To determine the mass $m_2$ of block 2, let's analyze the forces acting on both blocks and use Newton’s Second Law.

Step 1: Define the forces acting on both blocks

Block 1 (Hanging mass, $m_1$)

• Weight: $W_1 = m_1 g$
• Tension in the string: $T$
• Newton's Second Law:
  $m_1 g - T = m_1 a$

Block 2 (on an inclined plane, $m_2$)

• Weight components:
  • Perpendicular to the plane: $m_2 g \cos\theta$
  • Parallel to the plane: $m_2 g \sin\theta$
• Kinetic friction force:
  $f_k = \mu m_2 g \cos\theta$
• Tension in the string: $T$
• Newton's Second Law along the plane: $T - f_k - m_2 g \sin\theta = m_2 a$

Step 2: Express Tension $T$ from Block 1

From the equation for Block 1: $T = m_1 g - m_1 a$

Step 3: Substitute $T$ into Block 2's equation

$(m_1 g - m_1 a) - (\mu m_2 g \cos\theta) - (m_2 g \sin\theta) = m_2 a$

Step 4: Solve for $m_2$

$m_1 g - m_1 a - \mu m_2 g \cos\theta - m_2 g \sin\theta = m_2 a$

Factor out $m_2$: $m_1 g - m_1 a = m_2 g (\mu \cos\theta + \sin\theta) + m_2 a$

$m_2 (g (\mu \cos\theta + \sin\theta) + a) = m_1 g - m_1 a$

$m_2 = \frac{m_1 g - m_1 a}{g (\mu \cos\theta + \sin\theta) + a}$

Step 5: Plug in Values

• $m_1 = 0.650$ kg
• $g = 9.81$ m/s²
• $a = 0.400$ m/s²
• $\theta = 30.0^\circ$
• $\mu = 0.250$

Calculate the denominator: $g (\mu \cos\theta + \sin\theta) + a = 9.81 (0.250 \cos 30^\circ + \sin 30^\circ) + 0.400$

$= 9.81 (0.250 \times 0.866 + 0.5) + 0.400$

$= 9.81 (0.2165 + 0.5) + 0.400$

$= 9.81 \times 0.7165 + 0.400$

$= 7.0265 + 0.400$

$= 7.4265$

Calculate the numerator: $m_1 g - m_1 a = (0.650 \times 9.81) - (0.650 \times 0.400)$

$= 6.3765 - 0.26$

$= 6.1165$

Now solve for $m_2$:

$m_2 = \frac{6.1165}{7.4265}$

$m_2 \approx 0.824 \text{ kg}$

Final Answer:

$\boxed{0.824 \text{ kg}}$

Would you like a deeper breakdown of any part of the solution?

Related Questions:

1. What would happen if the coefficient of friction were increased?
2. How would the acceleration change if the angle $\theta$ was steeper?
3. What if the masses were equal—how would the system behave?
4. How does the tension in the rope change with different masses?
5. What is the normal force acting on block 2?

Tip:

In inclined plane problems, always resolve forces into components parallel and perpendicular to the surface—it simplifies calculations significantly. 🚀