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Question

Physics Question on Oscillations

Two springs are joined and attached to a mass of 16kg16\, kg. The system is then suspended vertically from a rigid support. The spring constant of the two springs are k1{{k}_{1}} and k2{{k}_{2}} respectively. The period of vertical oscillations of the system will be

A

18πk1+k2\frac{1}{8\pi }\sqrt{{{k}_{1}}+{{k}_{2}}}

B

8πk1+k2k1k28\pi \sqrt{\frac{{{k}_{1}}+{{k}_{2}}}{{{k}_{1}}{{k}_{2}}}}

C

π2k1k2\frac{\pi }{2}\sqrt{{{k}_{1}}-{{k}_{2}}}

D

π2k1k2\frac{\pi }{2}\sqrt{\frac{{{k}_{1}}}{{{k}_{2}}}}

Answer

8πk1+k2k1k28\pi \sqrt{\frac{{{k}_{1}}+{{k}_{2}}}{{{k}_{1}}{{k}_{2}}}}

Explanation

Solution

The two springs are in series. Therefore, the time period is
T=2πmk=2πm(k1+k2k1k2)T=2 \pi \sqrt{\frac{m}{k}}=2 \pi \sqrt{m\left(\frac{k_{1}+k_{2}}{k_{1} k_{2}}\right)}
As m=16kg;T=8πk1+k2k1k2m=16\, kg ; T=8 \pi \sqrt{\frac{k_{1}+k_{2}}{k_{1} k_{2}}}