Inclined plane

An oblique, oblique or inclined plane ( short respectively colloquially slope, skewness, slant or inclination ) is in the mechanics of a flat surface which is inclined to the horizontal. It is used to reduce the force required to change a height of the mass. However, the workload remains unchanged. The inclined plane is one such as the pulley and the screw to the simple machines.

In an inclined plane with an inclination angle of 45 ° (corresponding to an increase of 100 %), the range extends to lift a weight of for example, 10 m in the perpendicular to about 14.1 m along the inclined plane, which is the force reduced to 71 % (ignoring friction). If the inclination angle of 22.5 ° ( equal to a slope of 41.5%) halved the distance increases to approximately 22 m, the force is reduced to approximately 45% compared to direct lifting.

Everyday life

Applications of this principle are found for example in switchbacks in the mountains, ramps that were used in antiquity for the construction of buildings, bicycle or wheelchair ramps, etc. screws can also be viewed as a cylinder with a wound inclined plane.

The tool wedge also uses the principles of the inclined plane.

Physical Basics

In the following, the situation will be described in a quiescent mass balance on an inclined plane.

The gravitational force of a mass which is located on an inclined plane, has its point of focus in the crowd. It is disassembled to describe the problem in two components, the downgrade component of the weight force parallel to the surface of the inclined plane and the normal component of the force of gravity perpendicular to the surface. It is strictly distinguish between the real forces acting and the decomposition of the gravitational force into two components - the components are no forces acting. The normal force which acts from the bottom of the mass, is the contact force and stands perpendicular to the plane. Its point is not the center of gravity of the contact area, because the pressure is not constant. The magnitude of the normal force is equal to the sum of the normal component of the weight of a further force, which acts, the friction force also it is a contact force and acts on the center of gravity of the contact surface - however, is parallel to the plane and opposite to the direction of the slope output component of the weight force

Thus, the body remains at rest, the slope force must not be greater than the maximum static friction force latter is given by the coefficient of static friction and the magnitude of the normal force. The following applies:

If this condition is not met ( for example, because the inclination angle of the plane is too large or too small coefficient of static friction ), the mass begins to slip.

Does the mass of a speed or even act more forces, additional considerations and case distinctions must be made, which are not described here. The detailed mathematical description of the mass resting on an inclined plane is recorded in the next section.

Body is at rest

The following designations are used:

The weight can be divided into a component perpendicular to the inclined plane ( normal component ) and a component parallel to the inclined plane ( downgrade component).

At the contact surface between the body and an inclined plane, a normal force and a friction force acting

Since the body is at rest, the static friction force must just be the same size or larger than the slope descending component of the weight force:

With the static friction law:

Arises as a necessary condition:

When the inclination angle is too large or the friction coefficient is too small, no equilibrium is possible slipping of the body.

The coefficient of static friction (sometimes referred to as ) is greater than the coefficient of sliding friction in each case

Note that:

Is called.

Motion with air resistance

Hereinafter, the drag force is to be considered during the movement of the body on the inclined plane. In contrast to the above section of the body is no longer at rest. Effect is the air resistance and friction. The constant of the shape of the body and the density of the flowing medium ( for example: air) dependent. The following applies:

Where:

From the force approaches arise quite complex equations of motion - but these differential equations are solvable.

Downward movement

From the force approach:

Follows the differential equation:

With:

The following cases can be distinguished:

Approach:

Substituting into the differential equation is obtained taking into account:

And by comparing coefficients:

And

As a solution, we have:

Taking into account one obtains:

At the time the body comes to rest.

For the braking distance of the following applies:

The speed approaches although hyperbolically the rest, the braking distance is, however, infinitely long.

Upward movement

From the force approach:

Follows the differential equation:

With:

Approach:

Substituting into the differential equation is obtained taking into account:

And by comparing coefficients we obtain:

As a solution, we have:

At the time:

The body comes to rest, being negative.

For the braking distance of the following applies: