- #1

CAF123

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## Homework Statement

A light pulley can rotate freely about its axis of symmetry which is fixed in a horizontal position. A light inextensible string passes over the pulley. At one end the string carries a mass 4m, while the other end supports a second light pulley. A second string passes over this pulley and carries masses m and 4m at its ends. The whole system undergoes planar motion with the masses moving vertically. Find the acceleration of each of the masses.

## Homework Equations

[/B]

Newtonian/ lagrangian approach

## The Attempt at a Solution

The lagrangian approach is probably easier but I thought I'd try to crank out the answers using Newtonian mechanics. I would show a picture of the set up but the menu above does not seem to have an option unless I provide a URL. For the far left mass, I have $$T_1 - 4mg = 4m \ddot{x} = 4ma_1,$$ where ##a_1## is the acceleration of this mass relative to an inertial frame. Similarly, for the other masses attached to the second pulley, we have $$T_2 - mg = m_{2}a_2\,\,\,\,\text{and}\,\,\,\,T_2 - 4mg = 4ma_{3}$$ where ##a_{2,3}## are the accelerations of masses 2 and 3 relative to an inertial frame. Now, ##T_1 = 2T_2##. Write ##a_{i,P}## as the acceleration of the masses relative to the mobile pulley. So $$a_{i} = a_{i,P} + a_{P}$$ where ##a_P## is the acceleration of the mobile pulley relative to an inertial frame which is equal to -a_1. Rearranging the two equations for masses 2 and 3 above give $$T_2 = \frac{T_1}{2} = m(a_{2,P} - a_1 + g)\,\,\,\,\,\,\,\text{and}\,\,\,\,\,\,T_2 = \frac{T_1}{2} = 4m (a_{3,P} - a_1 +g)$$ Subsequently subbing in ##T_1 = 4m(a_1+g)## gives me 2 eqns with 3 unknowns.

A final constraint comes from the fact that the strings are all inextensible so that ##a_{2,P} = - a_{3,P}## Then ##a_{3} + a_1 = -(a_2 - a_1)## so that ##a_3 + a_2 + 2a_1 = 0##. We now have three eqns and three unknowns so can solve.

My answers are ##a_1 = -g/9, a_2 = -5g/9## and ##a_3 = 7g/9##. My answer for ##a_1## is correct but I seem to have the answers for ##a_2## and ##a_3## mixed up. Could be a mistake in the book but I just wanted to check that my reasoning above is sound - especially regarding what I have done relating accelerations from inertial frame to the non inertial accelerating frame of the mobile pulley.

Thanks!