Hello, I am wondering if someone can answer me this question: In an Atwood machine why do the tensions that handle both bodies are the same? (English is not my native language)

well, assuming that the string exerts no torque on the wheel and that the string is of zero mass, there is nothing to change the tension in the string as it runs over the wheel.

(In real life of course the wheel has mass and moment of inertia and the difference in tension on each side of the wheel accelerates the spin of the wheel. However this effect is assumed small because both the friction and moment of inertia of the wheel are considered small)

Of course! I'd be happy to explain that. In an Atwood machine, the tensions in the strings or ropes that handle both bodies are equal (assuming the ropes are massless and don't stretch) because the two masses are connected by the same rope or string.

To understand why the tensions are the same, let's break down the concept of tension. Tension is a force transmitted through a string or rope when it is pulled at both ends. When the rope or string is in equilibrium (not accelerating), the tension at any point in the string is equal in magnitude and opposite in direction.

In an Atwood machine, there are two masses connected by a single rope or string that passes over a pulley. The pulley helps change the direction of the tension force but doesn't affect its magnitude. As a result, the tension on the side of the rope connected to one mass is the same as the tension on the other side connected to the other mass. This equality of tension forces allows the masses to be in equilibrium, and it ensures that they experience the same acceleration (or lack of acceleration) in opposite directions.

To illustrate this, imagine two masses, m1 and m2, connected by a rope or string passing over a frictionless pulley. If we assume m1 is larger than m2, the heavier mass (m1) will tend to pull the rope downward, creating tension in that part of the rope or string. At the same time, the lighter mass (m2) will cause the rope to be pulled upward, creating tension on the other side of the rope. Since the rope is continuous and connected, the tensions on both sides must equalize, ensuring equilibrium and allowing the Atwood machine to function as intended.

I hope this explanation helps! Let me know if you have further questions.