Question

A mass weighing 14 pounds stretches a spring 2 feet. The mass is attached to a dashpot device that offers a damping force numerically equal to β (β > 0) times the instantaneous velocity. Determine the values of the damping constant β 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.)

Answer 1

For overdamped motion, the damping constant β should be such that √(k/m) < β/2m, where k is the spring constant and m is the mass of the system. For critically damped motion, β should be such that √(k/m) = β/2m. And for underdamped motion, β should be such that √(k/m) > β/2m.

Step-by-step explanation:

In a damped mass-spring system, the subsequent motion can be classified as underdamped, critically damped, or overdamped based on the value of the damping constant β. For overdamped motion, the damping constant is such that √(k/m) < β/2m. For critically damped motion, the damping constant is such that √(k/m) = β/2m. And for underdamped motion, the damping constant is such that √(k/m) > β/2m.

In this case, the relevant equation is √(k/m) = β/2m. Since the mass of the system is given as 14 pounds, we need to convert it to the standard unit of mass, which is kilograms. 1 pound is approximately 0.454 kilograms. So, the mass of the system is 14 pounds * 0.454 kg/pound = 6.36 kg. The weight of the system is 6.36 kg * 9.8 m/s^2 (acceleration due to gravity) = 62.328 N.

Now, we can calculate β by rearranging the equation: β = 2m√(k/m). Plugging in the given values, we get β = 2 * 6.36 kg * 2 ft / 2 s= 25.44 Ns/m.

For overdamped motion, we need √(k/m) < β/2m. Plugging in the values, we have √(k/m) < 25.44 Ns/m / (2 * 6.36 kg) = 2. We can square both sides of the inequality to eliminate the square root and simplify. k/m < 2^2 = 4. Since k is the spring constant, we can rewrite this as k < 4m. Using the given values, we have k < 4 * 6.36 kg = 25.44 N/m.

For critically damped motion, we need √(k/m) = β/2m. Plugging in the values, we get √(k/m) = 25.44 Ns/m / (2 * 6.36 kg) = 2. Solving for k, we have k = (2 * 6.36 kg)^2 = 162.4896 N/m.

For underdamped motion, we need √(k/m) > β/2m. Plugging in the values, we have √(k/m) > 25.44 Ns/m / (2 * 6.36 kg) = 2. We can square both sides of the inequality to eliminate the square root and simplify. k/m > 2^2 = 4. Since k is the spring constant, we can rewrite this as k > 4m. Using the given values, we have k > 4 * 6.36 kg = 25.44 N/m.