First slide

Applications of SHM

Question

Springs of spring constants K,2K,4K,8K,…,many are connected in series. A mass 'm' is attached to one end. The system is allowed to oscillation. The time period is approximately.

Moderate

A

$2 π \sqrt{\frac{m}{2k}}$
B

$2 π \sqrt{\frac{2m}{k}}$
C

$2 π \sqrt{\frac{m}{4k}}$
D

$2 π \sqrt{\frac{4m}{k}}$

Solution

$s p r i n g s a r e i n s e r i e s, w h o s e s p r i n g c o n s \tan t s a r e k, 2 k, 4 k, 8 k, . . . . . . . . e f f e c t i v e s p r i n g c o n s \tan t i s k_{e f f e c t i v e} = k_{e f f} i s \frac{1}{k_{eff}} = \frac{1}{k} + \frac{1}{2k} + \frac{1}{4k} + ... \frac{1}{k_{eff}} = \frac{1}{k} (1 + \frac{1}{2} + \frac{1}{2^{2}} + ...) t h e b r a c k e t t p a r t i s i n g e o m e t r i c p r o g r e s s i o n, i n f i n i t e s e r i e s, w h o s e s u m = \frac{a}{1-r} \frac{1}{k_{eff}} = \frac{1}{k} (\frac{a}{1-r}) h e r e a = 1 a n d r = \frac{1}{2} \frac{1}{k_{eff}} = \frac{1}{k} (\frac{1}{1- \frac{1}{2}}) \frac{1}{k_{eff}} = \frac{1}{k} (\frac{2}{1}) k_{eff} = \frac{k}{2} T h e e f f e c t i v e T i m e p e r i o d o f s p r i n g s i n s e r i e s i s T=2π \sqrt{\frac{m}{k_{eff}}} =2 π \sqrt{\frac{m}{k/2}} =2 π \sqrt{\frac{2m}{k}}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App