Question

A 10 kg mass is supported on a spring of stiffness 4 kN/m and has a dashpot which produces a resistance of 20 N at a velocity of 0.25 m/s. The damping ratio of the system is:

• A. 1
• B. 0.8
• C. 0.4
• D. 0.2

Hint:

Remember the formulas for damping coefficient \( c = F_d/v \) and critical damping coefficient \( c_c = 2 \sqrt{mk} \). The damping ratio \( \zeta = c/c_c \) is a dimensionless quantity that indicates the level of damping in a system.

Answer 1

The Correct Option is D

Step 1: Identify the given parameters.
We are given the following parameters:
Mass \( m = 10 \) kg
Spring stiffness \( k = 4 \) kN/m = 4000 N/m
Damping force \( F_d = 20 \) N at a velocity \( v = 0.25 \) m/s
Step 2: Determine the damping coefficient \( c \).
The damping force produced by a viscous damper is proportional to the velocity, \( F_d = cv \), where \( c \) is the damping coefficient. We can find \( c \) using the given values:
$$c = \frac{F_d}{v} = \frac{20 \text{ N}}{0.25 \text{ m/s}} = 80 \text{ Ns/m}$$ Step 3: Calculate the critical damping coefficient \( c_c \).
The critical damping coefficient \( c_c \) is the value of damping that results in the system returning to equilibrium as quickly as possible without oscillation. It is given by the formula:
$$c_c = 2 \sqrt{mk}$$Substituting the values of \( m \) and \( k \):$$c_c = 2 \sqrt{(10 \text{ kg})(4000 \text{ N/m})} = 2 \sqrt{40000 \text{ kg N/m}^2} = 2 \sqrt{40000 \text{ kg}^2/\text{s}^2} = 2 \times 200 \text{ kg/s} = 400 \text{ Ns/m}$$ (Note that 1 N = 1 kg m/s\(^2\), so kg N/m = kg\(^2\)/s\(^2\)) Step 4: Calculate the damping ratio \( \zeta \).
The damping ratio \( \zeta \) (zeta) is the ratio of the actual damping coefficient \( c \) to the critical damping coefficient \( c_c \):
$$\zeta = \frac{c}{c_c}$$Substituting the values of \( c \) and \( c_c \):$$\zeta = \frac{80 \text{ Ns/m}}{400 \text{ Ns/m}} = 0.2$$ Therefore, the damping ratio of the system is 0.2.