In an obscure location in the South Pacific an explorer in a

In an obscure location in the South Pacific an explorer in a small sail boat claims to have discovered a tiny volcanic island she calls the Isle of Abstractus. The volcano is shiny and seems to be made of obsidian which is a glass like substance. The map she made has 100 meter contour lines. The rim of the volcano crater happens to be one of the contour lines and the explorer estimates that the island is 250 m long from west to east and 100 meters wide from North to South. (e) Where is the steepest slope on the mountain (to the N, W, S, or E)? (F) Which direction (N, W, S, or E) is a ball most likely to roll in if it is released from the rim at point B but doesn\'t fall into the crater? (g) In what direction is the component of the gravitational force parallel to the surface of the volcano the greatest?

Solution

Contour lines are the lines joining points of same elevation on a geographical map. We will use this understanding to solve the given problems as follows:

e.) As the contour lines we have are of 100 metres, it implies that from one contour line to the other, the vertical distance covered is 100 metres. Now, as we can observe, the contour lines are closest on the west side of the island, as in, on the left of the point B in the given map. That would mean that there is 100 metres of elevation change between each of the lines over a very small horizontal distance, which in turn means that the slope is maximum there. Hence the slope is steepest to the West(W).

f.) As the rim itself is one of the contour lines, it is at an elevation greater than the next contour line, Plus, the surface would be sloping downwards towards the west there. Therefore the ball will roll towards West(W).

g.) Again, for any slope at angle, say , with the horizontal, the component of the gravitation force parallel to the surface is given as MgSin

Therefore the component is maximum for maximum or minimum vertical angle. That is for the maximum slope. Hence the magnitude of the component of the gravitational force would be maximum for the West direction.