This is a second-order differential equation problem involving damped harmonic motion with an external force. Let's go through the steps to find the position function $u(t)$ for the mass at any time $t$.

Step-by-Step Solution

1. Identify the given information:

   • Mass $m = 7 \, \text{kg}$
   • Spring stretch $\Delta x = 16 \, \text{cm} = 0.16 \, \text{m}$
   • External force $F_{\text{ext}} = 7 \sin(t/3) \, \text{N}$
   • Viscous force $F_{\text{viscous}} = 3 \, \text{N when } \frac{du}{dt} = 6 \, \text{cm/s} = 0.06 \, \text{m/s}$
   • Initial velocity $u'(0) = 2 \, \text{cm/s} = 0.02 \, \text{m/s}$
   • Gravitational acceleration $g = 9.8 \, \text{m/s}^2$

2. Calculate the spring constant $k$: The spring constant $k$ is calculated using Hooke's law: $mg = k \Delta x$ where $mg = 7 \times 9.8 = 68.6 \, \text{N}$. So, $k = \frac{68.6}{0.16} = 428.75 \, \text{N/m}$

3. Determine the damping constant $c$: Given the viscous force $F_{\text{viscous}} = c \cdot \frac{du}{dt}$, and $F_{\text{viscous}} = 3$ N when $\frac{du}{dt} = 0.06 \, \text{m/s}$, $c = \frac{3}{0.06} = 50 \, \text{Ns/m}$

4. Set up the differential equation: For a damped harmonic oscillator with an external force, the differential equation is: $m \frac{d^2u}{dt^2} + c \frac{du}{dt} + ku = F_{\text{ext}}$ Substituting the values, we get: $7 \frac{d^2u}{dt^2} + 50 \frac{du}{dt} + 428.75 u = 7 \sin\left(\frac{t}{3}\right)$

5. Simplify the equation: Dividing through by 7: $\frac{d^2u}{dt^2} + \frac{50}{7} \frac{du}{dt} + \frac{428.75}{7} u = \sin\left(\frac{t}{3}\right)$ which simplifies to: $\frac{d^2u}{dt^2} + \frac{50}{7} \frac{du}{dt} + 61.25 u = \sin\left(\frac{t}{3}\right)$

6. Solve the differential equation: This is a non-homogeneous second-order linear differential equation with constant coefficients. Solving this equation involves finding the complementary solution for the homogeneous equation and a particular solution for the non-homogeneous part.

Would you like the detailed steps for solving this differential equation, including finding the complementary and particular solutions?

Follow-up Questions:

1. What are the steps for finding the complementary solution for this type of equation?
2. How does the damping constant affect the motion of the mass?
3. What role does the external force $7 \sin(t/3)$ play in this system?
4. How would the solution change if the external force was different?
5. What are the physical interpretations of the complementary and particular solutions in this context?

Tip:

When solving differential equations involving oscillations and damping, always pay attention to units, as they play a critical role in correctly setting up the problem.