Question

A force of 9 N stretches a spring 1 m. A mass weighing 1.96 N is attached to the spring, and the system is then immersed in a medium that offers a damping force numerically equal to 1.2 times the instantaneous velocity. (a) Find the equation of motion if the mass is initially released from rest from a point 1 m above the equilibrium position. x(t) = m (b) Express the equation of motion in the form x(t) = Ae−λt sin ω2 − λ2 t + ϕ , which is given in (23) of Section 3.8. (Round ϕ to two decimal places.) x(t) = Incorrect: Your answer is incorrect. m (c) Find the first time at which the mass passes through the equilibrium position heading upward. (Round your answer to three decimal places.)

Answer 1

Calculating spring constant k,

Spring force= external force

kx=9N

k=9N/m ( as x=1m)

Calculating some basic term form what's given in the question:

Weight=1.96N

Mass,m=1.96/g=0.2kg (Taking g=9.8m/s²)

Damping force=-1.2 dx/dt

Spring force=-9x

a)

Using Newton 2nd law to balance the forces:

Putting known values:

Solving the above differential equation we get:

Now we have initial conditions:

x(0)=1m

v(0)=0m/s

From these conditions we got:

b)

Now writing this equation in the given form which is:

where

c)

Putting the above equation equal to zero:

Here we take n=2, as for n=1 the object will be going downward not upward, but for n=2 when the object reaches the equilibrium position it will be heading upward.

_________________________________________

Please upvote this answer if it helped you a bit and share your reviews if you feel like anything is missing or wrong so that I can improve that.