Two blods with masses \( \mathbf{M} \) and \( \mathbf{M} \), hang one under the other. For this problem, take the positive direction to be upuark, ans use g for the mogntude of the tee sul acceloration. (tigure 1) Part C Find \( T_{3} \), the tension in the lower rope. Express your answer in terms of some or all of the variables \( \mathrm{M}, \mathrm{M}, \mathrm{a} \), and g .

To find \( T_{3} \), the tension in the lower rope, we start by analyzing the forces acting on the lower mass, which experiences a gravitational force downward and the tension upward. The equation governing motion can be given by: \[ T_{3} - Mg = -Ma \] Since the mass is accelerating downward, we can rearrange this equation to find \( T_{3} \): \[ T_{3} = Mg - Ma \] This indicates that the tension in the lower rope not only counters the weight of the mass below it but also accounts for the amount of acceleration \( a \) in the system. Essentially, the lower block experiences a difference between the gravitational force and the force due to its acceleration. Thus, the final expression for the tension in the lower rope is: \[ T_{3} = M(g - a) \]