question_answer

Two springs have force constants \[{{\text{K}}_{\text{1}}}\]and \[{{\text{K}}_{\text{2}}}\left( {{\text{K}}_{\text{1}}}>\text{ }{{\text{K}}_{\text{2}}} \right)\]. On which spring is more work done when they are stretched by the same force?

Answer:

                \[{{\text{K}}_{\text{1}}}=\text{ F}/{{\text{x}}_{\text{1}}}\]and \[{{\text{K}}_{\text{2}}}=\text{ F}/{{\text{x}}_{\text{2}}}\] As \[{{\text{K}}_{\text{1}}}>{{\text{K}}_{\text{2}}}\] \[{{\text{x}}_{\text{1}}}<{{\text{x}}_{\text{2}}}\]                 \[\frac{{{W}_{1}}}{{{W}_{2}}}=\frac{\frac{1}{2}{{K}_{1}}x_{1}^{2}}{\frac{1}{2}{{K}_{2}}x_{2}^{2}}=\frac{(F/{{x}_{1}})x_{1}^{2}}{(F/{{x}_{2}})x_{2}^{2}}=\frac{{{x}_{1}}}{{{x}_{2}}}\] As \[{{\text{x}}_{\text{1}}}<{{\text{x}}_{\text{2}}}\] \[{{\text{W}}_{\text{1}}}<\text{ }{{\text{W}}_{\text{2}}}\]     or                 \[{{\text{W}}_{\text{2}}}>{{\text{W}}_{\text{1}}}\]