To solve questions 4.4 and 4.5 from the image, we need to consider the forces acting on both blocks, as well as the acceleration of the system. Here's the breakdown of the problem:

Given Data:

• Mass of Block 1 ($m_1$): 4 kg
• Mass of Block 2 ($m_2$): 8 kg
• Inclination Angle ($\theta$): 30°
• Coefficient of Kinetic Friction ($\mu_k$): 0.2

Question 4.4: Calculate the magnitude of the acceleration of the system.

Step-by-Step Solution:

1. Identify Forces on Each Block:

   • For block 1 ($m_1 = 4$ kg) on the incline:

     • Weight component along the incline: $m_1g \sin \theta$
     • Friction force: $f_1 = \mu_k N_1$, where $N_1 = m_1g \cos \theta$ (normal force)

   • For block 2 ($m_2 = 8$ kg) hanging vertically:

     • Gravitational force: $m_2g$

2. Equations for Block 1 (on the incline):

   • Weight component along the incline: $m_1g \sin \theta = 4 \times 9.8 \times \sin(30°) = 19.6 \, \text{N}$
   • Normal force: $N_1 = m_1g \cos \theta = 4 \times 9.8 \times \cos(30°) = 33.94 \, \text{N}$
   • Friction force: $f_1 = \mu_k N_1 = 0.2 \times 33.94 = 6.79 \, \text{N}$

3. Net Force on Block 1: $F_{1,\text{net}} = m_1g \sin \theta - f_1 = 19.6 - 6.79 = 12.81 \, \text{N}$

4. Net Force on Block 2:

   • The force due to gravity: $F_{2,\text{net}} = m_2g = 8 \times 9.8 = 78.4 \, \text{N}$

5. Total Net Force on the System: $F_{\text{net}} = F_{2,\text{net}} - F_{1,\text{net}} = 78.4 - 12.81 = 65.59 \, \text{N}$

6. Total Mass of the System: $m_{\text{total}} = m_1 + m_2 = 4 + 8 = 12 \, \text{kg}$

7. Acceleration of the System: $a = \frac{F_{\text{net}}}{m_{\text{total}}} = \frac{65.59}{12} = 5.47 \, \text{m/s}^2$

Answer to 4.4: The magnitude of the acceleration of the system is 5.47 m/s$^2$.

Question 4.5: How will the acceleration compare if the positions of the $4$ kg block and $8$ kg block are switched?

If we switch the positions of the blocks, the larger mass ($8$ kg) will be on the incline, and the smaller mass ($4$ kg) will be hanging. The friction force on the $8$ kg block will be higher due to its larger normal force, which will decrease the acceleration of the system because the net force driving the motion will be smaller.

Answer to 4.5: The acceleration of the system will be LESS THAN when the $4$ kg block is on the incline and the $8$ kg block is hanging.

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Additional Questions:

1. What is Newton's Second Law of Motion, and how is it applied in this scenario?
2. How would the acceleration change if the coefficient of friction was different?
3. What are the effects of the angle of inclination on the acceleration of the system?
4. How can we calculate the tension in the string connecting the two blocks?
5. How would the system's behavior change if there were no friction?

Tip: Understanding the role of friction is crucial in mechanics. It always acts opposite to the direction of motion and reduces the net force, thereby affecting acceleration.