Question 353: Oscillations

Question

A body of mass 64 g is made to oscillate turn by turn on two different springs $A$ and $B$. Spring $A$ and $B$ has force constant $4 \frac{\mathrm{~N}}{\mathrm{~m}}$ and $16 \frac{\mathrm{~N}}{\mathrm{~m}}$ respectively. If $T_1$ and $T_2$ are period of oscillations of springs $A$ and $B$ respectively, then $\frac{T_1+T_2}{T_1-T_2}$ will be

Accepted Answer

Given, $m=64 \mathrm{~g}=64 	imes 10^{-3} \mathrm{~kg}$,
$$k_A=4 \mathrm{~N} / \mathrm{m} 	ext { and } k_B=16 \mathrm{~N} / \mathrm{m}$$
The time period of oscillation of a spring,
$$\begin{aligned}
& T=2 \pi \sqrt{\frac{m}{k}} \
& \Rightarrow \quad T_A=T_1=2 \pi \sqrt{\frac{m}{k_A}}=2 \pi \sqrt{\frac{64 	imes 10^{-3}}{4}} \
& =2 \pi \sqrt{16 	imes 10^{-3}} \quad 	ext{... (i)}\
& 	ext { and } \quad T_B=T_2=2 \pi \sqrt{\frac{m}{k_B}}=2 \pi \sqrt{\frac{64 	imes 10^{-3}}{16}} \
& =2 \pi \sqrt{4 	imes 10^{-3}}\quad 	ext{... (ii)}
\end{aligned}$$
$$\begin{aligned}
& 	ext { Dividing Eq. (i) by Eq. (ii), we get } \
& \frac{T_1}{T_2}=\sqrt{\frac{16 	imes 10^{-3}}{4 	imes 10^{-3}}}=2 \
& \Rightarrow \quad T_1=2 T_2 \
& 	herefore \quad \frac{T_1+T_2}{T_1-T_2}=\frac{2 T_2+T_2}{2 T_2-T_2}=\frac{3 T_2}{T_2}=\frac{3}{1} 	ext { or } 3: 1
\end{aligned}$$

Answer

$1: 2$

Answer

$1: 3$

Answer

$3: 1$

Answer

$2: 1$

Oscillations

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