Centripetal force

The centripetal force (including centrifugal force ) is the component of the external force to the center of the circle of curvature, which has to act on a body so that it moves in a curved path in the inertial system.

Without this force, the body would move according to the law of inertia uniform in the direction of the instantaneous velocity vector ( the tangent vector of the path ), as observed for example in sparks that come off of a grinding wheel. The movement on a predetermined path, such as roller coasters, requires centripetal acceleration ( acceleration and radial ), from which the centripetal force can be calculated by multiplying by the mass.

The centripetal force is perpendicular to the velocity vector in the inertial frame. It thus differs from the centrifugal force, which must be taken into account only when describing the motion in an accelerated frame of reference.

Etymology and history of the concept

The term centripetal force is derived ( Latin for according to strive, go ) of petere from. He was introduced as vis centripeta of Isaac Newton. Newton, however, did not use the term in the modern sense, but in terms of an attractive central force. Coined the name ' Newton as opposed to the previously introduced by Christian Huygens centrifugal force.

Centripetal force and central force

While a central force is always directed to the same point, showing the centripetal force to the center of the current circle of curvature. Only a pure circular motion, the centripetal force is a central force. In the general case, ie for example in an elliptical planetary orbit, which decays directed at the focal point of a central force in the centripetal force is directed toward the center of curvature at this location, and a tangential component. The tangential component increases or decreases the speed of the planet and makes sure that he is near the Sun moves faster than in aphelion. In the adjacent figure, the decomposition of the external force are shown in centripetal and tangential component of the example of a point on the circumference of a wheel.

Examples

• When a car passes through a curve, it is only possible that a centripetal force is directed towards the inside of the curve act. It results from the sum of the lateral forces which arise between the tire and the road surface and acting on the vehicle. If this force ( eg ice ), so the car moves on a straight line, is thus carried out of the curve. The vehicle occupant moves on the same orbit as the car because the seat exerts a centripetal force on him.
• The earth moves ( approximately ) on a circular orbit around the sun. This circular movement is caused by the force exerted by the sun to the earth gravitational force that serves as a centripetal force. Strictly speaking, the Earth's orbit as the orbits of all the planets do not orbit, but an elliptical orbit. Shows the gravitational force on the sun as the center, which is located at one of the ellipse foci. This central force, however, differs slightly from the centripetal force from showing to the center of the local curvature of the path. The difference between central force and centripetal force is a tangential component, which ensures that the planet is near the Sun ( perihelion ) moves faster than in aphelion.
• Electrons move at right angles to a uniform magnetic field, so they are deflected by the Lorentz force perpendicular to the direction of movement and the magnetic field in a circular path. In this example, the Lorentz force is the centripetal force.
• When vortices is the centripetal force, the pressure gradient, ie in the vortex core low pressure area.

Mathematical derivation

If an object moves at a constant speed along a circular path, the speed at each moment perpendicular to the radius of the circle. The drawing illustrates these relationships for the times and

First, the correlations can be regarded purely geometrical: the arrow shown in blue in the sketch is created by parallel displacement of the arrow Their lengths are the same as those of the arrow thus applies to the lengths of the three arrows:

It follows from the similarity of triangles and thus

Or, after multiplication by

Division by the time results

Is now chosen sufficiently small, then:

• The distance traveled by the object path corresponds to a section on the circular path. Thus, the speed is the path velocity of the object.
• The acceleration is centripetal acceleration in the direction of the circle center point experienced by the object.

Therefore, the equation for small is to

Or

If the object is not only a point, but also has a mass as can be according to Newton 's laws of the magnitude of the centripetal force determine:

This centripetal force acting on each body with the mass of moving with the speed on a circular path with the radius.

Mass rotates at the angular speed around a stationary center, the web speed may be replaced by. It follows

And

Vectoring

The acceleration of a point which moves on an arbitrary track curve, the second derivative of the position vector from the origin of the inertial system to the point P with respect to time:

In general, the trajectory in the form of parameters in dependence on the path is given s. The time derivative can then be expressed by derivatives with respect to the path:

The centripetal acceleration to the local center of curvature is the first term of the equation:

Using the Frenet formulas can be the second derivative of the trajectory after it passes through the principal normal vector and the curvature radius express:

Thus one obtains the known relationship that the centripetal acceleration is proportional to the square of the speed and inversely proportional to the radius of the path:

In the special case of a purely circular movement, the vectors can be used for the distance and the angular velocity. Thus, the centripetal acceleration can be represented as a vector product:

General: