Choose the correct statements regarding the expression x(t)=A e-b t / 2 m cos (ω prime t+φ). Here, ω prime=√(k/m)-(b2/4 m2), (b/√k m) is much less than 1 .

Question

Choose the correct statements regarding the expression x(t)=A e-b t / 2 m cos (ω prime t+φ). Here, ω prime=√(k/m)-(b2/4 m2), (b/√k m) is much less than 1 . (A) x(t) is strictly periodic always. (B) x(t) is approximately periodic. (C) Amplitude of damped oscillation represented by above expression is constant. (D) None of the above

Tardigrade Admin · Accepted Answer

The given expression represents the displacement for a damped oscillator. In this amplitude for a damped oscillation varies as

A e^{- b t /2 m}

. But this oscillation is approximately periodic, if the damping is small. So, for the given displacement variation w.r.t. time is shown in the graph below.

From this, we can conclude that the amplitude of the damped oscillator decreases with time.

Q. Choose the correct statements regarding the expression $x (t) = A e^{- b t /2 m} cos (ω^{'} t + ϕ)$ . Here, $ω^{'} = \frac{k}{m} - \frac{b ^{2}}{4 m ^{2}}, \frac{b}{km}$ is much less than 1 .

$x (t)$ is strictly periodic always.

$x (t)$ is approximately periodic.

Amplitude of damped oscillation represented by above expression is constant.

None of the above

Q. Choose the correct statements regarding the expression x(t)=Ae−bt/2mcos(ω′t+ϕ). Here, ω′=mk​−4m2b2​​,km​b​ is much less than 1 .

x(t) is strictly periodic always.

x(t) is approximately periodic.

Amplitude of damped oscillation represented by above expression is constant.

None of the above

Q. Choose the correct statements regarding the expression $x (t) = A e^{- b t /2 m} cos (ω^{'} t + ϕ)$ . Here, $ω^{'} = \frac{k}{m} - \frac{b ^{2}}{4 m ^{2}}, \frac{b}{km}$ is much less than 1 .

$x (t)$ is strictly periodic always.

$x (t)$ is approximately periodic.