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Two long columns C1 and C2 are made of the same material. Column C1 has both the ends hinged while column C2 has one end hinged and the other end fixed. What is the ratio of the critical load for C1 to that for C2 according to Euler’s formula?
- a)2
- b)1/2
- c)4
- d)1/4
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
Two long columns C1 and C2 are made of the same material. Column C1 ha...
**Explanation:**
To understand the ratio of the critical load for the two columns, we need to first understand Euler's formula for columns.
**Euler's Formula:**
Euler's formula is used to determine the critical load for slender columns. It states that the critical load for a long column is given by:
Pcr = (π^2 * E * I) / L^2
Where:
- Pcr is the critical load
- E is the modulus of elasticity of the material
- I is the moment of inertia of the cross-sectional area
- L is the effective length of the column
**Analysis of Columns:**
1. Column C1 (Both Ends Hinged):
- Both ends of column C1 are hinged, which means they can rotate freely.
- In this case, the effective length of the column is twice the actual length of the column.
- Let L1 be the actual length of the column C1, then the effective length is 2L1.
2. Column C2 (One End Hinged and One End Fixed):
- One end of column C2 is hinged, which means it can rotate freely.
- The other end of column C2 is fixed, which means it cannot rotate.
- In this case, the effective length of the column is the actual length of the column.
- Let L2 be the actual length of the column C2, then the effective length is L2.
**Calculation:**
Now, let's calculate the ratio of the critical load for C1 to that for C2.
For column C1:
Pcr1 = (π^2 * E * I) / (2L1)^2
= (π^2 * E * I) / 4L1^2
For column C2:
Pcr2 = (π^2 * E * I) / L2^2
Taking the ratio of Pcr1 to Pcr2:
Pcr1/Pcr2 = [(π^2 * E * I) / 4L1^2] / [(π^2 * E * I) / L2^2]
= L2^2 / 4L1^2
Since L2 = 2L1 (as mentioned earlier), substituting the values:
Pcr1/Pcr2 = (2L1)^2 / 4L1^2
= 4L1^2 / 4L1^2
= 1
Therefore, the ratio of the critical load for C1 to that for C2 is 1.
Hence, the correct answer is option **b) 1/2**.
To understand the ratio of the critical load for the two columns, we need to first understand Euler's formula for columns.
**Euler's Formula:**
Euler's formula is used to determine the critical load for slender columns. It states that the critical load for a long column is given by:
Pcr = (π^2 * E * I) / L^2
Where:
- Pcr is the critical load
- E is the modulus of elasticity of the material
- I is the moment of inertia of the cross-sectional area
- L is the effective length of the column
**Analysis of Columns:**
1. Column C1 (Both Ends Hinged):
- Both ends of column C1 are hinged, which means they can rotate freely.
- In this case, the effective length of the column is twice the actual length of the column.
- Let L1 be the actual length of the column C1, then the effective length is 2L1.
2. Column C2 (One End Hinged and One End Fixed):
- One end of column C2 is hinged, which means it can rotate freely.
- The other end of column C2 is fixed, which means it cannot rotate.
- In this case, the effective length of the column is the actual length of the column.
- Let L2 be the actual length of the column C2, then the effective length is L2.
**Calculation:**
Now, let's calculate the ratio of the critical load for C1 to that for C2.
For column C1:
Pcr1 = (π^2 * E * I) / (2L1)^2
= (π^2 * E * I) / 4L1^2
For column C2:
Pcr2 = (π^2 * E * I) / L2^2
Taking the ratio of Pcr1 to Pcr2:
Pcr1/Pcr2 = [(π^2 * E * I) / 4L1^2] / [(π^2 * E * I) / L2^2]
= L2^2 / 4L1^2
Since L2 = 2L1 (as mentioned earlier), substituting the values:
Pcr1/Pcr2 = (2L1)^2 / 4L1^2
= 4L1^2 / 4L1^2
= 1
Therefore, the ratio of the critical load for C1 to that for C2 is 1.
Hence, the correct answer is option **b) 1/2**.
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Community Answer
Two long columns C1 and C2 are made of the same material. Column C1 ha...
**Explanation:**
According to Euler's formula, the critical load for a long column can be calculated using the equation:
P_cr = (π^2 * E * I) / (L^2)
Where:
- P_cr is the critical load
- E is the Young's modulus of the material
- I is the moment of inertia of the cross-sectional area of the column
- L is the effective length of the column
In this case, both columns C1 and C2 are made of the same material, so the value of E is the same for both columns. Let's assume it as E1.
The moment of inertia of the cross-sectional area is also the same for both columns, so the value of I is the same. Let's assume it as I1.
**Critical Load for C1:**
For column C1, both ends are hinged. In this case, the effective length of the column is equal to the actual length of the column, which we can assume as L1.
So the equation for the critical load of column C1 becomes:
P_cr1 = (π^2 * E1 * I1) / (L1^2)
**Critical Load for C2:**
For column C2, one end is hinged and the other end is fixed. In this case, the effective length of the column is 2 times the actual length of the column, which we can assume as L2.
So the equation for the critical load of column C2 becomes:
P_cr2 = (π^2 * E1 * I1) / (L2^2)
**Ratio of Critical Loads:**
To find the ratio of the critical load for C1 to that for C2, we can divide the equation for P_cr1 by the equation for P_cr2:
P_cr1 / P_cr2 = (π^2 * E1 * I1) / (L1^2) / (π^2 * E1 * I1) / (L2^2)
Simplifying this expression, we can cancel out the E1 and I1 terms, and we are left with:
P_cr1 / P_cr2 = (L2^2) / (L1^2)
Since L2 is equal to 2L1 (as the effective length is 2 times the actual length for a column with one end fixed), we can substitute this value into the equation:
P_cr1 / P_cr2 = (2L1^2) / (L1^2)
P_cr1 / P_cr2 = 2
Therefore, the ratio of the critical load for C1 to that for C2 is 2, which corresponds to option B.
According to Euler's formula, the critical load for a long column can be calculated using the equation:
P_cr = (π^2 * E * I) / (L^2)
Where:
- P_cr is the critical load
- E is the Young's modulus of the material
- I is the moment of inertia of the cross-sectional area of the column
- L is the effective length of the column
In this case, both columns C1 and C2 are made of the same material, so the value of E is the same for both columns. Let's assume it as E1.
The moment of inertia of the cross-sectional area is also the same for both columns, so the value of I is the same. Let's assume it as I1.
**Critical Load for C1:**
For column C1, both ends are hinged. In this case, the effective length of the column is equal to the actual length of the column, which we can assume as L1.
So the equation for the critical load of column C1 becomes:
P_cr1 = (π^2 * E1 * I1) / (L1^2)
**Critical Load for C2:**
For column C2, one end is hinged and the other end is fixed. In this case, the effective length of the column is 2 times the actual length of the column, which we can assume as L2.
So the equation for the critical load of column C2 becomes:
P_cr2 = (π^2 * E1 * I1) / (L2^2)
**Ratio of Critical Loads:**
To find the ratio of the critical load for C1 to that for C2, we can divide the equation for P_cr1 by the equation for P_cr2:
P_cr1 / P_cr2 = (π^2 * E1 * I1) / (L1^2) / (π^2 * E1 * I1) / (L2^2)
Simplifying this expression, we can cancel out the E1 and I1 terms, and we are left with:
P_cr1 / P_cr2 = (L2^2) / (L1^2)
Since L2 is equal to 2L1 (as the effective length is 2 times the actual length for a column with one end fixed), we can substitute this value into the equation:
P_cr1 / P_cr2 = (2L1^2) / (L1^2)
P_cr1 / P_cr2 = 2
Therefore, the ratio of the critical load for C1 to that for C2 is 2, which corresponds to option B.
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