Question

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 and what is its amplitude?

Accepted Answer

Step 1: Identify the key parameters of the problem. The mass of the pendulum bob is $$ m = 110 \; 	ext{kg} $$, the length of the pendulum is $$ L = 15.0 \; 	ext{m} $$, the initial displacement (amplitude) is $$ A_0 = 1.5 \; 	ext{m} $$, and the damping constant is $$ b = 0.010 \; 	ext{kg/s} . $$The time interval of interest is from 8:00 a.m. to 12:00 noon, which is $$ t = 4 \; 	ext{hours} = 14400 \; 	ext{seconds} . $$Step 2: Calculate the natural frequency of the pendulum. For a simple pendulum, the angular frequency $$ \omega_0  is $$given by $$ \omega_0 = \sqrt{\frac{g}{L}} $$, where $$ g  is $$the acceleration due to gravity ($$ g = 9.8 \; 	ext{m/s}^2 ). $$Substitute the values of $$ g $$ and $$ L  to $$find $$ \omega_0 . $$Step 3: Determine the damped angular frequency $$ \omega_d . $$The damped angular frequency is given by $$ \omega_d = \sqrt{\omega_0^2 - \left(\frac{b}{2m}ight)^2} . $$Substitute the values of $$ \omega_0 $$, $$ b $$, and $$ m  to $$calculate $$ \omega_d . $$Step 4: Calculate the number of oscillations completed in the given time. The period of the damped oscillation is $$ T_d = \frac{2\pi}{\omega_d} . $$The total number of oscillations is then $$ N = \frac{t}{T_d} . $$Substitute the values of $$ t $$ and $$ T_d  to $$find $$ N . $$Step 5: Determine the amplitude at noon. The amplitude of a damped oscillator decreases exponentially with time according to $$ A(t) = A_0 e^{-\frac{b}{2m}t} . $$Substitute the values of $$ A_0 $$, $$ b $$, $$ m $$, and $$ t  to $$calculate the amplitude $$ A(t)  at$$ $$ t = 14400 \; 	ext{seconds} .$$

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Simple Harmonic Motion (SHM)

Damping

Amplitude