Centripetal acceleration when moving around a circle: concept and formulas. Centrifugal and centripetal forces

In studying motion in physics, the concept of trajectory plays an important role. It is she who determines in many ways the type of movement of objects and, as a consequence, the form of formulas with which this movement is described. One of the common trajectories of movement is a circle. In this article, we consider what centripetal acceleration is when moving around a circle.

The concept of full acceleration

Before characterizing centripetal acceleration when moving around a circle, we consider the concept of full acceleration. Under it is supposed a physical quantity that simultaneously describes the change in the absolute value and the velocity vector. In mathematical form, this definition looks like this:

a¯ = dv¯ / dt

Acceleration is the total time derivative of speed.

As is known, the velocity v¯ of a body at each point of the trajectory is directed along the tangent. This fact allows us to represent it in the form of the product of the module v by the unit tangent vector u¯, that is:

v¯ = v * u¯

Then the full acceleration can be calculated as follows:

a¯ = d (v * u¯) / dt = dv / dt * u¯ + v * du¯ / dt

The quantity a¯ is the vector sum of two terms. The first term is directed along the tangent (as the speed of the body) and is called tangential acceleration. It determines the rate of change of the speed modulus. The second term is normal acceleration. Consider it in more detail later in the article.

Normal component acceleration

We write the expression obtained above for the normal acceleration component a _n ¯ in explicit form:

a _n ¯ = v * du¯ / dt = v * du¯ / dl * dl / dt = v ² / r * r _e ¯

Here dl is the path traveled by the body along the trajectory in time dt, r _e ¯ is the unit vector directed to the center of curvature of the trajectory, r is the radius of curvature. The resulting formula leads to several important features of the component a _n ¯ full acceleration:

• The value of a _n ¯ grows as the square of the velocity and decreases inversely with the radius, which distinguishes it from the tangential component. The latter is not equal to zero only in the case of a change in the velocity modulus.
• Normal acceleration is always directed towards the center of curvature, therefore it is called centripetal.

Thus, the main condition for the existence of a nonzero quantity a _n ¯ is the curvature of the trajectory. If such curvature does not exist (rectilinear displacement), then a _n ¯ = 0, since r-> ∞.

Centripetal acceleration in circular motion

A circle is a geometric line, all points of which are at the same distance from some point. The latter is called the center of the circle, and the distance mentioned is its radius. If the speed of the body during the rotation does not change in absolute value, then they speak of uniformly alternating motion around the circle. Centripetal acceleration in this case can be easily calculated using one of the two formulas below:

a _n = v ² / r;

a _n = ω ² * r

Where ω is the angular velocity, measured in radians per second (rad / s). The second equality is obtained thanks to the formula for the relationship between the angular and linear velocities:

v = ω * r

Centripetal and centrifugal forces

When the body moves uniformly around the circumference, centripetal acceleration arises due to the action of the corresponding centripetal force. Its vector is always directed towards the center of the circle.

The nature of this power can be very diverse. For example, when a person untwists a stone attached to a rope, then the tension of the rope holds him on his trajectory. Another example of the action of a centripetal force is the gravitational interaction between the Sun and planets. It is it that makes all planets and asteroids move in circular orbits. The centripetal force is not able to change the kinetic energy of the body, since it is directed perpendicularly to its speed.

Each person could pay attention to the fact that while turning the car, for example, to the left, passengers are pressed to the right edge of the vehicle interior. This process is the result of the centrifugal force of the rotational motion. In fact, this force is not real, because it is due to the inertial properties of the body and its desire to move along a straight path.

Centrifugal and centripetal forces are equal in magnitude and opposite in direction. If this were not the case, then the circular path of the body would be disturbed. If we take into account Newton’s second law, then it can be argued that during rotational motion, centrifugal acceleration is equal to centripetal.