# The Theory

Two equal masses, $M$, are hung from a rope that passes through two pulleys as shown. One mass hangs from a pulley in the middle of the rope and the other hangs from the end of the rope after the rope has passed through a pulley that is hanging from the ceiling. The other end of the rope is attached to the ceiling. In the actual situation the left-hand pulley which is attached to the first pass has a significant mass. Therefore, we will solve for the acceleration assuming the masses are different. If the masses are unequal, $m_1$ is the left=hand one and $m_2$ the right-hand one on the end of the rope: Newton’s second law for mass 1 is: $$2T -m_1g = m_1a/2$$ and for mass 2: $$m_2g -T = m_2a$$   Multiply the last equation by 2 $$2m_2g – 2T = 2m_2a$$ and add to the first one $$a = 2g \frac{2m_2 -m_1}{m_1+4m_2}$$ If $m_1$ = 0.130 kg, including the pulley,  and $m_2 = 0.100$ kg then $a = 2.641$ m/s$^2$.

## The Analysis

#### Predicted accelerations:

For mass 1: $a = -2.58$ m/s$^2$, and for mass 2 $a/2 = 1.295$ m/s$^2$

#### Estimated values from video analysis:

For mass 1: $a = -2.47$ m/s$^2$, and for mass 2 $a/2 = 1.32$ m/s$^2$.  The estimated uncertainty, judged by varying the ranges of the fit is about ±0.05 m/s2.