Which expression gives the tension in the string of an Atwood machine with masses m1 and m2 (m1 > m2)?

Which expression gives the tension in the string of an Atwood machine with masses m1 and m2 (m1 > m2)?

• A. T = (m1 − m2) g
• B. T = m1 g
• C. T = m2 g
• D. T = 2 m1 m2 g /(m1 + m2)

Answer: (D)

In an Atwood machine, the two masses share the same acceleration, with the heavier mass moving downward and the lighter mass moving upward. The string tension acts upward on the heavier mass and downward on the lighter mass, so we can write Newton’s second law for each mass. For the heavier mass: m1 g minus T equals m1 a (downward acceleration). For the lighter mass: T minus m2 g equals m2 a (upward acceleration). Adding these two equations eliminates T and gives (m1 − m2) g = (m1 + m2) a, so the acceleration is a = (m1 − m2) g /(m1 + m2). Now solve for the tension using one of the equations, say T = m1 g − m1 a. Substituting a yields T = m1 g − m1[(m1 − m2) g /(m1 + m2)] = [2 m1 m2 g/(m1 + m2)]. This is the tension in the string. Geometrically, the tension lies between m2 g and m1 g: it’s greater than the lighter weight’s pull but less than the heavier weight’s, and it reduces to m g when the masses are equal.

In an Atwood machine, the two masses share the same acceleration, with the heavier mass moving downward and the lighter mass moving upward. The string tension acts upward on the heavier mass and downward on the lighter mass, so we can write Newton’s second law for each mass.

For the heavier mass: m1 g minus T equals m1 a (downward acceleration). For the lighter mass: T minus m2 g equals m2 a (upward acceleration). Adding these two equations eliminates T and gives (m1 − m2) g = (m1 + m2) a, so the acceleration is a = (m1 − m2) g /(m1 + m2).

Now solve for the tension using one of the equations, say T = m1 g − m1 a. Substituting a yields T = m1 g − m1[(m1 − m2) g /(m1 + m2)] = [2 m1 m2 g/(m1 + m2)]. This is the tension in the string.

Geometrically, the tension lies between m2 g and m1 g: it’s greater than the lighter weight’s pull but less than the heavier weight’s, and it reduces to m g when the masses are equal.