circumferential arcs that make up the curved path.

The centripetal force

Now, a force is responsible for providing the 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 alter the speed of the satellite.

In such a case, gravity acts as a centripetal force, which is not a special or separate kind of force, but one that, 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 turning a curve, the role of centripetal force is played by static friction and for a stone tied to a rope that is rotated in circles, the tension in the rope is the force that forces the mobile to spin.

Formulas for centripetal acceleration

The centripetal acceleration is calculated by the expression:

ac = v ² / r

Diagram to calculate the centripetal acceleration in a mobile with MCU. Source: Source: Ilevanat

This expression will be derived below. By definition, acceleration is the change in speed over time:

The mobile uses a time Δt in the route, which is small, since the points are very close.

The figure also shows two position vectors r ₁ and r ₂, whose modulus is the same: the radius r of the circumference. The angle between the two points is Δφ. In green the arc traveled by the mobile stands out, denoted as Δl.

In the figure to the right, you see that the magnitude of Δv, the change in velocity, is roughly 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 adding the vectors that:

v ₁ + Δ v = v ₂ → Δ v = v ₂ - v ₁

Δ v is interesting because it is proportional to the centripetal acceleration. From the figure it can be seen that being the angle Δφ small, the vector Δ v is essentially perpendicular to both v ₁ and v ₂ and points to the center of the circle.

Although up to now 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, dispensing with the vector notation.

Something else: you need to make use of the definition of central angle, which is:

Δ φ = Δ l / r

Now both figures are compared, which are proportional since the angle Δ φ is common:

Dividing by Δt:

a _c = v ² / r

Exercise resolved

A particle moves in a circle with radius 2.70 m. At a certain moment its acceleration is 1.05 m / s ² in a direction that makes an angle of 32.0º with the direction of movement. Calculate your speed:

a) At that time

b) 2.00 seconds later, assuming constant tangential acceleration.

Reply

It is a varied circular movement, since the statement indicates that the acceleration has a given angle with the direction of the movement that 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 a _c and _t and are drawn in the following figure. The vector in green is the net acceleration vector or simply acceleration a.

A particle moves in a circular path in a counterclockwise direction and varied circular motion. Source: commons.wikimedia.org

a) Calculation of the acceleration components

a _c = a.cos θ = 1.05 m / s ². cos 32.0º = 0.89 m / s ² (in red)

a _t = a. sin θ = 1.05 m / s ². sin 32.0º = 0.57 m / s ² (in orange)

Calculation of the speed of the mobile

Since a _c = v ² / r, then:

v = v _or + a _t. t = 1.6 m / s + (0.57 x 2) m / s = 2.74 m / s

References

1. Giancoli, D. Physics. 2006. Principles with Applications. Sixth Edition. Prentice Hall. 107-108.
2. Hewitt, Paul. 2012. Conceptual Physical Science. Fifth Edition.Pearson.106 - 108.