Question
This question has Statement 1 and Statement 2. Of the four choices given after the Statements, choose the one that best describes the two Statements.
If two springs $${S_1}$$ and $${S_2}$$ of force constants $${k_1}$$ and $${k_2},$$ respectively, are stretched by the same force, it is found that more work is done on spring $${S_1}$$ than on spring $${S_2}.$$
STATEMENT 1 : If stretched by the same amount work done on $${S_1},$$ Work done on $${S_1}$$ is more than $${S_2}$$
STATEMENT 2 : $${k_1} < {k_2}$$
A.
Statement $$1$$ is false, Statement $$2$$ is true.
B.
Statement $$1$$ is true, Statement $$2$$ is false.
C.
Statement $$1$$ is true, Statement $$2$$ is true, Statement $$2$$ is the correct explanation for Statement $$1$$
D.
Statement $$1$$ is true, Statement $$2$$ is true, Statement $$2$$ is not the correct explanation for Statement $$1$$
Answer :
Statement $$1$$ is false, Statement $$2$$ is true.
Solution :
When force is same
$$\eqalign{
& W = \frac{1}{2}k{x^2} \cr
& W = \frac{1}{2}k\frac{{{F^2}}}{{{k^2}}}\,\,\left[ {\because \,F = kx} \right] \cr
& \therefore W = \frac{{{F^2}}}{{2x}} \cr
& {\text{As}}\,\,{W_1} > {W_2} \cr
& \therefore {k_1} < {k_2} \cr} $$
When extension is same
$$\eqalign{
& W \propto k\,\,\left( {\because x{\text{ is same}}} \right) \cr
& \therefore {W_1} < {W_2} \cr} $$
Statement I is false and statement 2 is true.