Tag: free, forced and damped oscillations

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Questions Related to free, forced and damped oscillations

Multiple choice physics oscillations introduction to sound free, forced and damped oscillations resonance

The equation of a damped simple harmonic motion is $ m \frac {d^2x}{dt^2} + b \frac {dx}{dt} + kx=0 . $ Then the angular frequency of oscillation is:

  1. $ \omega = ( \frac {k}{m}+\frac {b}{4m})^{1/2} $
  2. $ \omega = ( \frac {k}{m}-\frac {b}{4m})^{1/2} $
  3. $ \omega = ( \frac {k}{m}+\frac {b^2}{4m})^{1/2} $
  4. $ \omega = ( \frac {k}{m}-\frac {b^2}{4m^2})^{1/2} $
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The equation of motion is m*x'' + b*x' + k*x = 0. The angular frequency of the damped oscillation is omega = sqrt(k/m - (b/2m)^2) = sqrt(k/m - b^2/(4m^2)).

Multiple choice physics oscillations introduction to sound free, forced and damped oscillations resonance

The amplitude of a damped oscillator decreases to $0.9$ times to its original magnitude in $5s$. In another $10s$, it will decrease to $\alpha$ times to its original magnitude, where $\alpha$ equals.

  1. $0.7$
  2. $0.81$
  3. $0.729$
  4. $0.6$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A(5) = 0.9 * A0. A(15) = A0 * (exp(-5k))^3 = A0 * (0.9)^3 = 0.729 * A0. Thus alpha = 0.729.

Multiple choice physics oscillations introduction to sound free, forced and damped oscillations resonance

In damped oscillations, the amplitude after $50$ oscillations is $0.8\;a _0$, where $a _0$ is the initial amplitude, then the amplitude after $150$ oscillations is

  1. $0.512\;a _0$
  2. $0.280\;a _0$

  3. Zero
  4. $a _0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The amplitude, a, at time $t$ is given by $a=a _0\;exp(-\,\alpha t)$



$a _{50}=a _0\;exp(-\alpha\times 50T)=0.80\;a _0$



where $T$ is the period of oscillation



$a _{150}=a _0\;exp(-a\times 150T)$



$=a _0\;(0.8)^3=0.512\,a _0$

Multiple choice physics oscillations introduction to sound free, forced and damped oscillations resonance

When an oscillator completes $100$ oscillations its amplitude reduces to $\displaystyle\dfrac{1}{3}$ of its initial value. What will be its amplitude when it completes $200$ oscillations?

  1. $\displaystyle\dfrac{1}{8}$
  2. $\displaystyle\dfrac{2}{3}$
  3. $\displaystyle\dfrac{1}{6}$
  4. $\displaystyle\dfrac{1}{9}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation


Its is a damped oscillation, where amplitude of oscillation at time $t$ is given by $A = a _0e^{-\gamma t}$
where $a _0 = $ initial amplitude of oscillation
$\quad \gamma = $ damping constant
As per question, $\displaystyle\dfrac{a _0}{3} = a _0e^{-\gamma100/v}\quad                    ...(i)$
(where $v$ is the frequency of oscillation)
and $A = a _0e^{-\gamma200/v} \quad                ...(ii)$
From $(i)$; $\quad \displaystyle\dfrac{a _0}{3} = a _0e^{-\gamma\times100/v} \quad            ...(iii)$
Dividing equation $(ii)$ by $(iii)$, we have

$\quad \displaystyle\dfrac{A}{a _0(1/3)} = \displaystyle\dfrac{e^{-\gamma\times200/v}}{e^{-\gamma\times100/v}} = e^{-\gamma\times100/v} = \displaystyle\dfrac{1}{3}$

or $A = a _0\times\displaystyle\dfrac{1}{3}\times\displaystyle\dfrac{1}{3} = \displaystyle\dfrac{1}{9}a _0$


Multiple choice physics option b: engineering physics introduction to sound free, forced and damped oscillations resonance

In reality, a spring won't oscillate for ever.               will                the amplitude of oscillation until eventually the system is at rest.

  1. Frictional force, increase

  2. Viscous force, decrease

  3. Frictional force, decrease

  4. Viscous force, increase
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In reality, a spring won't oscillate forever. Frictional force will decrease the oscillation until eventually, the system is at rest.

Multiple choice physics option b: engineering physics introduction to sound free, forced and damped oscillations resonance

Undamped oscillations are practically impossible because

  1. there is always loss of energy.

  2. there is no force opposing friction.

  3. energy is not conserved in such oscillations.

  4. None of these.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Underdamped oscillation is practical because there always be resistive force present in reality which will try to make an oscillating body to lose its energy. This loss of energy makes the motion damped motion.

Multiple choice physics option b: engineering physics introduction to sound free, forced and damped oscillations resonance

If we wish to represent the equation for the position of the mass in terms of a differential equation, which one of these would be the most suitable?

  1. $ m \dfrac{d^2x}{dt^2} + b \dfrac{dx}{dt} + kx = 0$
  2. $ m \dfrac{d^2x}{dt^2} - b \dfrac{dx}{dt} + kx = 0$
  3. $ m \dfrac{d^2x}{dt^2} + b \dfrac{dx}{dt} - kx = 0$
  4. $ m \dfrac{d^2x}{dt^2} -b \dfrac{dx}{dt} - kx = 0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The force on body oscillating in resistive medium is 

$f = -kx - bv$
$\Rightarrow m\dfrac { { d }^{ 2 }x }{ d{ t }^{ 2 } } =-kx-b\dfrac { dx }{ dt } \ \Rightarrow m\dfrac { { d }^{ 2 }x }{ d{ t }^{ 2 } } +b\dfrac { dx }{ dt } +kx=0$
k = oscillating constant 
x = displacement of body from mean position 
b = constant depends on resistive medium 
v = velocity of object = $\dfrac{dx}{dt}$
m = mass of object .