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Question: A steel wire \(2\;{\text{m}}\) long is suspended from the ceiling. When a mass is hung at its lower ...

A steel wire 2  m2\;{\text{m}} long is suspended from the ceiling. When a mass is hung at its lower end, the increase in length recorded is 1  cm1\;{\text{cm}}. Determine the strain in the wire.
A. 5×10−35 \times {10^{ - 3}}
B. 65×10−365 \times {10^{ - 3}}
C. 6×10−36 \times {10^{ - 3}}
D. 7×10−37 \times {10^{ - 3}}

Explanation

Solution

The above problem can be solved by using the concept of the strain. The strain is defined as the ratio of change in the size of object to original size. It is the dimensionless quantity. It is proportional to the stress applied on the object up to the elastic limit of the material of the object.

Complete step by step answer:
Given:
The original length of the steel wire is l=2  ml = 2\;{\text{m}}.
The change in the length of the steel wire is Δl=1  cm=1  cm×1  m100  cm=1×10−2  m\Delta l = 1\;{\text{cm}} = 1\;{\text{cm}} \times \dfrac{{1\;{\text{m}}}}{{100\;{\text{cm}}}} = 1 \times {10^{ - 2}}\;{\text{m}}
The expression to calculate the strain in the steel wire is given as,
ε=Δll\varepsilon = \dfrac{{\Delta l}}{l}
Substitute 2  m2\;{\text{m}}for ll and 1×10−2  m1 \times {10^{ - 2}}\;{\text{m}} for Δl\Delta l in the above expression to find the strain in the steel wire.
ε=1×10−2  m2  m\varepsilon = \dfrac{{1 \times {{10}^{ - 2}}\;{\text{m}}}}{{2\;{\text{m}}}}
⇒ε=0.5×10−2\Rightarrow\varepsilon = 0.5 \times {10^{ - 2}}
∴ε=5×10−3\therefore\varepsilon = 5 \times {10^{ - 3}}

Thus, the strain in the steel wire is 5×10−35 \times {10^{ - 3}} and the option (A) is the correct answer.

Additional Information:
The ratio of stress to strain is called the modulus of elasticity. If the modulus of elasticity of the material is high then strain produced for some value of the applied stress becomes less. The modulus of elasticity is of three types: Young's modulus of elasticity, Bulk modulus and Modulus of rigidity. The strain may be of two types, one is the longitudinal strain and other is lateral strain. The ratio of lateral strain to longitudinal strain is called the Poisson’s ratio.

Note: The modulus of elasticity is the material property. It varies with the temperature of the object. The unit of original length and change in length must be the same to find the strain.