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In under damped vibrating system, if x1 and x2 are the successive values of the amplitude on the same side of the mean position, then the logarithmic decrement is equal to
- a)x1 / x2
- b)log (x1 / x2)
- c)loge (x1 / x2)
- d)log (x1.x2)
Correct answer is option 'B'. Can you explain this answer?
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The logarithmic decrement is a measure of the rate at which the amplitude of an underdamped vibrating system decreases over time. It is defined as the natural logarithm of the ratio of two successive amplitudes on the same side of the mean position.
Let's consider a vibrating system with an initial amplitude x1 and a subsequent amplitude x2, both on the same side of the mean position. The logarithmic decrement, denoted as δ, is given by the equation:
δ = ln(x1 / x2)
To understand why the correct answer is option 'B', let's evaluate the other options:
a) x1 / x2: This is simply the ratio of the two amplitudes, which does not represent the logarithmic decrement.
c) loge (x1 / x2): This is the natural logarithm of the ratio of the two amplitudes, which is equivalent to the logarithmic decrement. However, the logarithm should be in base 10, not base e, to match the standard definition of the logarithmic decrement.
d) log (x1.x2): This is the logarithm of the product of the two amplitudes, which is not equivalent to the logarithmic decrement.
Therefore, the correct answer is option 'B', which represents the natural logarithm of the ratio of the two amplitudes, matching the definition of the logarithmic decrement.
In summary, the logarithmic decrement is a measure of the rate at which the amplitude of an underdamped vibrating system decreases over time. It is equal to the natural logarithm of the ratio of two successive amplitudes on the same side of the mean position. The correct answer to the given question is option 'B', which represents the logarithm of the ratio of the amplitudes.
Let's consider a vibrating system with an initial amplitude x1 and a subsequent amplitude x2, both on the same side of the mean position. The logarithmic decrement, denoted as δ, is given by the equation:
δ = ln(x1 / x2)
To understand why the correct answer is option 'B', let's evaluate the other options:
a) x1 / x2: This is simply the ratio of the two amplitudes, which does not represent the logarithmic decrement.
c) loge (x1 / x2): This is the natural logarithm of the ratio of the two amplitudes, which is equivalent to the logarithmic decrement. However, the logarithm should be in base 10, not base e, to match the standard definition of the logarithmic decrement.
d) log (x1.x2): This is the logarithm of the product of the two amplitudes, which is not equivalent to the logarithmic decrement.
Therefore, the correct answer is option 'B', which represents the natural logarithm of the ratio of the two amplitudes, matching the definition of the logarithmic decrement.
In summary, the logarithmic decrement is a measure of the rate at which the amplitude of an underdamped vibrating system decreases over time. It is equal to the natural logarithm of the ratio of two successive amplitudes on the same side of the mean position. The correct answer to the given question is option 'B', which represents the logarithm of the ratio of the amplitudes.
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