Question

A spring block system is kept on a frictionless surface. Match the respective entries of column (I) with column (II) assuming minimum potential energy for the system as zero.
($K$ = Spring constant, $A$ = Amplitude, $m$ = Mass of block)
$\begin{array}{cc}\text{Column-I}& \text{Column-II}\\ \text{(A) If mass of the block is doubled}& \text{(p) Time period increases}\\ \text{(keeping}K\text{,}A\text{unchanged)}& \\ \text{(B) If the amplitude of oscillation is doubled}& \text{(q) Time period decreases}\\ \text{(keeping}K\text{,}m\text{unchanged)}& \\ \text{(C) If force constant is doubled}& \text{(r) Energy of oscillation increases}\\ \text{(keeping}m\text{,}A\text{unchanged)}& \\ \text{(D) If another spring of same force constant}& \text{(s) Energy of oscillation decreases}\\ \text{is attached parallel to the previous one}& \\ \text{(keeping}m\text{,}A\text{unchanged)}& \text{(t) Energy of oscillation remain constant}\end{array}$

• A. $\left(A\right)\to (r,t);\left(B\right)\to (r,t);\left(C\right)\to (p,q);\left(D\right)\to (q,t)$
• B. $\left(A\right)\to \left(s\right);\left(B\right)\to (p,q);\left(C\right)\to (q,r);\left(D\right)\to (p,t)$
• C. $\left(A\right)\to (p,t);\left(B\right)\to \left(r\right);\left(C\right)\to (q,r);\left(D\right)\to (q,r)$
• D. $\left(A\right)\to (q,t);\left(B\right)\to \left(s\right);\left(C\right)\to (p,q);\left(D\right)\to (q,r)$

Answer 1

The correct option is C $\left(A\right)\to (p,t);\left(B\right)\to \left(r\right);\left(C\right)\to (q,r);\left(D\right)\to (q,r)$

Since, time period is
$T=2\pi \sqrt{\frac{m}{K}}$
and energy of oscillation,
$E=\frac{1}{2}K{A}^2$
Options of column:
$\left(A\right)$ $K$ and $A$ are constant
$\Rightarrow E$ = constant
Since, $m\uparrow \therefore T\uparrow$
$\therefore \left(A\right)\to (p,t)$
$\left(B\right)$ $K$ and $m$ are constant.
$\therefore T$ = constant
Since, $A\uparrow \therefore E\uparrow$
$\therefore \left(B\right)\to \left(r\right)$
$\left(C\right)$ $m$ and $A$ are constant.
Since, $K\uparrow \therefore T\downarrow$ and $E\uparrow$
$\therefore \left(C\right)\to (q,r)$
$\left(D\right)$ Due to attaching another spring $K$ in parallel to previous one.
$K_{eq}=K+K=2K$
$\Rightarrow K\uparrow$ and $(m,A)$ are constant.
$\therefore T\downarrow$ and $E\uparrow$
$\therefore \left(D\right)\to (q,r)$
Thus, option $\left(c\right)$ is the correct answer.