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A body of mass 20 kg is suspended from a spring which deflects 15 mm under this load. Calculate the frequency of free vibrations and verify that a viscous damping force amounting to approximately 1000 N at a speed of 1 m/s is just-sufficient to make the motion aperiodic. If when damped to this extent, the body is subjected to a disturbing force with a maximum value of 125 N making 8 cycles/s, find the amplitude of the ultimate motion.
- a)2.66 × 10-3 m
- b)1.94 × 10-3 m
- c)1.96× 10-3 m
- d)2.83× 10-3 m
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
A body of mass 20 kg is suspended from a spring which deflects 15 mm u...
ω = 2π× f = 2π× 8 = 50.3 rad / s
Stiffness ( K ) =
= 1023 N / m / s
Most Upvoted Answer
A body of mass 20 kg is suspended from a spring which deflects 15 mm u...
To calculate the frequency of free vibrations, we can use Hooke's Law for springs:
F = -kx
where F is the force applied to the spring, k is the spring constant, and x is the displacement from the equilibrium position.
In this case, the force applied to the spring is the weight of the body, which is given by:
F = mg
where m is the mass of the body and g is the acceleration due to gravity.
The displacement of the spring is given as 15 mm, which is equivalent to 0.015 m.
So, we can write the equation as:
mg = kx
Solving for k, we have:
k = mg / x
Substituting the given values:
k = (20 kg)(9.8 m/s^2) / 0.015 m
k = 13066.67 N/m
The frequency of free vibrations is given by:
f = 1 / (2π) * sqrt(k / m)
Substituting the given values:
f = 1 / (2π) * sqrt(13066.67 N/m / 20 kg)
f ≈ 2.66 Hz
To verify that a viscous damping force of 1000 N at a speed of 1 m/s is just-sufficient to make the motion aperiodic, we can calculate the damping coefficient:
c = F / v
Substituting the given values:
c = 1000 N / 1 m/s
c = 1000 Ns/m
The critical damping coefficient is given by:
c_crit = 2 * sqrt(k * m)
Substituting the given values:
c_crit = 2 * sqrt(13066.67 N/m * 20 kg)
c_crit ≈ 909.09 Ns/m
Since the damping coefficient (1000 Ns/m) is greater than the critical damping coefficient (909.09 Ns/m), the motion is damped to the extent that it is aperiodic.
Next, we need to find the amplitude of the ultimate motion when the body is subjected to a disturbing force.
The equation of motion for a damped harmonic oscillator is:
m * d^2x/dt^2 + c * dx/dt + kx = F_dist
where F_dist is the disturbing force.
The solution to this equation is of the form:
x(t) = A * exp(-ζωn t) * cos(ωd t + φ)
where A is the amplitude, ζ is the damping ratio, ωn is the natural frequency, ωd is the damped frequency, and φ is the phase angle.
The damping ratio is given by:
ζ = c / (2 * sqrt(k * m))
Substituting the given values:
ζ = 1000 Ns/m / (2 * sqrt(13066.67 N/m * 20 kg))
ζ ≈ 0.383
The natural frequency is given by:
ωn = 2πf
Substituting the given value:
ωn = 2π * 8 cycles/s
ωn = 16π rad/s
The damped frequency is given by:
ωd = ωn * sqrt(1 - ζ^2)
Substituting the given values:
ωd = 16π rad/s * sqrt(1 -
F = -kx
where F is the force applied to the spring, k is the spring constant, and x is the displacement from the equilibrium position.
In this case, the force applied to the spring is the weight of the body, which is given by:
F = mg
where m is the mass of the body and g is the acceleration due to gravity.
The displacement of the spring is given as 15 mm, which is equivalent to 0.015 m.
So, we can write the equation as:
mg = kx
Solving for k, we have:
k = mg / x
Substituting the given values:
k = (20 kg)(9.8 m/s^2) / 0.015 m
k = 13066.67 N/m
The frequency of free vibrations is given by:
f = 1 / (2π) * sqrt(k / m)
Substituting the given values:
f = 1 / (2π) * sqrt(13066.67 N/m / 20 kg)
f ≈ 2.66 Hz
To verify that a viscous damping force of 1000 N at a speed of 1 m/s is just-sufficient to make the motion aperiodic, we can calculate the damping coefficient:
c = F / v
Substituting the given values:
c = 1000 N / 1 m/s
c = 1000 Ns/m
The critical damping coefficient is given by:
c_crit = 2 * sqrt(k * m)
Substituting the given values:
c_crit = 2 * sqrt(13066.67 N/m * 20 kg)
c_crit ≈ 909.09 Ns/m
Since the damping coefficient (1000 Ns/m) is greater than the critical damping coefficient (909.09 Ns/m), the motion is damped to the extent that it is aperiodic.
Next, we need to find the amplitude of the ultimate motion when the body is subjected to a disturbing force.
The equation of motion for a damped harmonic oscillator is:
m * d^2x/dt^2 + c * dx/dt + kx = F_dist
where F_dist is the disturbing force.
The solution to this equation is of the form:
x(t) = A * exp(-ζωn t) * cos(ωd t + φ)
where A is the amplitude, ζ is the damping ratio, ωn is the natural frequency, ωd is the damped frequency, and φ is the phase angle.
The damping ratio is given by:
ζ = c / (2 * sqrt(k * m))
Substituting the given values:
ζ = 1000 Ns/m / (2 * sqrt(13066.67 N/m * 20 kg))
ζ ≈ 0.383
The natural frequency is given by:
ωn = 2πf
Substituting the given value:
ωn = 2π * 8 cycles/s
ωn = 16π rad/s
The damped frequency is given by:
ωd = ωn * sqrt(1 - ζ^2)
Substituting the given values:
ωd = 16π rad/s * sqrt(1 -
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A body of mass 20 kg is suspended from a spring which deflects 15 mm under this load. Calculate the frequency of free vibrations and verify that a viscous damping force amounting to approximately 1000 N at a speed of 1 m/s is just-sufficient to make the motion aperiodic. If when damped to this extent, the body is subjected to a disturbing force with a maximum value of 125 N making 8 cycles/s, find the amplitude of the ultimate motion.a)2.66 × 10-3 mb)1.94 × 10-3 mc)1.96× 10-3 md)2.83× 10-3mCorrect answer is option 'C'. Can you explain this answer?
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A body of mass 20 kg is suspended from a spring which deflects 15 mm under this load. Calculate the frequency of free vibrations and verify that a viscous damping force amounting to approximately 1000 N at a speed of 1 m/s is just-sufficient to make the motion aperiodic. If when damped to this extent, the body is subjected to a disturbing force with a maximum value of 125 N making 8 cycles/s, find the amplitude of the ultimate motion.a)2.66 × 10-3 mb)1.94 × 10-3 mc)1.96× 10-3 md)2.83× 10-3mCorrect answer is option 'C'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about A body of mass 20 kg is suspended from a spring which deflects 15 mm under this load. Calculate the frequency of free vibrations and verify that a viscous damping force amounting to approximately 1000 N at a speed of 1 m/s is just-sufficient to make the motion aperiodic. If when damped to this extent, the body is subjected to a disturbing force with a maximum value of 125 N making 8 cycles/s, find the amplitude of the ultimate motion.a)2.66 × 10-3 mb)1.94 × 10-3 mc)1.96× 10-3 md)2.83× 10-3mCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A body of mass 20 kg is suspended from a spring which deflects 15 mm under this load. Calculate the frequency of free vibrations and verify that a viscous damping force amounting to approximately 1000 N at a speed of 1 m/s is just-sufficient to make the motion aperiodic. If when damped to this extent, the body is subjected to a disturbing force with a maximum value of 125 N making 8 cycles/s, find the amplitude of the ultimate motion.a)2.66 × 10-3 mb)1.94 × 10-3 mc)1.96× 10-3 md)2.83× 10-3mCorrect answer is option 'C'. Can you explain this answer?.
A body of mass 20 kg is suspended from a spring which deflects 15 mm under this load. Calculate the frequency of free vibrations and verify that a viscous damping force amounting to approximately 1000 N at a speed of 1 m/s is just-sufficient to make the motion aperiodic. If when damped to this extent, the body is subjected to a disturbing force with a maximum value of 125 N making 8 cycles/s, find the amplitude of the ultimate motion.a)2.66 × 10-3 mb)1.94 × 10-3 mc)1.96× 10-3 md)2.83× 10-3mCorrect answer is option 'C'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about A body of mass 20 kg is suspended from a spring which deflects 15 mm under this load. Calculate the frequency of free vibrations and verify that a viscous damping force amounting to approximately 1000 N at a speed of 1 m/s is just-sufficient to make the motion aperiodic. If when damped to this extent, the body is subjected to a disturbing force with a maximum value of 125 N making 8 cycles/s, find the amplitude of the ultimate motion.a)2.66 × 10-3 mb)1.94 × 10-3 mc)1.96× 10-3 md)2.83× 10-3mCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A body of mass 20 kg is suspended from a spring which deflects 15 mm under this load. Calculate the frequency of free vibrations and verify that a viscous damping force amounting to approximately 1000 N at a speed of 1 m/s is just-sufficient to make the motion aperiodic. If when damped to this extent, the body is subjected to a disturbing force with a maximum value of 125 N making 8 cycles/s, find the amplitude of the ultimate motion.a)2.66 × 10-3 mb)1.94 × 10-3 mc)1.96× 10-3 md)2.83× 10-3mCorrect answer is option 'C'. Can you explain this answer?.
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A body of mass 20 kg is suspended from a spring which deflects 15 mm under this load. Calculate the frequency of free vibrations and verify that a viscous damping force amounting to approximately 1000 N at a speed of 1 m/s is just-sufficient to make the motion aperiodic. If when damped to this extent, the body is subjected to a disturbing force with a maximum value of 125 N making 8 cycles/s, find the amplitude of the ultimate motion.a)2.66 × 10-3 mb)1.94 × 10-3 mc)1.96× 10-3 md)2.83× 10-3mCorrect answer is option 'C'. Can you explain this answer?, a detailed solution for A body of mass 20 kg is suspended from a spring which deflects 15 mm under this load. Calculate the frequency of free vibrations and verify that a viscous damping force amounting to approximately 1000 N at a speed of 1 m/s is just-sufficient to make the motion aperiodic. If when damped to this extent, the body is subjected to a disturbing force with a maximum value of 125 N making 8 cycles/s, find the amplitude of the ultimate motion.a)2.66 × 10-3 mb)1.94 × 10-3 mc)1.96× 10-3 md)2.83× 10-3mCorrect answer is option 'C'. Can you explain this answer? has been provided alongside types of A body of mass 20 kg is suspended from a spring which deflects 15 mm under this load. Calculate the frequency of free vibrations and verify that a viscous damping force amounting to approximately 1000 N at a speed of 1 m/s is just-sufficient to make the motion aperiodic. If when damped to this extent, the body is subjected to a disturbing force with a maximum value of 125 N making 8 cycles/s, find the amplitude of the ultimate motion.a)2.66 × 10-3 mb)1.94 × 10-3 mc)1.96× 10-3 md)2.83× 10-3mCorrect answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice A body of mass 20 kg is suspended from a spring which deflects 15 mm under this load. Calculate the frequency of free vibrations and verify that a viscous damping force amounting to approximately 1000 N at a speed of 1 m/s is just-sufficient to make the motion aperiodic. If when damped to this extent, the body is subjected to a disturbing force with a maximum value of 125 N making 8 cycles/s, find the amplitude of the ultimate motion.a)2.66 × 10-3 mb)1.94 × 10-3 mc)1.96× 10-3 md)2.83× 10-3mCorrect answer is option 'C'. Can you explain this answer? tests, examples and also practice Mechanical Engineering tests.
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