ec the point B. 7.3 Bone mineral density is often measured, related to the stren
ID: 2074783 • Letter: E
Question
ec the point B. 7.3 Bone mineral density is often measured, related to the strength and stiffness of bone. The measurement can also be used as a diagnostic tool for osteoporosis. Consider a rectangular piece in the xy-plane of a Cartesian xyz-coordinate system, with varying density = (x,y). For the corner points ABCD of the specimen the den. sity is given (expressed in [kg dm-3), see figure. Prove, that it is impossible for the gradient of the density field in the figure to be constant. Po- 1.02 1.09 PA-1.00 Pe- 1.05
Explanation / Answer
Gradient of any scaler quantity is given by the change of that scaler quantity along the coordinate directions (in our case those are x and y coordinates) . if gradient of any scaler quantity is constant that means that quantitiy is independent of the coordinates directions. That means value of the scaler quantity along the directions will remain same or change in the value of the quatity between two points will remain same for the whole span (x-y plan) because increase or decrease in value is constant.
in our case scaler quantitiy is density and its value is given at 4 points.
In X-direction, difference between density of points A - B and points C - D is 0.05 and 0.07 respectively. Here, diffrence is not same so increase in the value of density in the x-direction is not same for two values of y coordinate. so it is not constant in x direction.
In Y-direction, difference between density of points A - D and points B - C is 0.02 and 0.04 repectively. Here, diffrence is not same so increase inthe value of density in the y-direction is same for two values of x coordinate. so it is not constant in y direction.
So, it is not possible for the gradient of this density function to be constant over the span.
feel free to drop comment if you have any doubt.
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