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Question: A uniform metal rod fixed at its ends of \( 2m{m^2} \) cross-section is cooled from \( 40^\circ C \)...

A uniform metal rod fixed at its ends of 2mm22m{m^2} cross-section is cooled from 40∘C40^\circ C to 20∘C20^\circ C . The coefficient of the linear expansion of the rod is 12×10−612 \times {10^{ - 6}} per degree Celsius and its young’s modulus of elasticity is 1011N/m2{10^{11}}N/{m^2} . The energy stored per unit volume of the rod is
(A) 2880J/m32880J/{m^3}
(B) 1500J/m31500J/{m^3}
(C) 5760J/m35760J/{m^3}
(D) 1440J/m31440J/{m^3}

Explanation

Solution

Here we will use the formula which expresses the energy stored per unit volume that is the product of stress and strain and will place the given values by simplifying which we will get the required value.

Complete step by step solution:
By using the relation,
ΔI=IαΔΘ\Delta I = I\alpha \Delta \Theta
Similarly, strain is the ratio of the change in the current with respect to the current.
Strain =ΔII=αΔΘ= \dfrac{{\Delta I}}{I} = \alpha \Delta \Theta
Stress =Y×strain=YαΔΘ= Y \times strain = Y\alpha \Delta \Theta
Now, the energy stored per unit volume =12×stress×strain= \dfrac{1}{2} \times stress \times strain
Place the formula in the above expression-
Energy =12×Y×(αΔΘ)2= \dfrac{1}{2} \times Y \times {(\alpha \Delta \Theta )^2}
Place the given values in the above equation –
Energy =12×1011×(12×10−6×20)2= \dfrac{1}{2} \times {10^{11}} \times {(12 \times {10^{ - 6}} \times 20)^2}
Simplify the above terms using the laws of power and exponent and also use that the common factors from the numerator and the denominator cancel each other.
Energy =2880J/m2= 2880J/{m^2}
Hence, from the given multiple choices, the option A is the correct answer.

Additional Information:
Prime factorization is the process of finding which prime numbers can be multiplied together to make the original number, where prime numbers are the numbers greater than 11 and which are not the product of any two smaller natural numbers. For Example: 2, 3, 5, 7,......2,{\text{ 3, 5, 7,}}...... 22 is the prime number as it can have only 11 factor. Factors are the number 11 and the number itself.

Note :
First of all find the correlation between the known and unknown terms and frame the correct formula to get the resultant required value. Be good in multiples and division. Always remember that the common factors from the numerator and the denominator cancel each other.