Verified Answer :
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In a science museum, a 110 kg brass pendulum bob swings at the end of a 15.0-m-long wire. The pendulum is started at exactly 8:00 a.m. every morning by pulling it 1.5 m to the side and releasing it. Because of its compact shape and smooth surface, the pendulum's damping constant is only 0.010 kg/s.
At exactly 12:00 noon, how many oscillations will the pendulum have completed?
ANSWER:
The damping time constant is
m/b=11000s.n t2 4 x 3600 \(\sqrt{9.8/ 15 - 0.01 x^{2}/ 4 x 110 x^{2} /2 \pi\) = 1852.5
The pendulum will complete \(\boxed{1872}\) oscillations between \(8:00\,{\text{am}}\) to \(12:00\,{\text{pm}}\) .
Further Explanation:
Given:
The length of the simple pendulum is \(15\,{\text{m}}\) .
The pendulum is started at \(8:00\,{\text{am}}\) .
The pendulum is stopped at \(12:00\,{\text{pm}}\) .
Concept:
The frequency of oscillation of the simple pendulum is given by:
\(f=\frac{1}{{2\pi}}\sqrt{\frac{g}{l}}\)
(1)
Here, \(f\) is the frequency of oscillation of pendulum, \(g\) is the acceleration due to gravity and \(l\) is the length of the pendulum.
The value of acceleration due to gravity on the surface of Earth is \(9.8\,{{\text{m}}\mathord{\left/{\vphantom{{\text{m}}{{{\text{s}}^{\text{2}}}}}}\right.\kern-
ulldelimiterspace}{{{\text{s}}^{\text{2}}}}}\)
Substitute the values in equation (1).
\(\begin{aligned}f&=\frac{1}{{2\pi}}\sqrt{\frac{{9.8}}{{15}}}\\&=\frac{1}{{2\pi}}\left({0.808}\right)\\&\approx0.13\,{\text{Hz}}\\\end{aligned}\)
It means that pendulum makes \(0.13\) oscillation in \(1\,\sec\) .
The number of seconds from \(8:00\,{\text{am}}\) to \(12:00\,{\text{pm}}\) is:
\(\begin{aligned}t&=\left({12 - 8}\right)\left({3600\,}\right)\,{\text{s}}\\&=4\times3600\,{\text{s}}\\&=14400\,{\text{s}}\\\end{aligned}\)
So number of oscillations completed by the pendulum is:
\(\begin{aligned}n7&=f\times t\\&=0.13\times14400\\&=1872\,{\text{oscillations}}\\\end{aligned}\)
Thus, the pendulum will complete \(\boxed{1872}\) oscillations from \(8:00\,{\text{am}}\) to \(12:00\,{\text{pm}}\) .
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Answer details:
Grade: High School
Subject: Physics
Chapter: Simple Pendulum
Keywords:
Pendulum, oscillations, 8:00 am, 12:00 pm, length of pendulum 15m, 110kg, brass pendulum, bob swings, frequency of pendulum, number of oscillations.