
A copper wire is held at the two ends by rigid supports. At , the wire is just taut with negligible tension. The speed of transverse waves in this wire at is . Then the value of 'x' is. ( Take Young's modulus, , coefficient of linear expansion, and density, )
(1)
(2)
(3)
(4)
Answer
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Hint:We will utilize the concept of thermal expansion or compression of given copper wire. The expression for force developed in the wire provides us with the relationship between Young's modulus, cross-sectional area, thermal expansion coefficient, and change in the wire temperature.
Complete step by step answer:
Given:
The temperature at which wire is taut with negligible tension is .
The speed of transverse waves in the given wire is .
The temperature at the speed of transverse waves v is .
The value of Young's modulus of copper wire is .
The coefficient of linear expansion of copper wire is .
The density of copper wire is .
We have to evaluate the value of 'x'
Let us write the expression for the change in copper wire length when its temperature changed from to .
Here l is the length of the given wire and is the temperature change.
We know that the expression for force of thermal expansion or compression of the given copper wire can be written as:
Here A is the cross-sectional area of the wire.
Let us write the expression for the speed of transverse of the given copper wire.
Substitute for F in the above expression.
Substitute for V, for , for , for , and for and for in the above expression.
From the above expression, we get:
Therefore, we can say that the value of x is 4, and option (1) is correct.
Note:We can remember Pascal's conversion to Newton-meter and converting Newton into its base units (kilogram, meter, and second). We also have to be extra careful while rearranging the final expression.
Complete step by step answer:
Given:
The temperature at which wire is taut with negligible tension is
The speed of transverse waves in the given wire is
The temperature at the speed of transverse waves v is
The value of Young's modulus of copper wire is
The coefficient of linear expansion of copper wire is
The density of copper wire is
We have to evaluate the value of 'x'
Let us write the expression for the change in copper wire length when its temperature changed from
Here l is the length of the given wire and
We know that the expression for force of thermal expansion or compression of the given copper wire can be written as:
Here A is the cross-sectional area of the wire.
Let us write the expression for the speed of transverse of the given copper wire.
Substitute
Substitute
From the above expression, we get:
Therefore, we can say that the value of x is 4, and option (1) is correct.
Note:We can remember Pascal's conversion to Newton-meter and converting Newton into its base units (kilogram, meter, and second). We also have to be extra careful while rearranging the final expression.
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