Classical Mechanics
# Damped Oscillators

In the above diagram, a mass \( m (= 3 \text{ kg}) \) 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.

Assume that the gravitational force is negligible relative to the damping force \( F_d \) and the force exerted by the spring \( F_s. \)

In the above diagram, a mass \( m (= 4 \text{ kg}) \) 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?

Assume that the gravitational force is negligible relative to the damping force \( F_d \) and the force exerted by the spring \( F_s. \)

In the above diagram, a mass \( m (= 5 \text{ kg}) \) 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?

Assume that the gravitational force is negligible relative to the damping force \( F_d \) and the force exerted by the spring \( F_s. \)