Question

Equation $F=-bv-kx$ represents equation of a damped oscillation for a particle of 2 kg mass where $b=\ln 2\frac{N.S.}{m}$ and $k=100N/m$ then time after which energy of oscillations will be reduced to half of initial is :

• A. $ln2$ sec
• B. $2$ sec
• C. $2ln2$ sec
• D. $1$ sec

Answer 1

The correct option is B $2$ sec

For damped oscillation for a particle of $m=2㎏$.

$F=-bV-kx$

$\therefore$ Amplitude ca be written as $A=A_0{e}^{-\frac{bt}{2m}}$

Energy $E\propto A^2$

$\therefore$ Energy $E=E_0{e}^{-\frac{bt}{m}}$

Let after time $t_0$ its energy becomes half of initial energy.

$\therefore \frac{E_0}{2}=E_0{e}^{-\frac{b{t}_0}{m}}$

$\therefore ln2=\frac{b{t}_0}{m}$

$\therefore t_0=\frac{m\times ln2}{b}$

$\therefore t_0=\frac{2\times ln2}{ln2}=2sec$

$\therefore t_0=2sec$