Damped Simple Harmonic Motion MCQ Quiz in বাংলা - Objective Question with Answer for Damped Simple Harmonic Motion - বিনামূল্যে ডাউনলোড করুন [PDF]

Concept:

Damped simple harmonic Motion:

• A damped oscillation means an oscillation that fades away with time.
• Examples include a swinging pendulum, a weight on a spring, and also a resistor-inductor-capacitor (RLC) circuit.
• The damped harmonic oscillator is a good model for many physical systems because most systems both obey Hooke's law when perturbed about an equilibrium point and also lose energy as they decay back to equilibrium.
• These two conditions are sufficient to obey the equation of motion of the damped harmonic oscillator.

Overdamped oscillation:

• When the damping force is greater than the critical damping force.
• This results in the oscillation decaying quickly.​
   

Critically damped oscillation:

• When the damping force is equal to the critical damping force.
• This results in the oscillation decaying at the fastest possible rate.
   

Underdamped oscillation:

•  When the damping force is less than the critical damping force.
• This results in the oscillation decaying slowly.​

Explanation:

From the above figure, the transition stage between aperiodic and damped oscillatory motion is critically damped simple harmonic motion.

Concept:

Damped Oscillation:

When the motion of a simple pendulum dies out due to air drag and friction, the oscillator and its motion are damped.

→The damping force depends on the nature of the surrounding medium. If the block is immersed in a liquid, the magnitude of damping will be much greater and the dissipation of energy much faster.

→The damping force(F_d) is generally proportional to the velocity of the bob. F_d = -bv where b is the damping constant or decay constant which depends on the medium.

→The amplitude of the damped oscillator decreases gradually with time and can be mathematically expressed as: A₀e^{\(\frac{-bt}{2m}\)}

mass(m) = 500 g 

decay constant (b) = 20 g/s

Let Initial Amplitude be A₀

Amplitude after time 't' (A) = \(\frac{A_0}{2}\)

The Amplitude of damped oscillator is given as   A₀e\(\frac{-bt}{2m}\) 

∴ A = A0e\(\frac{-bt}{2m}\) 

∴\(\frac{A_0}{2}\)= A0e\(\frac{-20t}{2×500}\) 

\(\frac{1}{2}\) = e\(\frac{-t}{50}\) 

Taking natural log both the sides

∴ ln(\(\frac{1}{2}\)) = ln(e^{\(\frac{-t}{50}\)})

ln(\(\frac{1}{2} \)) = \(\frac{-t}{50}\) ln(e)

ln(2) = \(\frac{t}{50}\)    [ ln(a) = -ln(\(\frac{1}{a}\)) & ln(e) = 1 ]  

t = 50× ln(2)

t = 50× 0.693

∴ t = 34.65 s

Hence, the correct option is (1)

Concept Used:

The displacement of a damped oscillator is given by:

x(t) = A₀ exp(-bt) cos(ωt + ϕ)

where A₀ is the initial amplitude, and the amplitude decays exponentially as:

A(t) = A₀ exp(-bt)

We need to find the time t when A(t) = (1 / e¹²) A₀.

Calculation:

Given:

A(t) = A₀ exp(-bt)

Decay constant, b = 0.2

⇒ (1 / e¹²) A₀ = A₀ exp(-0.2t)

Taking natural logarithm on both sides:

⇒ ln(1 / e¹²) = ln(exp(-0.2t))

⇒ -12 = -0.2t

⇒ t = 12 / 0.2

⇒ t = 60 / 10

⇒ t = 6 s

∴ The correct answer is 6 s.

Concept:

• Damping Force: The damping force is given by F = -αx², where α is a proportionality constant and x is the distance from the origin.
• Equation of Motion: According to Newton’s second law, ma = -αx², where a is the acceleration of the particle.
• Acceleration and Force Relation: Acceleration a is the second derivative of displacement x with respect to time: a = d²x/dt².
• Use of Energy Method: The total mechanical energy (kinetic + potential) is considered. Initial kinetic energy K_i is K_i = 1/2 m v₀².
• Work Done by Damping Force: Work done W by the variable damping force is W = ∫₀^x_f F dx = -∫₀^x_f αx² dx.

Calculation:

Since, a = vdv/dx

⇒ \(\rm \int_{V_i}^{V_f} V d v=\int_{X_i}^{X_f} a d x\)

Given:- vi = v0

⇒ Vf = 0

⇒ Xi = 0

⇒ Xf = x

⇒ From Damping Force, a = -αx2/m

⇒ \(\rm \int_{V_o}^O V d v=-\int_O^X \frac{a x^2}{m} d x\)

⇒ \(\rm -v_0^2/2 = (-α/m) [x^3/3]\)

⇒ \(\rm x = [3mv_0^2/2α]1/3\)

Thus, most suitable answer could be (3) as mass ‘m’ is not given in any options.

∴ The correct option is 3

CONCEPT:

Damped Simple Harmonic Motion:

• In damped oscillations, the energy of the system is dissipated continuously; but, for small damping, the oscillations remain approximately periodic.
• The dissipating forces are generally the frictional forces.
• To understand the effect of such external forces on the motion of an oscillator, let us consider a system of the spring-mass system as shown in the figure.
• Here a block of mass m connected to an elastic spring of spring constant k oscillates vertically.
• However, in practice, the surrounding medium (air) will exert a damping force on the motion of the block, and the mechanical energy of the spring-mass system will decrease.
• The energy loss will appear as the heat of the surrounding medium (and the block also).
• The damping force depends on the nature of the surrounding medium.
• If the block is immersed in a liquid, the magnitude of damping will be much greater and the dissipation of energy much faster. The damping force is generally proportional to the velocity of the bob.
•  If the damping force is denoted by Fd, then,

⇒ Fd = -bv

Where b = damping constant, and v = velocity of the block

• The damping constant depends on characteristics of the medium (viscosity, for example) and the size and shape of the block, etc.
• The displacement of the block from its mean position of the spring-mass system at any time t under the influence of a damping force is given as,

⇒ x(t) = Ae-bt/2m.cos(ωt + ϕ)

Where m = mass of the block, A = amplitude, and ω = angular frequency

• The angular frequency of the damped oscillator given by,

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

• For a damped oscillator, the amplitude is not constant but depends on time.
• The amplitude of the damped oscillator at any time t is given as,

⇒ A' = Ae-bt/2m

• The total energy of the damped oscillator at any time t is given as,

\(\Rightarrow E(t)=\frac{1}{2}kA^2e^{-bt/m}\)

• Note that small damping means that the dimensionless ratio \(\frac{b}{\sqrt{km}}\) is much less than 1.

EXPLANATION:

• We know that the total energy of the damped spring-mass system at any time t is given as,

\(\Rightarrow E(t)=\frac{1}{2}kA^2e^{-bt/m}\)     -----(1)

Where k = spring constant, A = initial amplitude, b = damping constant, and m = mass

• By equation 1 it is clear that the total energy of the system decreases exponentially with time. Hence, option 1 is correct.

CONCEPT:

• Oscillation: To and fro movement about a mean position in a regular rhythm is called oscillation.
• Damped Oscillation: When the amplitude of oscillations of a simple harmonic system decreases with time, the oscillations are called damped simple harmonic oscillations.
• Some of the examples of damped oscillations are: the oscillations of the bob of a simple pendulum in the air, stretched guitar string, and a child swinging on a swing.

EXPLANATION:

• Option 1 and 2: In the damped oscillation, the amplitude of the oscillation decreases with time and ultimately becomes zero.
• Typical diagram of amplitude in a damped oscillation

• Option 3: The reason for decreasing amplitude with time is an external force that reduces the amplitude.
• In the oscillations of a bob or other SHM, air friction or other damp forces are present and they act as a damped force.
  • These forces decrease the amplitude of oscillations with time.
• So, The correct answer is option 2.

CONCEPT:

• Damped oscillation: The oscillation of a body whose amplitude goes on decreasing with time are defined as a damped oscillation.

• Forced oscillation: The oscillation in which a body oscillates under the influence of an external periodic force is known as forced oscillation.
• The external agent which exerts the periodic force is called the driver and the oscillating system under consideration is called the driver body.

EXPLANATION:

• The oscillation of a particle with fundamental frequency under the influence of restoring force is defined as free oscillations. Therefore statement 1 is incorrect.

• The oscillation of a body whose amplitude goes on decreasing with time is defined as a damped oscillation. Therefore statement 2 is correct.
• The oscillation in which a body oscillates under the influence of an external periodic force is known as forced oscillation. Therefore statement 3 is incorrect.

CONCEPT:

• Damped oscillation: The oscillation of a body whose amplitude goes on decreasing with time are defined as a damped oscillation.
• Forced oscillation: The oscillation in which a body oscillates under the influence of an external periodic force is known as forced oscillation.
• The external agent which exerts the periodic force is called the driver and the oscillating system under consideration is called the driver body.

EXPLANATION:

• In forced oscillation, the amplitude of the oscillator decreases due to damping forces but on account of the energy gained from the external source, it remains constant. Therefore only statement 1 is correct.
• In damped oscillation, the amplitude of oscillation decreases exponentially due to damping forces like frictional force, viscous force, hysteresis, etc. Due to a decrease in amplitude, the energy of the oscillator also goes on decreasing exponentially.

CONCEPT:

• In the damped simple harmonic motion, the energy of the oscillator dissipates continuously.
• But for small damping, the oscillations remain approximately periodic.

EXPLANATION:

• Guitar string stops oscillating a few seconds after being plucked, because of the friction in the air. So it is an example of damped oscillation.
• Shock absorbers in automobiles, when a car hit any surface shock absorber suppress the vibrations for a pleasant and safe ride. So it is also an example of damped oscillation
• Door with a damper, it stops the door to avoid unnecessary movement of the door. So it is also an example of damped oscillation
• So all are the examples of damped oscillation. So option 4 is correct.

CONCEPT:

Damped vibration (ωd): 

• When the energy of the vibrating system is gradually dissipating (transient) by friction and other resistances, the vibrations are said to be damped.
  • The vibrations gradually cease and the system rests in its equilibrium position.
• Damped vibration of a system is defined as:

\(\Rightarrow {ω _d}\; = \;\sqrt {{{\left( {{ω _n}} \right)}^2}\; - \;{{\left( {\frac{c}{{2m}}} \right)}^2}}\)

Where ω_n = natural frequency  

• Damping coefficient: The measure of the effectiveness of damper, reflects the ability of damper to which it can resist the motion is called the damping coefficient. It is denoted by c.

EXPLANATION:

The general case of damped harmonic motion

\({F_{net}} = m\frac{{{d^2}x}}{{d{t^2}}} + c\frac{{dx}}{{dt}} + kx = 0\)

Where k = the stiffness of spring, c = damping coefficient and m = mass

• Damped vibration of a system is defined as:

\(\Rightarrow {ω _d}\; = \;\sqrt {{{\left( {{ω _n}} \right)}^2}\; - \;{{\left( {\frac{c}{{2m}}} \right)}^2}}\)              -------------- (1)

Damping ratio (ζ): It is defined as the ratio of actual damping to the critical damping.    

\(\Rightarrow ζ = \frac{{actual\;damping}}{{critical\;damping}} = \frac{c}{{{c_c}}}\)

\(\rightarrow \xi \; = \;\frac{c}{{{c_c}}}\; = \;\frac{c}{{2m{ω _n}}}\; = \;\frac{c}{{2\sqrt {km} }} \Rightarrow \frac{c}{{2m}}\; = \;\xi {ω _n}\)

Substitute the value of ζ in equation 1, we get

\(\Rightarrow {ω _d}\; = \;\sqrt {{{\left( {{ω _n}} \right)}^2}\; - \;{{\left( {\frac{c}{{2m}}} \right)}^2}} \; = \;\sqrt {{{\left( {{ω _n}} \right)}^2}\; - \;{{\left( {\xi {ω _n}} \right)}^2}} \; = \;\sqrt {({1\; - \;{\xi ^2}})} {ω _n} \)