Two Pulley Systems

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If a light string passes over two smooth light pulleys, then the tension in the string will be the same throughout. If one of the pulleys if fixed to the ceiling and the string passes around both pulleys in the arrangement below, we can applyto find the acceleration of each mass and the tension in the string.

Assuming the 6 kg mass accelerates down, as each of the lengths attached to its pulley increases by the length os the string between the 2 kg mass and its pulley must increase byThis means that the acceleration of the 2 kg mass is twice the acceleration of the 6 kg mass.

Applyingto the 6 kg mass gives

(1)

Applying F=ma to the 2 kg mass gives

(2)

(1) +2 times (2) givesthen from (2)

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