Amass of 3 slugs (this is the english unit of mass, a pound is a force) is attached to a vertical spring with a spring constant k = 9 lbs/ft. the spring has a dampening constant of beta = 6lbs * s/ft. the mass is pulled 1 ft below its equilibrium position and released. determine the equation of motion for the mass.

equation of motion for the mass is x(t) = e^αt ( C1 cos √{α² - ω²} t + C2 sin  √{α² - ω²} t )

Explanation:

Given data

mass = 3 slugs = 3 * 32.14 = 96.52 lbs

constant k = 9 lbs/ft

Beta = 6lbs * s/ft

mass is pulled =  1 ft below

to find out

equation of motion for the mass

solution

we know that The mass is pulled 1 ft below so

we will apply here differential equation of free motion i.e

dx²/dt² + 2 α dx/dt + ω² x =0     1

here 2 α  =  Beta / mass

so 2 α  = 6 / 96.52

α  = 0.031

α²  = 0.000961         2

and

ω² = k/mass

ω² = 9 /96.52

ω² = 0.093     3

we can say that from equation 2 and 3 that α² - ω²  = -0.092239

this is less than zero

so differential equation is

x(t) = e^αt ( C1 cos √{α² - ω²} t + C2 sin  √{α² - ω²} t )

equation of motion for the mass is x(t) = e^αt ( C1 cos √{α² - ω²} t + C2 sin  √{α² - ω²} t )

b. a lightning bug, plug of grass, and a worm