Question 4

A light string passing over a smooth light pulley connects two blocks of masses $$m_1$$ and $$m_2$$ (where $$m_2>m_1$$). If the acceleration of the system is $$\frac{g}{\sqrt{2}}$$, then the ratio of the masses $$\frac{m_1}{m_2}$$ is:

Solution

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$$a = \frac{(m_2 - m_1)g}{m_1 + m_2} = \frac{g}{\sqrt{2}}$$.

$$\frac{m_2 - m_1}{m_1 + m_2} = \frac{1}{\sqrt{2}}$$.

Let $$r = m_1/m_2$$: $$\frac{1-r}{1+r} = \frac{1}{\sqrt{2}}$$.

$$\sqrt{2}(1-r) = 1+r \Rightarrow \sqrt{2} - \sqrt{2}r = 1 + r \Rightarrow r(1+\sqrt{2}) = \sqrt{2}-1 \Rightarrow r = \frac{\sqrt{2}-1}{\sqrt{2}+1}$$.

The correct answer is Option (2): $$\frac{\sqrt{2}-1}{\sqrt{2}+1}$$.

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