SATHEE: Oscillations - Result Question 48

Oscillations - Result Question 48

52. A body of mass M, executes vertical SHM with periods $t_{1}$ and $t_{2}$ , when separately attached to spring A and spring B respectively. The period of SHM, when the body executes SHM, as shown in the figure is $t_{0}$ . Then

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(a) $t_{0}^{- 1} = t_{1}^{- 1} + t_{2}^{- 1}$

(b) $t_{0} = t_{1} + t_{2}$

(d) $t_{0}^{- 2} = t_{1}^{- 2} + t_{2}^{- 2}$

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Answer:

Correct Answer: 52. (d)

Solution:

(d) The time period of spring mass system,

$T = 2 π \sqrt{\frac{m}{K}}$

$∴ t_{1} = 2 π \sqrt{\frac{m}{k_{1}}}$

$\begin{matrix} (ii) & t_{2} = 2 π \sqrt{\frac{m}{k_{2}}} \end{matrix}$

When springs are connected in parallel then $K_{eff} = k_{1} + k_{2}$

So, $t_{0} = 2 π \sqrt{\frac{m}{k_{e f f}}} \Rightarrow 2 π \sqrt{\frac{m}{(k_{1 + k_{2}})}}$

from (i), $\frac{1}{t_{1}^{2}} = \frac{1}{4 π^{2}} \times \frac{k_{1}}{m}$

from (ii), $\frac{1}{t_{2}^{2}} = \frac{1}{4 π^{2}} \times \frac{k_{2}}{m}$

from (iii), $\frac{1}{t_{0}^{2}} = \frac{1}{4 π^{2}} \times \frac{(k_{1} + k_{2})}{m}$

from above expressions,

$\frac{1}{t_{0}^{2}} = \frac{1}{t_{1}^{2}} + \frac{1}{t_{2}^{2}}$

$t = 2 π \sqrt{\frac{m}{k}}$

$\Rightarrow k =$ Const. $t^{- 2}$

Here the springs are joined in parallel. So

$k_{0} = k_{1} + k_{2}$

where $k_{0}$ is resultant force constant

$∴$ Const. $t_{0}^{- 2} =$ Const. $t_{1}^{- 2} +$ Const. $t_{2}^{- 2}$

or, $t_{0}^{- 2} = t_{1}^{- 2} + t_{2}^{- 2}$

Oscillations - Result Question 48

52. A body of mass M, executes vertical SHM with periods $t_{1}$ and $t_{2}$ , when separately attached to spring A and spring B respectively. The period of SHM, when the body executes SHM, as shown in the figure is $t_{0}$ . Then

Answer:

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Oscillations - Result Question 48

52. A body of mass M, executes vertical SHM with periods t1 and t2, when separately attached to spring A and spring B respectively. The period of SHM, when the body executes SHM, as shown in the figure is t0. Then

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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52. A body of mass M, executes vertical SHM with periods $t_{1}$ and $t_{2}$ , when separately attached to spring A and spring B respectively. The period of SHM, when the body executes SHM, as shown in the figure is $t_{0}$ . Then