Question

Equation $F = - bv - kx$ represents equation of a damped oscillation for a particle of $2kg$ mass where $b = \ln 2\dfrac{{N.S.}}{m}$and $k = 100N/m$ then time after which energy of oscillations will be reduced to half of initial is:
A) $\ln 2\sec $
B) $2\sec $
C) $2\ln 2\sec $
D) $1\sec $

Answer 1

Hint: The total energy of a harmonic oscillator always remains constant. The reduction in the energy or the amplitude of an oscillator is known as damping. The oscillations thus produced are thus said to be damped. The damping forces oppose the motion of the body and its direction is also opposite to the velocity.

Complete step by step solution:
Given that the mass of the particle for damped oscillation is $m = 2kg$
Also the given equation is
$F = - bv - kx$
The amplitude can be written as
$A = {A_0}{e^{ - \dfrac{{bt}}{{2m}}}}$---(i)
Energy $E \propto {A^2}$
Also the energy decays exponentially in a damped harmonic oscillator. Therefore the expression for energy can be written as
$E = {E_0}{e^{ - \dfrac{{bt}}{m}}}$---(ii)
Where ‘t’ is the time constant of energy and ‘m’ is the mass
Let after a time ${t_0}$the energy becomes half of the initial energy. The equation (ii) can be written as
$ \Rightarrow \dfrac{{{E_0}}}{2} = {E_0}{e^{ - \dfrac{{b{t_0}}}{m}}}$
$ \Rightarrow \dfrac{1}{2} = {e^{ - \dfrac{{b{t_0}}}{m}}}$
$ \Rightarrow \ln 2 = \dfrac{{b{t_0}}}{m}$
$ \Rightarrow {t_0} = \dfrac{{\ln 2 \times m}}{b}$---(iii)
Given that $b = \dfrac{{\ln 2}}{m}N.S.$
Substitute the value of ‘b’, in equation (iii),
${t_0} = \dfrac{{2 \times \ln 2}}{{\ln 2}}$
$ \Rightarrow {t_0} = 2\sec $
$\therefore $the time after which the energy of oscillations will be reduced is $ = 2\sec $

Option B is the right answer.

Note: It is to be remembered that the damping force does negative work on the system and the energy is lost in this type of oscillation. The magnitude of a damping force in a damping oscillator is directly proportional to the velocity of the body. In a damping motion, the frictional force present in the system acts such that the amplitude of the oscillations decreases with time. It is to be noted that the term damping and friction are different. Friction can be the cause of damping in a system.