Question

MASS SPRING SYSTEMS problem (Differential Equations)

A mass weighing 6 pounds, attached to the end of a spring, stretches it 6 inches.

If the weight is released from rest at a point 4 inches below the equilibrium position, and the entire system is immersed in a liquid that imparts a damping force numerically equal to 3 times the instantaneous velocity, solve:

a. Deduce the differential equation that models the mass-spring system.
b. Calculate the displacements of the mass ? (?) at all times “?”
c. Make a graph that shows the motion

Thank you for the help!

Answer 1

The equation of motion is Since 6 pounds stretches it (the spring) by 6 inches, it means (spring constant). Now here denotes the distance from the mean (equilibrium) position of the spring system.

So the system of equation now reads, The initial condition reads, All the units are in pounds (for mass) and inches (for length).

Solving the above equation one gets, Note the exp factor overall which is due to the damping present in the system.