Physics

Problem

Two spring–mass systems, I and II, oscillate independently on smooth horizontal surfaces. The following data are provided: • System I: mass

m_1 = 0.40\,\text{kg}

, spring constant

k_1 = 64\,\text{N\,m}^{-1}

, amplitude

A_1 = 0.05\,\text{m}

• System II: mass

m_2 = 0.20\,\text{kg}

, spring constant

k_2 = 36\,\text{N\,m}^{-1}

, amplitude

A_2 = 0.04\,\text{m}

Answer the following: a. (i) Derive an expression for the ratio

\dfrac{v_{\text{max},I}}{v_{\text{max},II}}

of the maximum speeds of the two systems in terms of

m_1, m_2, k_1, k_2, A_1,

and

A_2

. b. (ii) Using the given numerical data, evaluate this ratio. c. (iii) State which system attains the larger maximum speed and justify your answer with reference to your calculation.

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