The plumb-line of a weight resting on a slope
How a sphere's contact angle governs the weight it bears on an oblique surface
The page defines how the pendant, or plumb-line, of a heavy body behaves when the body rests on a slope. Diagrams of a sphere on inclined planes show that a vertical pendant a e carries the whole weight e, while the oblique pendants b e, c e and d e carry progressively less, the missing weight discharging onto the slope d f n. Two successive definitions state the rule geometrically: when the line through the pendant to the sphere's centre meets the semidiameter at the point of contact at a right angle, the slope feels the full natural weight. An acute angle (as at a c d) adds accidental weight, while an obtuse angle annuls the accidental weight and lightens the load on the slope.
On this page
Vertical versus oblique pendants
The vertical pendant a e sustains all of the weight e, which the pendant b e cannot do, and still less c e and d e. Whatever weight is lacking to each of these oblique pendants discharges onto the obliquity d f n.
Pendant parallel to the slope gives a right-angle contact
When the line of the weight's pendant is parallel to the line of the slope, the contact of the spherical body is right-angled: the semidiameter of the sphere meets the line of the slope at a right angle.
Right, acute and obtuse contact angles
When the line from the pendant to the sphere's centre is at a right angle to the semidiameter reaching the point of contact, the slope feels the full natural weight. If the angle becomes acute, as a c d, the slope takes part natural and part accidental weight; if obtuse, the accidental weight is annulled and the natural weight is lightened.