A mass on a driver is oscillating. The mass of the mass is 2.5330kg, the spring

Question

A mass on a driver is oscillating. The mass of the mass is 2.5330kg, the spring constant of the spring is 300N/m, the damping constant is 1N/(m/s), the frequency of the driver is 1.72Hz, and the amplitude of the driver is 0.50cm. At equilibrium, the mass sits at 23m, and at its highest point, the mass reaches 35m.

a. What is the potential energy of the spring, the kinetic energy of the mass, and the total energy of the mass and spring when the mass is at its highest displacement?

b. When the mass is at its equilibrium position (still oscillating), what is its kinetic energy? What is its potential energy (spring)? What is its total energy of the system?

c. What can you say about the total energy of the oscillating mass and spring system in this case?

Resonator 1 30 mass 25 2.5330 kg 20 spring constant 300 N/m frequency 1.732 Hz damping constant 15 -10 1.0 N/(m/s) Gravity: ON OFF Ruler Reset All DRIVER frequency amplitude OFF 0.50 cm 1.72 Hz

Explanation / Answer

As given in problem,

Mass of mass = mm (say) = 2.5330 kg

Spring constant = Ks (say) = 300 N/m

Damping constant = dc (say) = 1 N/ (m/s)

Frequency of driver = fd (say) = 1.72 Hz

Amplitude of driver = Ad (say) = 0.50 cm

Equilibrium position = xe (say) = 23 m

Highest position = xh (say) = 35 m

a. At the highest displacement in a damping medium,

Potential energy of spring = (1/2) * Ks* (xh - xe )2....(1),

Substituting respective values in (1), we have,

Potential energy of spring = 0.5 * 300 * (35-23)2 = 21600 J = 21.6 KJ

Kinetic energy of mass = (1/2) * mm * v2 , where v = velocity of mass

Kinetic energy of mass = 0 (as at the highest point, the velocity of mass becomes zero momentarily before it changes direction and starts coming down)

Thus, total energy of mass and spring at the highest displacement= Potential energy of spring = 21.6 KJ

b. At equilibrium position and still oscillating,

Kinetic energy of mass = 0 (as at the equilibrium point during oscillation, the velocity of mass becomes zero momentarily before it changes direction and starts going up)

The potential energy will be the same as in a. and the value will be 21.6 KJ, however the magnitude will be negative as the final distance during reaching equilibrium is lesser than the highest distsnce (from where it started falling).

c. The total energy of the oscillating mass and spring system depends only on the potential energy of the spring since the kinetic energy of the mass at the highest displacement and equilibrium points are zero, during oscillation.

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