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here is the solution for ITEM 4: This problem involves a system of pulleys and masses. We will use Newton's Second Law and the constraint of the string's length. Given masses: • m_1 = 2 kg • m_2 = 12 kg (movable pulley) • m_3 = 3 kg We will use g = 9.8 m/s^2 for the acceleration due to gravity. a) Acceleration of each mass. Step 1: Write down the equations of motion for each mass. Let T be the tension in the string. We define the downward direction as positive for acceleration. For mass m_1: m_1 g - T = m_1 a_1 (1) For mass m_3: m_3 g - T = m_3 a_3 (2) For mass m_2 (the movable pulley): It is pulled downwards by gravity and upwards by two segments of the string, each with tension T. m_2 g - 2T = m_2 a_2 (3) Step 2: Determine the constraint relationship between the accelerations. The total length of the string is constant. If y_1, y_2, y_3 are the vertical positions of m_1, m_2, m_3 from a fixed reference (e.g., the ceiling), the length constraint is: y_1 + 2y_2 + y_3 = constant Differentiating this equation twice with respect to time gives the relationship between accelerations: a_1 + 2a_2 + a_3 = 0 (4) Step 3: Solve the system of equations for the accelerations. From (1), T = m_1(g - a_1). From (2), T = m_3(g - a_3). Equating these two expressions for T: m_1(g - a_1) = m_3(g - a_3) Substitute m_1=2 and m_3=3: 2(g - a_1) = 3(g - a_3) 2g - 2a_1 = 3g - 3a_3 3a_3 - 2a_1 = g (A) From (4), we can express a_3 = -a_1 - 2a_2. Substitute this into (A): 3(-a_1 - 2a_2) - 2a_1 = g -3a_1 - 6a_2 - 2a_1 = g -5a_1 - 6a_2 = g (B) Now, substitute T = m_1(g - a_1) into (3): m_2 g - 2m_1(g - a_1) = m_2 a_2 Substitute m_1=2 and m_2=12: 12g - 2(2)(g - a_1) = 12a_2 12g - 4g + 4a_1 = 12a_2 8g + 4a_1 = 12a_2 Divide by 4: 2g + a_1 = 3a_2 From this, express a_1: a_1 = 3a_2 - 2g (C) Substitute (C) into (B): -5(3a_2 - 2g) - 6a_2 = g -15a_2 + 10g - 6a_2 = g -21a_2 = -9g a_2 = (9g)/(21) = (3g)/(7) Now find a_1 using (C): a_1 = 3((3g)/(7)) - 2g = (9g)/(7) - (14g)/(7) = -(5g)/(7) And find a_3 using (A): 3a_3 - 2(-(5g)/(7)) = g 3a_3 + (10g)/(7) = g 3a_3 = g - (10g)/(7) = -(3g)/(7) a_3 = -(g)/(7) Step 4: Calculate the numerical values for the accelerations. Using g = 9.8 m/s^2: a_1 = -(5 × 9.8)/(7) = -5 × 1.4 = -7 m/s^2 a_2 = (3 × 9.8)/(7) = 3 × 1.4 = 4.2 m/s^2 a_3 = -(9.8)/(7) = -1.4 m/s^2 The negative signs indicate acceleration in the upward direction, as we defined downward as positive. The acceleration of each mass is: • a_1 = 7 m/s^2 (upwards) • a_2 = 4.2 m/s^2 (downwards) • a_3 = 1.4 m/s^2 (upwards) b) Tension in the connecting string. Step 5: Calculate the tension T. We can use equation (1): T = m_1 g - m_1 a_1 Substitute the values for m_1 and a_1: T = 2 kg × 9.8 m/s^2 - 2 kg × (-7 m/s^2) T = 19.6 N + 14 N T = 33.6 N The tension in the connecting string is 33.6 N. What's next?