Вопрос
The frequency of free oscilla tions of a mathematical pendulum is 8 Hz. Wh at will be the frequency of if both the m ass of the load and the length of the thread are inc reased by 4 times? Select one answer: a. 16 Hz b. 4 Hz c. 8 Hz d.2Hz
Решения
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мастер · Репетитор 5 летЭкспертная проверка
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The frequency of a mathematical pendulum is given by the formula:<br /><br />\[ f = \frac{1}{2\pi} \sqrt{\frac{g}{L}} \]<br /><br />where \( f \) is the frequency, \( g \) is the acceleration due to gravity, and \( L \) is the length of the pendulum.<br /><br />In this case, both the mass of the load and the length of the thread are increased by 4 times. However, the frequency of a pendulum is independent of its mass. Therefore, increasing the mass of the load will not affect the frequency.<br /><br />The length of the thread is increased by 4 times, which means \( L \) becomes \( 4L \). Substituting this into the formula, we get:<br /><br />\[ f' = \frac{1}{2\pi} \sqrt{\frac{g}{4L}} = \frac{1}{2\pi} \cdot \frac{1}{2} \sqrt{\frac{g}{L}} = \frac{1}{2} \cdot \frac{1}{2\pi} \sqrt{\frac{g}{L}} = \frac{1}{2} \cdot f \]<br /><br />Since the original frequency \( f \) is 8 Hz, the new frequency \( f' \) will be:<br /><br />\[ f' = \frac{1}{2} \cdot 8 \text{ Hz} = 4 \text{ Hz} \]<br /><br />Therefore, the correct answer is:<br /><br />b. 4 Hz
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