Modern Physics

# Centripetal acceleration: definition, formulas, calculations, exercises

The acceleration centripetal the c , also called radial or normal, is the acceleration of a moving object loads to describe a circular path. Its magnitude is 2 / r , where r is the radius of the circle, it is directed towards the center of the circle and is responsible for keeping the mobile in its path.

The dimensions of centripetal acceleration are the length per unit of 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.

That’s what happens to a car that tries to turn on a flat, icy road, in which the friction between the tread and the wheels is insufficient for the car to turn. So the only possibility left is to move in a straight line and that’s why it comes out of the curve.

## Circular movements

When an object moves in a circle, centripetal acceleration is always directed radially towards the center of the circle, perpendicular to the path followed.

Since velocity is always tangent to the trajectory, velocity and centripetal acceleration are perpendicular. Therefore, velocity and acceleration do not always have the same direction.

In these circumstances, the cell phone has the possibility to describe the circle with constant or variable speed. The first case is known as Uniform Circular Movement or MCU by its acronym, the second case will be a Variable Circular Movement.

In both cases, centripetal acceleration is responsible for keeping the rover rotating, making sure that the speed only varies in direction and direction.

However, to have a variable circular motion, another component of acceleration in the same direction as velocity would be needed, responsible for increasing or decreasing velocity. 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 the countless arcs of circumference that make up the curved path.

## Centripetal force

Now a force is responsible for providing acceleration. For a satellite orbiting Earth, it’s the force of gravity. And since gravity always acts perpendicular to the trajectory, it doesn’t change the satellite’s speed.

In this case, gravity acts as a centripetal force , which is not a special or separate class of force, but 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 that turns, the role of centripetal force is interpreted 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 rotate.

## Formulas for Centripetal Acceleration

Centripetal acceleration is calculated by the expression:

ac = 2 / r

Diagram for calculating centripetal acceleration on a mobile phone with MCU. Source: Source: Ilevanat [CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0)]

This expression will be deduced below. By definition, acceleration is the change in velocity over time: Mobile employs a time Δ t on the path, which is small because the points are very close.

The figure also shows two position vectors of 1 and 2 , whose order of magnitude is the same: the radius r of the circle. The angle between the two points is Δφ. Green highlights the arc traversed by the cell phone, denoted Δl.

In the figure on the right is that the magnitude of Δ v , the change in velocity, is approximately proportional to .DELTA.L, since the angle Δφ is small. But the change in velocity is precisely related to acceleration. The triangle shows, by the sum of the vectors that:

1 + Δ v = 2 → Δ v = v 2 – 1

Δ v is interesting since it is proportional to centripetal acceleration. From the figure, note that the angle Δφ is small, the vector Δ v is essentially perpendicular to 1 and 2 and points to the center of the circle.

Related:   Vector Magnitude: What It Is and Examples

Although so far the vectors are highlighted in bold, due to the effects of the geometric nature below, we work with the modules or magnitudes of these vectors, regardless of the vector notation.

Another thing: you need to use the center angle definition, which is:

Δ φ = Δ l / r

Now the two numbers are compared, which are proportional as the angle Δ φ is common: Dividing by Δt: c = v 2 / r

## Exercise solved

A particle moves in a circle with a radius of 2.70 m. At any given time, its acceleration is 1.05 m / s 2 in a direction that makes an angle of 32.0° with the direction of motion. Calculate your speed:

a) At that moment

b) 2.00 seconds later, assuming constant tangential acceleration.

### Response

It is a varied circular motion, as the statement indicates that the acceleration has a certain angle with the direction of motion that is neither 0° (it could not be a circular motion) nor 90° (it would be a uniform circular motion).

Therefore, the two components – radial and tangential – coexist. They are designated by c and t and appear drawn in the figure below. The green vector is the net acceleration vector or simply the acceleration a.

#### a) Calculation of acceleration components

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

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

##### Mobile speed calculation

As a c = 2 / r , then: v = v or + a t . t = 1.6 m / s + (0.57 x 2) m / s = 2.74 m / s