Classical Mechanics
# Damped Oscillators

$m = 400 \text{ g}$ is attached to a spring with spring constant $k = 75 \text{ N/m}.$ Since the floor is made of a special material, the friction between the floor and the mass exerts a damping force with damping constant $b = 3 \text{ kg/s}$ which is proportional to the velocity $v$ of the mass. What is the approximate ratio (in percentage terms) of the amplitude of the damped oscillations to the initial amplitude at the end of $1$ cycle?

In the above diagram, a mass of$m (= 3 \text{ kg})$ is attached to a spring with spring constant $k = 51 \text{ N/m}.$ A vane, attached to the mass $m$ and immersed in a liquid, exerts a damping force $F_d$ with damping constant $b = 6 \text{ kg/s}$ which is proportional to the velocity $v$ of the vane and mass. Find the approximate period of the motion.

In the above diagram, a massAssume that the gravitational force is negligible relative to the damping force $F_d$ and the force exerted by the spring $F_s.$

$m (= 4 \text{ kg})$ is attached to a spring with spring constant $k = 20 \text{ N/m}.$ A vane, attached to the mass $m$ and immersed in a liquid, exerts a damping force $F_d$ with damping constant $b = 4 \text{ kg/s}$ which is proportional to the velocity $v$ of the vane and mass. Approximately how long does it take for the amplitude of the damped oscillation to drop to half its initial value?

In the above diagram, a massAssume that the gravitational force is negligible relative to the damping force $F_d$ and the force exerted by the spring $F_s.$

$m (= 5 \text{ kg})$ is attached to a spring with spring constant $k = 85 \text{ N/m}.$ A vane, attached to the mass $m$ and immersed in a liquid, exerts a damping force $F_d$ with damping constant $b = 3 \text{ kg/s}$ which is proportional to the velocity $v$ of the vane and mass. Approximately how long does it take for the mechanical energy of the oscillator to drop to $\frac{1}{3}$ of its initial value?

In the above diagram, a massAssume that the gravitational force is negligible relative to the damping force $F_d$ and the force exerted by the spring $F_s.$

$m = 3 \text{ kg}$ is attached to a spring with spring constant $k = 15 \text{ N/m}.$ Since the floor is made of a special material, the friction between the floor and the mass exerts a damping force with damping constant $b$ which is proportional to the velocity $v$ of the mass. If the mass is oscillating with time period $T = 3.14 \text{ s},$ what is the approximate damping constant?

In the above diagram, a mass of