To solve this problem, we can use Newton's Second Law, $F = ma$, and the principles of tension and forces in a pulley system. Here’s the step-by-step approach:

Step 1: Identify the forces acting on each bucket

Let:

• $m_b = 7.0 \, \text{kg}$ (the mass of each bucket),
• $a = 1.5 \, \text{m/s}^2$ (the acceleration),
• $m_r$ (the mass of the rock, which we need to find),
• $g = 9.8 \, \text{m/s}^2$ (acceleration due to gravity).

There are two buckets: one with the rock and one without. Since the system is accelerating, the forces are unbalanced.

Step 2: Write the equation of forces for each bucket

1. For the bucket with the rock:

   • The weight pulling down on this bucket is $(m_b + m_r)g$.
   • The tension in the rope pulling upwards is $T$.

   Using Newton’s second law: $(m_b + m_r)g - T = (m_b + m_r)a$

2. For the bucket without the rock:

   • The weight pulling down on this bucket is $m_b g$.
   • The tension in the rope pulling upwards is $T$.

   Using Newton’s second law for this bucket: $T - m_b g = m_b a$

Step 3: Combine the two equations

From the second equation, solve for $T$: $T = m_b g + m_b a$

Substitute this expression for $T$ into the first equation: $(m_b + m_r)g - (m_b g + m_b a) = (m_b + m_r)a$

Step 4: Simplify the equation

$(m_b + m_r)g - m_b g - m_b a = (m_b + m_r)a$

$m_r g = m_r a + 2m_b a$

Factor out $m_r$ on the left: $m_r(g - a) = 2m_b a$

Step 5: Solve for $m_r$

$m_r = \frac{2m_b a}{g - a}$

Substitute the known values ($m_b = 7.0 \, \text{kg}$, $a = 1.5 \, \text{m/s}^2$, and $g = 9.8 \, \text{m/s}^2$):

$m_r = \frac{2(7.0)(1.5)}{9.8 - 1.5}$

$m_r = \frac{21.0}{8.3}$

$m_r \approx 2.53 \, \text{kg}$

Conclusion:

The mass of the rock is approximately $2.53 \, \text{kg}$.

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Would you like further clarification or details on any part of the solution?

Here are five related questions:

1. How does the acceleration affect the tension in the rope?
2. What would happen if the mass of the rock was less than calculated?
3. How would the system behave if both buckets had rocks of equal mass?
4. What other factors might influence the equilibrium in this system?
5. How does the choice of pulley affect the system's dynamics?

Tip: In pulley systems, always break the forces into their components to simplify the analysis.