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For a thin spherical shell subjected to internal pressure, the ratio of volumetric strain to diametrical strain is _____.
- a)5 : 4
- b)3 : 2
- c)2 : 1
- d)3 : 1
Correct answer is option 'D'. Can you explain this answer?
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Solution:
The volumetric strain (εv) is defined as the change in volume per unit original volume, while the diametrical strain (εd) is defined as the change in diameter per unit original diameter.
Let's consider a thin spherical shell with an internal pressure.
Step 1: Deriving the formulas
The formula for volumetric strain is given by:
εv = ΔV / V
where ΔV is the change in volume and V is the original volume.
The change in volume can be calculated using the formula:
ΔV = 4/3π (R2^3 - R1^3)
where R1 is the initial radius and R2 is the final radius.
Substituting the value of ΔV in the volumetric strain formula, we get:
εv = 4/3π (R2^3 - R1^3) / (4/3π R1^3)
Simplifying the equation, we get:
εv = (R2^3 - R1^3) / R1^3
The formula for diametrical strain is given by:
εd = ΔD / D
where ΔD is the change in diameter and D is the original diameter.
The change in diameter can be calculated using the formula:
ΔD = 2(R2 - R1)
Substituting the value of ΔD in the diametrical strain formula, we get:
εd = 2(R2 - R1) / 2R1
Simplifying the equation, we get:
εd = (R2 - R1) / R1
Step 2: Evaluating the ratio
To find the ratio of volumetric strain to diametrical strain, we divide the equation for volumetric strain by the equation for diametrical strain:
(εv / εd) = [(R2^3 - R1^3) / R1^3] / [(R2 - R1) / R1]
Simplifying the equation, we get:
(εv / εd) = [(R2^3 - R1^3) / (R2 - R1)] * (R1 / R1)
(εv / εd) = (R2^3 - R1^3) / (R2 - R1)
Since the shell is thin, R2 is very close to R1, so we can approximate R2 - R1 as ΔR.
(εv / εd) = (R1^3 + R1^2ΔR + R1ΔR^2 - R1^3) / ΔR
(εv / εd) = (R1^2ΔR + R1ΔR^2) / ΔR
(εv / εd) = (R1^2 + R1ΔR) / 1
(εv / εd) = R1(R1 + ΔR) / ΔR
As ΔR approaches zero (since the shell is thin), the ratio becomes:
(εv / εd) = R1 / 0
Therefore, the ratio of volumetric strain to diametrical strain is undefined if the shell is infinitely thin. However, in practice, for thin shells,
The volumetric strain (εv) is defined as the change in volume per unit original volume, while the diametrical strain (εd) is defined as the change in diameter per unit original diameter.
Let's consider a thin spherical shell with an internal pressure.
Step 1: Deriving the formulas
The formula for volumetric strain is given by:
εv = ΔV / V
where ΔV is the change in volume and V is the original volume.
The change in volume can be calculated using the formula:
ΔV = 4/3π (R2^3 - R1^3)
where R1 is the initial radius and R2 is the final radius.
Substituting the value of ΔV in the volumetric strain formula, we get:
εv = 4/3π (R2^3 - R1^3) / (4/3π R1^3)
Simplifying the equation, we get:
εv = (R2^3 - R1^3) / R1^3
The formula for diametrical strain is given by:
εd = ΔD / D
where ΔD is the change in diameter and D is the original diameter.
The change in diameter can be calculated using the formula:
ΔD = 2(R2 - R1)
Substituting the value of ΔD in the diametrical strain formula, we get:
εd = 2(R2 - R1) / 2R1
Simplifying the equation, we get:
εd = (R2 - R1) / R1
Step 2: Evaluating the ratio
To find the ratio of volumetric strain to diametrical strain, we divide the equation for volumetric strain by the equation for diametrical strain:
(εv / εd) = [(R2^3 - R1^3) / R1^3] / [(R2 - R1) / R1]
Simplifying the equation, we get:
(εv / εd) = [(R2^3 - R1^3) / (R2 - R1)] * (R1 / R1)
(εv / εd) = (R2^3 - R1^3) / (R2 - R1)
Since the shell is thin, R2 is very close to R1, so we can approximate R2 - R1 as ΔR.
(εv / εd) = (R1^3 + R1^2ΔR + R1ΔR^2 - R1^3) / ΔR
(εv / εd) = (R1^2ΔR + R1ΔR^2) / ΔR
(εv / εd) = (R1^2 + R1ΔR) / 1
(εv / εd) = R1(R1 + ΔR) / ΔR
As ΔR approaches zero (since the shell is thin), the ratio becomes:
(εv / εd) = R1 / 0
Therefore, the ratio of volumetric strain to diametrical strain is undefined if the shell is infinitely thin. However, in practice, for thin shells,
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For a thin spherical shell subjected to internal pressure, the ratio of volumetric strain to diametrical strain is _____.a)5 : 4b)3 : 2c)2 : 1d)3 : 1Correct answer is option 'D'. Can you explain this answer?
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