A 1-kilogram mass is attached to a spring whose constant is 21 N/m, and the enti
ID: 1262882 • Letter: A
Question
A 1-kilogram mass is attached to a spring whose constant is 21 N/m, and the entire system is then submerged in a liquid that imparts a damping force numerically equal to 10 times the instantaneous velocity. Determine the equations of motion if the following is true. (a) the mass is initially released from rest from a point 1 meter below the equilibrium position X(t) = m (b) the mass is initially released from a point 1 meter below the equilibrium position with an upward velocity of 13 m/s X(t)= mExplanation / Answer
for spring mass problems there equations for the different situations are given as follows
no driving force, undamped m y''(t) + k y(t) = 0
no driving force, damped m y''(t) + alpha y'(t) + k y(t) = 0
with driving force, undamped m y''(t) + k y(t) = F(t)
with driving force, damped m y''(t) + alpha y'(t) + k y(t) = F(t)
where m = mass; k = spring constant; alpha is your fictional constant, it has units of mass per unit time. normally is given as ( like in this problem) "submerged in a liquid that imparts a damping force numerically equal to 10 times the instantaneous velocity" so its 10 for this problem.
all of these can be solved by hand however its easy if you get mathmatica or something to solve it for you
this problem is dampaned and no driving force
m y''(t) + alpha y'(t) + k y(t) = 0
m = 1kg
alpha = 10 kg/s
k = 21 N/m
i plugd this into mathmatica and the path function y(t) is
y(t) = E^(-10.5 t) C[1] + E^(-2 t) C[2]
where C[1] and C[2] are your integration constants to solve for them you need initial values which are normally the starting position and velocity y(0) = -1 and y'(0) = 0
however mathmatica can solve with boundary conditions so
y(t) = 1/3 E^(-10.5 t) (-1 + 4 E^(6 t))
b)
y(t) = 1/3 E^(-10.5 t) (-5 + 2 E^(6 t))
from the numbers it seems the damping values are high enough for the mass to just move up and down once before all the energy is bleed out of the system
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