Masses and are connected to a system of strings and pulleys as shown. The strings are massless and inextensible, and the pulleys are massless and frictionless. Find the acceleration of .
Problem Source : An Introduction to Mechanics
We first look for the relationship between the tension of the two strings. Consider the tension force exert on the lower pulley with mass ,
Although there is acceleration on the pulley, it’s mass is . So, we can conclude that
Let the acceleration of be , then the acceleration of the lower pulley will be .
If is the length of the string, we have
Differentiating twice we have
Now, we have all the details needed to solve this problem, with two variables and two equations that study the forces on and .
We can also find the tension force on the string . Substitute on the second equation,