Answer-
After 76 swings the angle through which it swings less than 1°
Solution-
From the question,
Angle of the first of swing = 30° and then each succeeding oscillation is through 95% of the angle of the one before it.
So the angle of the second swing = [tex](30\times \frac{95}{100})^{\circ}[/tex]
Then the angle of third swing = [tex](30\times (\frac{95}{100})^2)^{\circ}[/tex]
So, this follows a Geometric Progression.
[tex](30,\ 30\cdot \frac{95}{100},\ 30\cdot (\frac{95}{100})^2............,0)[/tex]
a = The initial term = 30
r = Common ratio = [tex]\frac{95}{100}[/tex]
As we have to find the number swings when the angle swept by the pendulum is less than 1°.
So we have the nth number is the series as 1, applying the formula
[tex]T_n=ar^{n-1}[/tex]
Putting the values,
[tex]\Rightarrow 1=30(\frac{95}{100})^{n-1}[/tex]
[tex]\Rightarrow \frac{1}{30} =(\frac{95}{100})^{n-1}[/tex]
Taking logarithm of both sides,
[tex]\Rightarrow \log \frac{1}{30} =\log (\frac{95}{100})^{n-1}[/tex]
[tex]\Rightarrow \log \frac{1}{30} =(n-1)\log (\frac{95}{100})[/tex]
[tex]\Rightarrow -1.5=(n-1)(-0.02)[/tex]
[tex]\Rightarrow 1.5=(n-1)(0.02)[/tex]
[tex]\Rightarrow n-1=\dfrac{1.5}{0.02}[/tex]
[tex]\Rightarrow n-1=75[/tex]
[tex]\Rightarrow n=76[/tex]
Therefore, after 76 swings the angle through which it swings less than 1°
