A heavy ball is suspended from the ceiling of a motor car through a light string. A transverse pulse travels at speed of $60 c m / \sec$ on the string when the car is at rest and $62 c m / \sec$ when the car accelerates on a horizontal road. The acceleration of the car is _________ $m / \sec^{2}$ .

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answer is 3.65.

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Detailed Solution

$I n i t i a l l y, T = m g F i n a l l y, T^{1} = \sqrt{{(m g)}^{2} + {(m a)}^{2}} = m \sqrt{g^{2} + a^{2}} G i v e n : V = 60 c m / \sec, V^{1} = 62 c m / \sec W e k n o w, V = \sqrt{\frac{T}{μ}} ∴ \frac{V^{1}}{V} = \sqrt{\frac{T^{1}}{T}} \Rightarrow \frac{62}{60} = \sqrt{\frac{m {(g^{2} + a^{2})}^{1 / 2}}{m g}} \Rightarrow 1 + \frac{2}{60} = {(1 + \frac{a^{2}}{g^{2}})}^{1 / 4} \Rightarrow {(1 + \frac{1}{30})}^{4} = 1 + \frac{a^{2}}{g^{2}} \Rightarrow (1 + \frac{4}{30}) = 1 + \frac{a^{2}}{g^{2}} \Rightarrow a^{2} = \frac{2}{15} g^{2} \Rightarrow a = \sqrt{\frac{2}{15}} \times g \Rightarrow a = 3.65 m / \sec^{2}$