Final answer:
The range of the ratio pi/po for which the magnitude of ce is greater at r=a and r=b in a thick-walled cylinder is b^2/a^2 < pi/po and a < b.
Step-by-step explanation:
In a thick-walled cylinder, the magnitude of ce represents the radial stress at a specific radius. The radial stress is caused by the internal and external pressures acting on the cylinder. The ratio of internal pressure (pi) to external pressure (po) determines the magnitude of ce.
To determine the range of the ratio pi/po for which the magnitude of ce is greater at r=a and r=b, we need to consider the stress distribution equations for a thick-walled cylinder.
The stress distribution in a thick-walled cylinder can be described by the Lamé equations:
σr = (pi * a^2 - po * b^2) / (a^2 - b^2)
σθ = (pi * a^2 * b^2) / (a^2 - b^2)
τrθ = 0
Where σr is the radial stress, σθ is the tangential stress, and τrθ is the shear stress.
At r=a and r=b, the radial stress σr is equal to ce. Therefore, we can equate ce to the expression for σr:
ce = (pi * a^2 - po * b^2) / (a^2 - b^2)
To find the range of pi/po for which ce is greater at r=a and r=b, we need to analyze the behavior of ce with respect to the ratio pi/po.
Let's consider the case where r=a. In this case, the expression for ce becomes:
ce = (pi * a^2 - po * b^2) / (a^2 - b^2)
Similarly, for r=b, the expression for ce becomes:
ce = (pi * a^2 - po * b^2) / (a^2 - b^2)
For ce to be greater at both r=a and r=b, the numerator of the expression for ce must be positive, and the denominator must be negative. This implies that pi * a^2 - po * b^2 > 0 and a^2 - b^2 < 0.
Simplifying the inequalities, we get:
pi * a^2 > po * b^2
a^2 < b^2
From the first inequality, we can conclude that the ratio pi/po must be greater than b^2/a^2. From the second inequality, we can conclude that a must be less than b.
Therefore, the range of the ratio pi/po for which the magnitude of ce is greater at r=a and r=b is:
b^2/a^2 < pi/po
a < b
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