## Wednesday, March 24, 2010

### Lifting the Empire State Building with Pulleys

I did pull-downs at the gym the other day and felt unusually strong.  I was lifting 300 lbs like it was nothing, and then, despite my ego’s warning, I decided to check out the equipment because it seemed too easy.  It turns out the weights had not one but two pulleys on them!  This means I was actually only lifting one quarter of the total weight.  How many pulleys would a person need to lift the Empire State Building?

Pulleys can provide additional force to help move an object.  By attaching a pulley to a weight as shown here and pulling the rope with some force, you get double that force on the weight because there are two ropes pulling up on it.  If you add N pulleys to the weight, you will get 2N times as much force.  The trade-off for adding a pulley is that you now have to pull the rope twice the distance that you want to lift the weight.  For example, if you’re using a single pulley to lift an object 1.0 m off the ground, you need to pull the rope 2.0 m.  If you want to do the same thing but with two pulleys attached to the object, then you have to pull the rope 4.0 m.

The Empire State Building weighs about 7.3×108 lbs (~365,000 tons).  A reasonably strong person can lift 100 lbs.  To lift the Empire State Building, we’ll need the force applied by the pulleys to balance the weight of the building,

(weight of the Empire State Building) = 2 · (# of pulleys) · (force applied).

From this, we can solve for the number of pulleys,

# of pulleys = (weight of the Empire State Building) / [2 · (force applied)]
= (7.3×108 lbs) / [2 · (100 lbs)]
= 3.7×106 pulleys.

You’d need almost 4 million pulleys to lift the Empire State Building.  Even if you could fit them all, you’d still have a lot of rope to pull.  Just to lift the building 1 m off the ground, you’d need to pull the rope 4,500 miles, roughly the distance between New York and Athens, Greece.