Block and Tackle

Question

A block and tackle with a mechanical advantage of 48 is used to lift a piano 11 m to the third floor of a building. Although the arrangement of pulleys in the block and tackle makes it easy to lift the piano, it takes a long time because of the length of rope that must be pulled to lift the piano a small amount. What is this length, or input distance, of the rope that must be pulled?

Answer 1

The mechanical advantage equals the ratio of the load versus the input force applied.

F_load

MA = ----------

F_input

So a mechanical advantage of 48 means that your block and tackle can lift an object which is 48 times heavier than the amount of force that you apply to it.

But the mechanical advantage is also directly proportional to the ratio of the input string velocity versus the velocity of the load.

(take velocity to be speed in this context, that is the amount of displacement within one unit of time.)

V_input

MA = ----------

V_load

(this becomes clear if you draw the free-body diagram for the weight being lifted in blocks and tackles with different numbers of pulleys. By adding each moving pulley the force ratio as well as the velocity ratio double.)

So, knowing that MA = 48 and the load displacement d_load = 11 m, we can use the last formula:

d_input

48 = ----------

11

That is, d_input = 48 × 11 = 528 meters

which is the final answer.