Correct Answer - Option 1 : 1
CONCEPT:
-
Oscillatory motion is regularly repeating the periodic motion, In an oscillatory motion, an object oscillates about an equilibrium point with a maximum and minimum displacement due to the action of restoring force.
- The amplitude of oscillations decreases in a medium due to the presence of resistive force in the medium and is known as damping.
- The resistive force decreases the energy of the oscillator while oscillating through the medium.
- The decrease in energy of the oscillator due to damping is given by
\( ⇒ E(t) = E(0) e^{\frac{-bt}{2m}}\)
Where E(t) = Energy for the time t,E(0) = Initial energy, b = Damping coefficient, t = time, and m = mass
CALCULATION:
Given - m = 150 g = 0.150 Kg, K = 75 Nm-1 and b = 35gs-1, E(t) = \(\frac{3E(0)}{4}\)
- The decrease in the energy of the oscillator is given by
\( ⇒ E(t) = E(0) e^{\frac{-bt}{2m}}\)
Substituting the given values in the above equation
\( ⇒ \frac{3 E(0)}{4} = E(0) e^{-\frac{35\;\times \;t}{2\;\times\; 0.15}}\)
\( ⇒ \frac{3}{4} = e^{-\frac{35\;\times\; t}{0.3}}\)
Taking the logarithm on both the sides
⇒ log(\(\frac{3}{4}\)) = log\(e^{-\frac{35\;\times\; t}{300}}\)
\(⇒ log 3 - log 4 =-\frac{35\;\times\; t}{300}\)
\(⇒ -0.1249 =-\frac{35\;\times \;t}{300}\)
The above equation can be rewritten for t as
\(⇒ t = \frac{0.1249\;\times\; 300}{35} = 1.07 \,Sec\)
- Hence option 1 is the answer
