### Related Formula Springs in parallel: k_p = k_1 + k_2$k_p = k_1 + k_2$ Springs in series: frac1k_s = frac1k_p + frac1k_3$\frac{1}{k_s} = \frac{1}{k_p} + \frac{1}{k_3}$ Time period: T = 2pi sqrtfracMk_eq$T = 2\pi \sqrt{\frac{M}{k_{eq}}}$ ### Core Logic Evaluate the equivalent spring constant of the given setup by first resolving the parallel configuration, then the series configuration. ### Step 1: Equivalent Spring Constant Based on the standard layout of the problem (as implied by the solution text), let's assume one section is 2k$2k$ and k$k$ in parallel... Wait, the solution implies frac2k cdot k3k + k$\frac{2k \cdot k}{3k} + k$, which represents a series combination of 2k$2k$ and k$k$, followed by a parallel combination with another k$k$. Let's trust the official PDF's algebraic step: k_eq = frac2k cdot k3k + k$k_{eq} = \frac{2k \cdot k}{3k} + k$. This evaluates to: k_eq = frac2k3 + k = frac5k3$k_{eq} = \frac{2k}{3} + k = \frac{5k}{3}$ ### Step 2: Time Period Calculation T = 2pi sqrtfracMk_eq$T = 2\pi \sqrt{\frac{M}{k_{eq}}}$ T = 2pi sqrtfracM5k/3 = 2pi sqrtfrac3M5k$T = 2\pi \sqrt{\frac{M}{5k/3}} = 2\pi \sqrt{\frac{3M}{5k}}$ ### Step 3: Matching the Format Bring the 2 inside the square root as 4: T = pi sqrtfrac4 times 3M5k = pi sqrtfrac12M5k$T = \pi \sqrt{\frac{4 \times 3M}{5k}} = \pi \sqrt{\frac{12M}{5k}}$ Comparing with pi sqrtfracalpha M5k$\pi \sqrt{\frac{\alpha M}{5k}}$: alpha = 12$\alpha = 12$ ### Pattern Recognition Whenever an external coefficient like 2 (from 2pi$2\pi$) is pushed inside the square root to match a format, it squares. Don't forget 2 to 4$2 \to 4$, turning 3M$3M$ into 12M$12M$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Oscillations