Answer:
33.33 seconds
Explanation:
[tex]N=\dfrac{1}{e}N_0[/tex]
[tex]N_0[/tex] = Initial length pulled = 20 cm
b = Damping constant = 0.015 kg/s
k = Spring constant = 4 N/m
m = Mass of glider = 250 g
Time period is given by
[tex]T=2\pi\sqrt{\dfrac{m}{k}}\\\Rightarrow T=2\pi\sqrt{\dfrac{0.25}{4}}\\\Rightarrow T=1.57079632679\ s[/tex]
Using exponential decay formula
[tex]N=N_0e^{\dfrac{-bt}{m}}[/tex]
Final amplitude = Initial times decay
[tex]\dfrac{1}{e}0.2=0.2e^{\dfrac{-0.015t}{2\times 0.25}}\\\Rightarrow 0.2=0.2e^{\frac{-0.015t}{2\cdot \:0.25}+1}\\\Rightarrow e^{\frac{-0.015t}{2\cdot \:0.25}+1}=1\\\Rightarrow \ln \left(e^{\frac{-0.015t}{2\cdot \:0.25}+1}\right)=\ln \left(1\right)\\\Rightarrow \left(\frac{-0.015t}{2\cdot \:0.25}+1\right)\ln \left(e\right)=\ln \left(1\right)\\\Rightarrow \frac{-0.015t}{2\cdot \:0.25}+1=\ln \left(1\right)\\\Rightarrow -\frac{0.015t}{0.5}=-1\\\Rightarrow -0.000225t=-0.0075\\\Rightarrow t=33.33\ s[/tex]
The time taken is 33.33 seconds
