In a single degree of freedom vibrating system with only viscous damping, the critical damping coefficient is 350 Ns/m and the damping coefficient is 35 Ns/m. The logarithmic decrement of system is,

The correct option is 0.63.

1. Calculate the Damping Ratio:
The damping ratio ($\zeta$) is defined as the ratio of the actual viscous damping coefficient ($c$) to the critical damping coefficient ($c_c$):

$\zeta =\frac{c}{c_c}$

Given the values:
Damping coefficient, $c=35\text{Ns/m}$
Critical damping coefficient, $c_c=350\text{Ns/m}$

Substituting these values into the damping ratio equation:

$\zeta =\frac{35}{350}=0.1$

2. Calculate the Logarithmic Decrement:
The logarithmic decrement ($\delta$) represents the rate at which the amplitude of free underdamped vibrations decays over time. The relationship between the damping ratio and the logarithmic decrement is given by:

$\delta =\frac{2\pi \zeta }{\sqrt{1-{\zeta }^2}}$

Substituting the damping ratio ($\zeta =0.1$) into the equation:

$\delta =\frac{2\times \pi \times 0.1}{\sqrt{1-{\left(0.1\right)}^2}}$

$\delta =\frac{0.2\pi }{\sqrt{1-0.01}}$

$\delta =\frac{0.2\pi }{\sqrt{0.99}}$

Using the approximations $\pi \approx 3.1416$ and $\sqrt{0.99}\approx 0.9950$:

$\delta =\frac{0.2\times 3.1416}{0.9950}$

$\delta \approx 0.6315$

Rounding to two decimal places, we get:

$\delta \approx 0.63$