# At exactly 12:00 noon, how many oscillations will the pendulum have completed? And what is its amplitude?

In a science museum, a 110 kg brass pendulum bob swings at the end of a 13.2 m -long wire. The pendulum is started at exactly 8:00 a.m. every morning by pulling it 1.7 m to the side and releasing it. Because of its compact shape and smooth surface, the pendulum's damping constant is only 0.010kg/s.

A. At exactly 12:00 noon, how many oscillations will the pendulum have completed?

B. And what is its amplitude?

(A) The expression of time period of the pendulum is,

T=2 \pi \sqrt{\frac{l}{g}}

The expression of the frequency is,

f=\frac{1}{T}

Substitute 2 \pi \sqrt{\frac{l}{g}} for T.

f=\frac{1}{2 \pi} \sqrt{\frac{g}{l}}

Substitute 13.2 \mathrm{~m} for \mathrm{l} and 9.8 \mathrm{~m} / \mathrm{s}^{2} for \mathrm{g} to find frequency.

f=\frac{1}{2 \pi} \sqrt{\frac{9.8 \mathrm{~m} / \mathrm{s}^{2}}{13.2 \mathrm{~m}}}

=0.137 oscillation/ \mathrm{s}

The time interval between 8:00 AM to 12.00 PM is,

t=4 \mathrm{~h}

The number of oscillations completed in time t is, N=f t

Substitute 0.137 oscillation/s for f and 4 \mathrm{~h} for t.

N=(0.137 oscillation/ \mathrm{s})\left(4 \mathrm{~h}\left(\frac{3600 \mathrm{~s}}{1 \mathrm{~h}}\right)\right)

=1973 oscillation

The number of oscillations of the pendulum is 1973

(B) The expression of the amplitude at time t is,

x(t)=A \exp \left(-\frac{b t}{2 m}\right)

Substitute 110 \mathrm{~kg} for \mathrm{m}, 1.7 \mathrm{~m} for \mathrm{A}, 0.010 \mathrm{~kg} / \mathrm{s} for \mathrm{b}, and 4 \mathrm{~h} for \mathrm{t}.

x(t) =(1.7 \mathrm{~m}) \exp \left(-\frac{(0.010 \mathrm{~kg} / \mathrm{s})\left(4 \mathrm{~h}\left(\frac{3600 \mathrm{~s}}{1 \mathrm{~h}}\right)\right)}{2(110 \mathrm{~kg})}\right)

=0.883 \mathrm{~m}\left(\frac{10^{-2} \mathrm{~m}}{1 \mathrm{~m}}\right)

=88.3 \mathrm{~cm}

The amplitude is 88.3 \mathrm{~cm}.

### Raymond Puzio

Raymond Puzio has a PhD in Physics from Yale University. I have been creating PlanetPhysics with Aaron Krowne and Ben Loftin since 2005.