Final answer:
To calculate the linear acceleration of the left box in a two-box pulley system, we set up equations based on Newton's laws for rotation and motion, considering the gravitational force, the tension in the rope, and the moment of inertia of the pulley, and then solve the equations to find the acceleration.
Step-by-step explanation:
To find the linear acceleration of the left box in a two-box pulley system, we need to account for both the gravitational force acting on the boxes and the moment of inertia of the pulley. The system contains masses m₁ (3.5 kg) on the left and m₂ (1.7 kg) on the right, with a pulley of mass m₃ (2.9 kg) and radius r (0.14 m).
We start with Newton's second law for rotational motion, which states that the sum of the torques (Τ) is equal to the moment of inertia (I) of the pulley times its angular acceleration (α): Τ = Iα. The torque is also the difference in the tension of the two sides of the rope times the radius of the pulley (r), Τ = (T₁ - T₂)r, where T₁ and T₂ are the tensions in the rope on each side.
The moment of inertia of a solid disk is I = (1/2)m₃r². Also, the angular acceleration is related to the linear acceleration (a) by the formula α = a/r. The linear forces acting on each mass are the weight and the tension, with Newton's second law for the linear motion being F = ma.
Using these relationships, we can set up the following equations:
By solving these three equations simultaneously, we can find the linear acceleration (a) of the left box, considering that the tensions are internal forces and cancel out when considering the system as a whole.