To solve this problem, we need to determine the new string length required for the pendulum to perform 40 oscillations in 10 seconds, given that it currently performs 20 oscillations in the same duration with a 30 cm string.
- The formula for the time period \((T)\) of a simple pendulum, assuming small oscillations, is given by: \(T = 2\pi \sqrt{\frac{L}{g}}\)
where \(L\) is the length of the pendulum and \(g\) is the acceleration due to gravity. - The time period \((T)\) is the time for one complete oscillation. Thus, the frequency \((f)\) of oscillation, which is the number of oscillations per unit time, is given by: \(f = \frac{1}{T}\)
- Let \(T_1\) and \(f_1\) be the time period and frequency for the initial conditions: \(f_1 = \frac{20}{10} = 2 \text{ Hz}\)
- The relationship between frequency and length for pendulum oscillations is: \(f \propto \frac{1}{\sqrt{L}}\)
- Let \(L_2\) be the new length required for 40 oscillations in 10 seconds, giving us \(f_2 = \frac{40}{10} = 4 \text{ Hz}\)
- The proportionality between the frequencies and the square roots of the lengths is: \(\frac{f_1}{f_2} = \sqrt{\frac{L_2}{L_1}}\)
- Substitute known values to find \(L_2\):
- \(\frac{2}{4} = \sqrt{\frac{L_2}{30}}\)
- Simplify: \(\frac{1}{2} = \sqrt{\frac{L_2}{30}}\)
- Square both sides: \(\frac{1}{4} = \frac{L_2}{30}\)
- Arrange to find \(L_2\): \(L_2 = 30 \times \frac{1}{4} = 120 \text{ cm}\)
- Thus, the required string length for the pendulum to perform 40 oscillations in 10 seconds is 120 cm.
Hence, the correct answer is 120 cm.
Similar Questions
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$x(t) = A \sin (at + \delta)$
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Identify the correct match below :
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Important Questions from Oscillations and Waves
- Two tuning forks $A$ and $B$ are sounded together giving rise to $8$ beats in $2\text{ s}$. When fork $A$ is loaded with wax, the beat frequency is reduced to $4$ beats in $2\text{ s}$. If the original frequency of tuning fork $B$ is $380\text{ Hz}$ then original frequency of tuning fork $A$ is _________ $\text{Hz}$.
- The velocity of sound in air is doubled when the temperature is raised from $0^\circ\text{C}$ to $\alpha^\circ\text{C}$. The value of $\alpha$ is ________.
- The equation of a transverse wave is $y = y_0 \sin 2\pi(ft - \frac{x}{\lambda})$. If the maximum particle velocity be four times that of wave velocity then
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- 5 beats/second are heard when a turning fork is sounded with a sonometer wire under tension, when the length of the sonometer wire is either 0.95 m or 1 m. The frequency of the fork will be
