Question

A mass weighing 10 pounds stretches a spring 2 feet. The mass is attached to a dashpot device that offers a damping force numerically equal to ẞ (B> 0) times the instantaneous velocity. Determine the values of the damping constant & so that the subsequent motion is overdamped, critically damped, and underdamped. (If an answer is an interval, use interval notation. Use g = 32 ft/s² for the acceleration due to gravity.) (a) overdamped (b) critically damped (c) underdamped

Answer 1

(a) Overdamped: ẞ > 24√5 lb⋅s/ft

(b) Critically damped: ẞ = 24√5 lb⋅s/ft

(c) Underdamped: ẞ < 24√5 lb⋅s/ft

Step-by-step explanation:

For overdamped motion, the damping ratio ζ > 1. Using the formula ζ = ẞ/(2√(mk)), where m is the mass (10 lb/g), k is the spring constant (k = F/x = 10 lb/2 ft = 5 lb/ft), and g is the acceleration due to gravity (32 ft/s²), the critical value of ẞ can be calculated using ζ > 1, which gives ẞ > 24√5 lb⋅s/ft.

For critically damped motion, ζ = 1. Utilizing the same formula with ζ = 1, the value of ẞ equals 24√5 lb⋅s/ft. This condition occurs when the damping force is precisely balanced with the restoring force of the spring, allowing the system to return to equilibrium without oscillation in the shortest time.

For underdamped motion, ζ < 1. Substituting ζ < 1 into the formula, ẞ < 24√5 lb⋅s/ft. This scenario results in oscillatory motion, where the system returns to equilibrium with some oscillations before settling.

Answer 2

The values of the damping constant ẞ for each type of motion:

(a) Overdamped: ẞ > 20√2 lb·s/ft

(b) Critically damped: ẞ = 20√2 lb·s/ft

(c) Underdamped: 0 < ẞ < 20√2 lb·s/ft

Spring Constant (k):

Weight stretches the spring 2 feet: F = kx = 10 lb

k = F/x = 10 lb / 2 ft = 5 lb/ft

Damping Coefficient (ẞ):

Mass (m): 10 lb / 32 ft/s² = 5/16 slugs

Critical damping coefficient (ẞc): 2√(mk) = 2√((5/16)(5)) = 20√2 lb·s/ft

Damping Conditions:

Overdamped: ẞ > ẞc, system returns to equilibrium slowly without oscillation.

Critically damped: ẞ = ẞc, system returns to equilibrium as quickly as possible without oscillation.

Underdamped: 0 < ẞ < ẞc, system oscillates with decreasing amplitude before reaching equilibrium.