We ramp up the level of difficulty and turn to the magic of combining pulleys to get establish a ``mechanical advantage''. A typical set-up is shown here. It is analogous to the Atwood machine in that two masses are connected by a cord but here the cord loops around several times between wheels on two displaced ``friction-less'' axis. Eventually the cord is anchored to the axel of the lower pulley.
To analyze this problem the first thing to notice is that if we make the usual assumptions that the pulley wheels are mass-less and friction-less and the cord too is mass-less, then as in the case of the Atwood machine, there is just a single tension, T, in the cord. We figure the tension by applying Newton's law.
In this case we have to much friction to obtain sensible results for the acceleration of the device so we just consider what will be the conditions for balancing the masses.
We apply Newtons second law to the mass hanging at the end of the string.
Since it is not accelerating we conclude that no net force is acting on it
Now because the cord loops around the effects of this tension is felt
multiple times. If we focus on a single loop going round a pulley
on the lower axis then each of the strands of the cord pull upward by
a force of magnitude, T, on the lower pulley. Thus the pulley experiences
a force 2T from a single loop. In fact each cord in tension between
the lower and upper axis exerts an upwards force of magnitude T. The total
force on the lower pulley system and mass is therefore
Where N=5 (here) is the total number of cord-segments in tension between
the pulleys. We can now write Newton's second law for the
brick attached to the lower pulley:
eliminating T by subtracting N times Eq. 23 from Eq. 25
yields:
We check out the masses and see that indeed they are in this ratio to each other.