Question 271: Oscillations

Question

All the springs in fig. (a), (b) and (c) are identical, each having force constant K . Mass attached to each system is ' $m$ '. If $T_a, T_b$ and $T_c$ are the time periods of oscillations of the three systems respectively, then

Accepted Answer

$$\begin{aligned}
&\begin{aligned}
\mathrm{T}_{\mathrm{a}} & =2 \pi \sqrt{\frac{\mathrm{~m}}{\mathrm{~K}}} \
\mathrm{~T}_{\mathrm{b}} & =2 \pi \sqrt{\frac{\mathrm{~m}}{\frac{\mathrm{~K}}{2}}} \
	herefore \quad \mathrm{~T}_{\mathrm{b}} & =2 \pi \sqrt{\frac{2 \mathrm{~m}}{\mathrm{~K}}}=\sqrt{2} \mathrm{~T}_{\mathrm{a}} \Rightarrow \mathrm{~T}_{\mathrm{a}}=\frac{\mathrm{T}_{\mathrm{b}}}{\sqrt{2}} \quad	ext{.... (i)}\
\mathrm{~T}_{\mathrm{c}} & =2 \pi \sqrt{\frac{\mathrm{~m}}{2 \mathrm{~K}}}=\frac{\mathrm{T}_{\mathrm{a}}}{\sqrt{2}} \Rightarrow \mathrm{~T}_{\mathrm{a}}=\sqrt{2} \mathrm{~T}_{\mathrm{c}} \quad	ext{.... (ii)}
\end{aligned}\
&	ext { From (i) and (ii), }\
&\frac{T_b}{\sqrt{2}}=\sqrt{2} T_c \Rightarrow T_b=2 T_c
\end{aligned}$$

Answer

$\mathrm{T}_{\mathrm{a}}=\sqrt{2} \mathrm{~T}_{\mathrm{b}}$

Answer

$\mathrm{T}_{\mathrm{a}}=\frac{\mathrm{T}_{\mathrm{c}}}{\sqrt{2}}$

Answer

$T_b=2 T_a$

Answer

$T_b=2 T_c$

Oscillations

All the springs in fig. (a), (b) and (c) are identical, each having force constant K . Mass attached to each system is ' m '. If T_a, T_b and T_c are the time periods of oscillations of the three systems respectively, then

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