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 )

in simple terms, the principle states that the buoyancy force on an object is equal to the weight of the fluid displaced by the object, or the density of the fluid multiplied by the submerged volume times the gravitational acceleration, g. this is also known as upthrust.

i think the answer is a, hope this is right, that would be my best guess.