Write the model ODE and the implied initial values, and then solve the IVP to answer each of the f
ollowing:
[2*8=16pts]
(a) A 3kg mass is attached to a spring whose
constant is 6 N/m, and the entire system is submerged in a liquid which imparts a
damping force equal to 6 times the instantaneous velocity. The weight is released from equilibrium w
ith upward velocity of 4
m/s. Find the equation of motion of the mass
To find the equation of motion for the mass, we need to set up the model ODE (Ordinary Differential Equation) for the system.
Let's denote:
- m: mass of the object (3 kg)
- k: spring constant (6 N/m)
- v: velocity of the mass
- F_d: damping force
- g: acceleration due to gravity (approximately 9.8 m/s^2)
The equation of motion for the mass can be described as:
m * dv/dt + F_d + k * x = m * g
where dx/dt = v represents the velocity, and x represents the displacement from the equilibrium position.
Since the damping force is given by 6 times the instantaneous velocity, F_d = 6 * v.
Using these values, we can rewrite the equation of motion as:
3 * dv/dt + 6 * v + 6 * x = 3 * 9.8
Now, for the initial conditions of the problem, we are given that the weight is released from equilibrium with an upward velocity of 4 m/s. This means that at t = 0, v = 4 m/s and x = 0.
So, our initial conditions are:
v(0) = 4 m/s
x(0) = 0
Now we have the model ODE and the implied initial values. To solve this IVP (initial value problem), we can use methods like numerical integration or Laplace transforms.
By solving the equation of motion with the given initial conditions, we can obtain the equation that describes the motion of the mass over time.