Question

If the length of a wire increases by $ 1\,mm $ under $ 1\,kg $ weight, what will be the increase under $ 2\,kg\,? $ Under $ 100\,kg\,? $

Answer 1

Hint
In the question the length of wire is increasing under the weight of the wire. So, the length of the wire and weight is given. By substituting the known values in the equation of spring constant we get the value of extension under the weight.
The expression for finding the spring constant is
 $ $ $ K = \dfrac{F}{e} $
Where, $ K $ be the spring constant, $ F $ be the force, and $ e $ be the extension of the wire.

Complete step by step answer
Given that $ {m_1} = 1\,Kg,\,{m_2} = 2\,Kg,\,{m_3} = 100\,Kg\, $ ,
We know that, $ F = mg $
Where $ m $ be the mass and g be the acceleration due to gravity.
 $ K = \dfrac{{mg}}{e}.........\left( 1 \right) $
Here Let us assume $ g = 10 $
 $ \begin{gathered}
  F = {m_1} \times g \\
  F = 1 \times 10 \\
  F = 10\,N \\
\end{gathered} $
Substitute the known in the equation $ \left( 1 \right) $
 $ K = \dfrac{{10}}{1} = 10 $
Now we can find the value of weight under $ 2\,kg $ we get,
Given that mass $ {m_2} = 2\,kg $
 $ F = {m_2} \times g $
Substitute the known values in the above equation, we get
 $ F = 2 \times 10 $ $ $
 $ F = 20\,N $
Now we want the value of extension, so we change the equation $ \left( 1 \right) $ , we get
 $ e = \dfrac{F}{K} $
Substitute the value of $ F $ and $ e $ in the above equation we get,
 $ e = \dfrac{{20}}{{10}}\,mm $
 $ e = 2\,mm $ .
Hence the value of extension under $ 2\,kg $ is $ 2\,mm. $
Now we can find the value of weight under $ 100\,kg $ we get,
Given that $ mass\,{m_3} = 100\,kg $
 $ F = {m_3} \times g $
 $ F = 100 \times 10 $
 $ F = 1000\,N $
Now we can find the extension under the weight $ 100\,kg $ , we get
 $ e = \dfrac{F}{K} $
 $ e = \dfrac{{1000}}{{10}} $
 $ e = 100\,mm $
Therefore, the increase in weight under $ 100\,kg $ is $ 100\,mm. $

Note
In the question, we know that the length of the wire is increasing so it is added to the length. First, we have to find the actual force and then equate the weight of the wires we get the value of extension. In other cases, the weight of the wire is given in the question we directly substitute in the equation of spring constant. But here we don’t know the extension so we find the force and then calculate the extension of the wire.