Downhill force

Inclined plane with the angle of inclination α.
Red is the weight and its breakdown into the components. They are (if the body remains at rest) neutralized by the contact forces between the body and the surface, which are shown in green at their point of application.

The downhill force is the component of the weight that is directed downhill on an inclined plane .

The weight of a body is broken down into the downhill force parallel to the inclined plane and into a component perpendicular to it. ${\ displaystyle F _ {\ text {G}}}$ ${\ displaystyle F _ {\ text {GH}}}$ ${\ displaystyle F _ {\ text {GN}}}$

{\ displaystyle F _ {\ text {GH}} = F _ {\ text {G}} \ cdot \ sin \ alpha = m \, g \ sin \ alpha}

{\ displaystyle F _ {\ text {GN}} = F _ {\ text {G}} \ cdot \ cos \ alpha = m \, g \ cos \ alpha}

The downhill force increases with the inclination angle of the plane and is at a maximum at 90 °, namely equal to the weight of the body. The normal force component, on the other hand, is maximum at 0 ° and decreases as the angle of inclination increases. ${\ displaystyle \ alpha}$

The block remains at rest as long as the downhill force counteracts an equally large static friction force . If the plate is set too steeply, the block slips downwards when the downhill force is greater than the maximum static friction force . Here is the dimensionless coefficient of static friction . The normal force on the pad resulting from the equilibrium of forces perpendicular to the inclined plane: . ${\ displaystyle F _ {\ text {GH}}}$ ${\ displaystyle F _ {\ text {R}}}$ ${\ displaystyle F _ {\ text {R, max}} = \ mu _ {H} \ cdot F _ {\ text {N}}}$ ${\ displaystyle \ mu _ {H}}$ ${\ displaystyle F _ {\ text {N}}}$ ${\ displaystyle F _ {\ text {N}} = F _ {\ text {GN}}}$

This results in the maximum angle up to which sticking is possible:

{\ displaystyle \ tan \ alpha _ {\ text {max}} = \ mu _ {H}}

.

example

A vehicle traveling downhill is accelerated by the downhill force. At the same time, the normal force and thus the grip on the ground decrease with increasing gradient. Assuming the usual static friction between the vehicle and the road, the braking distance is increased for both of the reasons mentioned.

Individual evidence

↑ H. Steger: Mechanical engineering for electrical engineers, part 2 . Teubner, 1991, ISBN 978-3-519-06735-1 , pp. 69 . ( limited preview in Google Book search)

[1] H. Steger: Mechanical engineering for electrical engineers, part 2 . Teubner, 1991, ISBN 978-3-519-06735-1 , pp. 69 . ( limited preview in Google Book search)