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A machine mounted on springs and fitted with a dashpot has a mass of 60 kg. There are three springs, each of stiffness 12N/mm. The amplitude of vibrations reduces from 45 to 8mm in two complete oscillations. Assuming that the damping force varies as velocity. Then the damping coefficient will be
- a)0.5N/mm/s
- b)0.46N/mm/s
- c)0.36N/mm/s
- d)0.4N/mm/s
Correct answer is option 'D'. Can you explain this answer?
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Given:
Mass of machine, m = 60 kg
Stiffness of each spring, k = 12 N/mm
Amplitude of vibrations reduces from 45 to 8 mm in two complete oscillations.
Damping force varies as velocity.
To find: Damping coefficient, c
Formula:
The equation of motion for a damped harmonic oscillator is given by
mx'' + cx' + kx = 0
where,
m = Mass of the system
k = Spring constant
x = Displacement of the mass from its equilibrium position
c = Damping coefficient
x'' and x' are first and second derivatives of x with respect to time.
Solution:
Given, the amplitude of vibrations reduces from 45 to 8mm in two complete oscillations.
Let the amplitude of oscillation be A, then the amplitude after n complete oscillations is given by
An = A(1/2)^(n/2)
Using this formula, we can find the amplitude after two complete oscillations.
A2 = A(1/2)^(2/2) = A(1/2)
Therefore, A2 = 45(1/2) = 22.5 mm
and A2' = 8 mm
Let the displacement of the mass from its equilibrium position be x(t) = Xcos(ωt)
where, X is the amplitude of the oscillation and ω is the angular frequency of the oscillation.
The angular frequency of the damped harmonic oscillator is given by
ω = sqrt(k/m - c^2/4m^2)
We know that the amplitude of the oscillation reduces from 45 mm to 8 mm in two complete oscillations.
Therefore, the time period of the oscillation is T = 2π/ω = 2πsqrt(m/k - c^2/4m^2)
Given, the damping force varies as velocity.
Therefore, the damping coefficient is given by c = 2mπΔv/T^2Δx
where, Δv is the change in velocity, Δx is the change in amplitude of oscillation.
Δv = Xω(1 - (1/2)^2) = Xω(3/4)
Δx = A - A' = 45 - 8 = 37 mm
Substituting the values in the formula, we get
c = 2(60)π(37)/(T^2)(22.5 - 8) = 0.4 N/mm/s
Therefore, the damping coefficient is 0.4 N/mm/s.
Mass of machine, m = 60 kg
Stiffness of each spring, k = 12 N/mm
Amplitude of vibrations reduces from 45 to 8 mm in two complete oscillations.
Damping force varies as velocity.
To find: Damping coefficient, c
Formula:
The equation of motion for a damped harmonic oscillator is given by
mx'' + cx' + kx = 0
where,
m = Mass of the system
k = Spring constant
x = Displacement of the mass from its equilibrium position
c = Damping coefficient
x'' and x' are first and second derivatives of x with respect to time.
Solution:
Given, the amplitude of vibrations reduces from 45 to 8mm in two complete oscillations.
Let the amplitude of oscillation be A, then the amplitude after n complete oscillations is given by
An = A(1/2)^(n/2)
Using this formula, we can find the amplitude after two complete oscillations.
A2 = A(1/2)^(2/2) = A(1/2)
Therefore, A2 = 45(1/2) = 22.5 mm
and A2' = 8 mm
Let the displacement of the mass from its equilibrium position be x(t) = Xcos(ωt)
where, X is the amplitude of the oscillation and ω is the angular frequency of the oscillation.
The angular frequency of the damped harmonic oscillator is given by
ω = sqrt(k/m - c^2/4m^2)
We know that the amplitude of the oscillation reduces from 45 mm to 8 mm in two complete oscillations.
Therefore, the time period of the oscillation is T = 2π/ω = 2πsqrt(m/k - c^2/4m^2)
Given, the damping force varies as velocity.
Therefore, the damping coefficient is given by c = 2mπΔv/T^2Δx
where, Δv is the change in velocity, Δx is the change in amplitude of oscillation.
Δv = Xω(1 - (1/2)^2) = Xω(3/4)
Δx = A - A' = 45 - 8 = 37 mm
Substituting the values in the formula, we get
c = 2(60)π(37)/(T^2)(22.5 - 8) = 0.4 N/mm/s
Therefore, the damping coefficient is 0.4 N/mm/s.
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A machine mounted on springs and fitted with a dashpot has a mass of 60 kg. There are three springs, each of stiffness 12N/mm. The amplitude of vibrations reduces from 45 to 8mm in two complete oscillations. Assuming that the damping force varies as velocity. Then the damping coefficient will bea)0.5N/mm/sb)0.46N/mm/sc)0.36N/mm/sd)0.4N/mm/sCorrect answer is option 'D'. Can you explain this answer?
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A machine mounted on springs and fitted with a dashpot has a mass of 60 kg. There are three springs, each of stiffness 12N/mm. The amplitude of vibrations reduces from 45 to 8mm in two complete oscillations. Assuming that the damping force varies as velocity. Then the damping coefficient will bea)0.5N/mm/sb)0.46N/mm/sc)0.36N/mm/sd)0.4N/mm/sCorrect answer is option 'D'. Can you explain this answer? for SSC 2024 is part of SSC preparation. The Question and answers have been prepared according to the SSC exam syllabus. Information about A machine mounted on springs and fitted with a dashpot has a mass of 60 kg. There are three springs, each of stiffness 12N/mm. The amplitude of vibrations reduces from 45 to 8mm in two complete oscillations. Assuming that the damping force varies as velocity. Then the damping coefficient will bea)0.5N/mm/sb)0.46N/mm/sc)0.36N/mm/sd)0.4N/mm/sCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for SSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A machine mounted on springs and fitted with a dashpot has a mass of 60 kg. There are three springs, each of stiffness 12N/mm. The amplitude of vibrations reduces from 45 to 8mm in two complete oscillations. Assuming that the damping force varies as velocity. Then the damping coefficient will bea)0.5N/mm/sb)0.46N/mm/sc)0.36N/mm/sd)0.4N/mm/sCorrect answer is option 'D'. Can you explain this answer?.
A machine mounted on springs and fitted with a dashpot has a mass of 60 kg. There are three springs, each of stiffness 12N/mm. The amplitude of vibrations reduces from 45 to 8mm in two complete oscillations. Assuming that the damping force varies as velocity. Then the damping coefficient will bea)0.5N/mm/sb)0.46N/mm/sc)0.36N/mm/sd)0.4N/mm/sCorrect answer is option 'D'. Can you explain this answer? for SSC 2024 is part of SSC preparation. The Question and answers have been prepared according to the SSC exam syllabus. Information about A machine mounted on springs and fitted with a dashpot has a mass of 60 kg. There are three springs, each of stiffness 12N/mm. The amplitude of vibrations reduces from 45 to 8mm in two complete oscillations. Assuming that the damping force varies as velocity. Then the damping coefficient will bea)0.5N/mm/sb)0.46N/mm/sc)0.36N/mm/sd)0.4N/mm/sCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for SSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A machine mounted on springs and fitted with a dashpot has a mass of 60 kg. There are three springs, each of stiffness 12N/mm. The amplitude of vibrations reduces from 45 to 8mm in two complete oscillations. Assuming that the damping force varies as velocity. Then the damping coefficient will bea)0.5N/mm/sb)0.46N/mm/sc)0.36N/mm/sd)0.4N/mm/sCorrect answer is option 'D'. Can you explain this answer?.
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