A) \[\frac{1}{2\pi }\sqrt{\frac{k}{4M}}\]
B) \[\frac{1}{2\pi }\sqrt{\frac{4k}{M}}\]
C) \[\frac{1}{2\pi }\sqrt{\frac{k}{7M}}\]
D) \[\frac{1}{2\pi }\sqrt{\frac{7k}{M}}\]
Correct Answer: B
Solution :
[b] Springs on the left of the block are in series, hence their equivalent spring constant is \[{{K}_{1}}=\frac{(2k)(2k)}{2k+2k}=k\] Springs on the right of the block are in parallel, hence their equivalent spring constant is \[{{k}_{2}}=k+2k=3k\] Now again both \[{{K}_{1}}\]and \[{{K}_{2}}\] are in parallel \[\therefore \,{{K}_{eq}}={{k}_{1}}+{{k}_{2}}=k+3k=4k\] Hence, frequency is \[f=\frac{1}{2\pi }\sqrt{\frac{{{K}_{eq}}}{M}}=\frac{1}{2\pi }\sqrt{\frac{4k}{M}}\]
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