Altitude As Gravitational Potential Energy
An example is the (nearly) uniform gravitational field near the Earth's surface. It has a potential energy
where U is the gravitational potential energy and h is the height above the surface. This means that gravitational potential energy on a contour map is proportional to altitude. On a contour map, the two-dimensional negative gradient of the altitude is a two-dimensional vector field, whose vectors are always perpendicular to the contours and also perpendicular to the direction of gravity. But on the hilly region represented by the contour map, the three-dimensional negative gradient of U always points straight downwards in the direction of gravity; F. However, a ball rolling down a hill cannot move directly downwards due to the normal force of the hill's surface, which cancels out the component of gravity perpendicular to the hill's surface. The component of gravity that remains to move the ball is parallel to the surface:
where θ is the angle of inclination, and the component of FS perpendicular to gravity is
This force FP, parallel to the ground, is greatest when θ is 45 degrees.
Let Δh be the uniform interval of altitude between contours on the contour map, and let Δx be the distance between two contours. Then
so that
However, on a contour map, the gradient is inversely proportional to Δx, which is not similar to force FP: altitude on a contour map is not exactly a two-dimensional potential field. The magnitudes of forces are different, but the directions of the forces are the same on a contour map as well as on the hilly region of the Earth's surface represented by the contour map.
Read more about this topic: Scalar Potential
Famous quotes containing the words altitude, potential and/or energy:
“On a level plain, simple mounds look like hills; and the insipid flatness of our present bourgeoisie is to be measured by the altitude of its “great intellects.””
—Karl Marx (1818–1883)
“Not many appreciate the ultimate power and potential usefulness of basic knowledge accumulated by obscure, unseen investigators who, in a lifetime of intensive study, may never see any practical use for their findings but who go on seeking answers to the unknown without thought of financial or practical gain.”
—Eugenie Clark (b. 1922)
“After the planet becomes theirs, many millions of years will have to pass before a beetle particularly loved by God, at the end of its calculations will find written on a sheet of paper in letters of fire that energy is equal to the mass multiplied by the square of the velocity of light. The new kings of the world will live tranquilly for a long time, confining themselves to devouring each other and being parasites among each other on a cottage industry scale.”
—Primo Levi (1919–1987)