Find the time period of the oscillation of mass m in figures-i 2-m-k1-k2ii 2-mk1-k2-k1k2iii 2-mk1-k2-k1k2

A) Equivalent spring constant k=k_1+k_2 (parallel) T=2π√(M/k)=2π√(m/k_1+k_2) b) Let us, displace the block m towards left through ...

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Standard XII

Physics

Springs in Parallel and Series

Find the time...

Question

Find the time period of the oscillation of mass m in figures.

(i) $2 π \sqrt{\frac{m}{k_{1} + k_{2}}}$

(ii) $2 π \sqrt{\frac{m (k_{1} + k_{2})}{k_{1} k_{2}}}$

(iii) $2 π \sqrt{\frac{m (k_{1} - k_{2})}{k_{1} k_{2}}}$

(x) - (i); (y) - (ii); (z) - (ii)

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(x) - (i); (y) - (i); (z) - (ii)

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(x) - (ii); (y) - (i); (z) - (iii)

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(x) - (i); (y) - (iii); (z) - (ii)

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Solution

The correct option is B (x) - (i); (y) - (i); (z) - (ii)
a) Equivalent spring constant $k = k_{1} + k_{2}$ (parallel)

$T = 2 π \sqrt{\frac{M}{k}} = 2 π \sqrt{\frac{m}{k_{1} + k_{2}}}$

b) Let us, displace the block m towards left through displacement `x'

Resultant force $F = F_{1} + F_{2} = (k_{1} + k_{2}) x$

Acceleration $(F / m) = \frac{(k_{1} + k_{2}) x}{m}$

Time period $T = 2 π \sqrt{\frac{d i s p l a c e m e n t}{A c c e l e r a t i o n}} = 2 π \sqrt{\frac{\frac{x}{x (k_{1} + k_{2})}}{m}} = 2 π \sqrt{\frac{m}{k_{1} + k_{2}}}$

The equivalent spring constant $k = k_{1} + k_{2}$

c) In series combination, let equivalent spring constant be k.

So, $\frac{1}{k} = \frac{1}{k_{1}} + \frac{1}{k_{2}} = \frac{k_{2} + k_{1}}{k_{1} k_{2}} \Rightarrow k = \frac{k_{1} k_{2}}{k_{1} + k_{2}}$

$T = 2 π \sqrt{\frac{M}{k}} = 2 π \sqrt{\frac{m (k_{1} + k_{2})}{k_{1} k_{2}}}$

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Find the time period of the oscillation of mass m in figures. (i) 2π√mk1+k2 (ii) 2π√m(k1+k2)k1k2 (iii) 2π√m(k1−k2)k1k2

Find the time period of the oscillation of mass m in figures.

(i) $2 π \sqrt{\frac{m}{k_{1} + k_{2}}}$

(ii) $2 π \sqrt{\frac{m (k_{1} + k_{2})}{k_{1} k_{2}}}$

(iii) $2 π \sqrt{\frac{m (k_{1} - k_{2})}{k_{1} k_{2}}}$