Springs of constants K, $$2K$$, $$4K$$, $$8K$$, $$16K$$,… are connected in series. The mass ‘m’ kg is attached to the lower end of the last spring and the system is allowed to vibrate. The frequency of oscillation is

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Infinity Learn · Accepted Answer

K, $$2K$$, $$4K$$, $$8K$$, $$16K$$,… are connected in $$series1Keffective=1K+12K+14K+$$−−→−geometric $$progression(gp)$$  sum to infinite series of gp $$=a1-r$$; a is first term; r is common ratio $$1Keff=1K1+12+122+$$...  here first term $$a=1$$ and common ratio $$r=12$$    $$1Keff=1Ka1-r$$  substitute the values $$1Keff=1K11-12$$ $$1Keff=1K21Keff=K2$$ Frequency $$n=12$$$$π$$Keffm  ; substitute obtained value of Keff $$n=12$$$$π$$$$K2mn=12$$$$π$$$$K2m$$

Springs of constants K, 2K, 4K, 8K, 16K,… are connected in series. The mass ‘m’ kg is attached to the lower end of the last spring and the system is allowed to vibrate. The frequency of oscillation is

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