Let’s address each question in turn.

First question:
We are looking for the acceleration $a$ of the cart in terms of the masses $m_1$ and $m_2$, the spring constant $k$, and the displacement $x$.

• The system consists of a spring and a cart. The force exerted by the spring is $F = -kx$.
• Newton's second law gives:
  $F_{\text{net}} = (m_1 + m_2)a$, where $m_1$ is the mass of the cart and $m_2$ is the mass of the hanging weight.

Equating the force from the spring and the net force:
$-kx = (m_1 + m_2)a$

Solving for $a$:
$a = -\frac{k}{m_1 + m_2}x$

Correct answer:
$\boxed{a = -\frac{k}{m_1 + m_2}x}$

Second question:
We are tasked with calculating the period of oscillation.

The period of oscillation for a spring-mass system is given by:
$T = 2\pi \sqrt{\frac{m_{\text{eff}}}{k}}$

Here, $m_{\text{eff}}$ is the effective mass, which for a system with a cart and a hanging mass is:
$m_{\text{eff}} = m_1 + m_2$

Substitute the given values:

• $m_1 = 0.250 \, \text{kg}$
• $m_2 = 0.061 \, \text{kg}$
• $k = 4.1 \, \text{N/m}$

$m_{\text{eff}} = 0.250 + 0.061 = 0.311 \, \text{kg}$

$T = 2\pi \sqrt{\frac{0.311}{4.1}}$

Simplify:
$T = 2\pi \sqrt{0.07585}$

$T = 2\pi (0.2754)$

$T \approx 1.73 \, \text{seconds}$

Final answer:
$\boxed{1.73 \, \text{seconds}}$

Would you like additional details or explanations? Here are five related questions you might find helpful:

1. How does the effective mass impact the oscillation period?
2. Why is the displacement $x$ directly proportional to acceleration in harmonic motion?
3. Can the spring constant $k$ change for different materials?
4. What is the significance of the negative sign in the acceleration equation?
5. How does adding more mass to the cart affect the oscillation period?

Tip: Always confirm whether a spring obeys Hooke’s law ($F = -kx$) for small displacements before applying these equations!