Question

Which of the following expression represent the angular velocity of damped harmonic oscillator 
1. \(\omega = \sqrt{\frac{k}{m} - \frac{b^{2}}{4m^{2}}}\)
2. \(\omega = \sqrt{\frac{k}{m}}\)
3. ω² = \(\frac{k}{m}\)
4. \(\omega =\frac{2\pi}{T}\)

Answer 1

Correct Answer - Option 1 : \(\omega = \sqrt{\frac{k}{m} - \frac{b^{2}}{4m^{2}}}\)

CONCEPT:

• 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 equation of motion damped oscillator is given by

⇒ ma = -kx - bv

\(⇒ m \frac{d ^{2}x}{dt^{2}}+b\frac{dx}{dt}+kx = 0\)

• The solution of the above differential is given by

\(⇒ x(t) = Ae^\frac{-bt}{2m} Cos(ω t +\phi) \)

EXPLANATION:

• The expression of ω for a damped oscillator is given by

\(\Rightarrow \omega = \sqrt{\frac{k}{m} - \frac{b^{2}}{4m^{2}}}\)

• The term \(\frac{b^{2}}{4m^{2}}\)represents damping. hence option 1 is the answer
• Options two and three are the same equations \(\omega = \sqrt{\frac{k}{m}}\)written in alternative forms and does not include the damping term, hence option 2 and three are not the answers
• Option 4 does not include the term of damping hence option 4 is incorrect