Pulleys and counterweights have been used for centuries to gain mechanical advantages in war machines and industrial equipment. The physics behind them is fairly simple, but their importance is paramount.
In our modern world, pulleys and counterweights are used in a wide variety of applications. Being the engineer that you are, you likely realize that every time you step on an elevator, you are engaging what is likely a system of pulleys and counterweights. In terms of machine design, pulleys are perhaps the easiest way to gain a mechanical advantage. Another common way to increase your output force in modern mechanics would be levers, which pulleys function very similar to.
Without getting into all of the various pulley set-ups, at their core, pulleys have a wheel and a rope. A pulley with only one fixed wheel simply reverses the direction of the input force relative to the mass being pulled. With only one wheel, in order to lift a mass of 100 kg, you would need to exert a 100 kg force equivalent (1000 Newtons) on the other end of the rope. When more wheels, or blocks, are added into the simple machine that is a pulley system, you gain more and more mechanical advantage. With a system of 2 wheels, you could lift a 100 kg weight by only exerting 50 kg of force equivalent to the rope. Calculating input forces gets a little more complicated the more blocks you add as well as through different set-up variations. However, even the most complex pulley systems can be understood by adding up lengths of the different stretches of rope and generating equations from derivations.
The most important thing to remember is that pulleys aren’t magical systems that simply transform a small force into a larger force. This may seem true if you focus on the input and output, but the mass of what is being lifted is always translated into the fixed points where the pulleys are attached to your rigid system. Rather than transforming forces, we can say that a system of pulleys and ropes manages forces to maximize how much mass you can lift or move.
Now that we have a basic grasp of pulley systems, we can dive into counterweight systems to gain even more mechanical advantage in machine design. Counterweights are both used in pulley systems, like elevators, as well as lever systems, like a crane. In terms of crane counterweights, it is all about moments. To keep a crane arm rigid while lifting a 100 kg mass that is 10 m away from the fulcrum, you would need a 1000 kg counterweight 1 m from the fulcrum on the other side to stabilize the crane arm. In cranes, counterweights typically maintain a constant mass, so in order to adjust for different loads, the counterweight usually has the ability to be moved to adjust the moment. This is obviously a very rudimentary explanation of crane counterweights.
In terms of pulley systems, a counterweight simply helps to apply an input force to lift an object. The easiest way to demonstrate this would be to look at an elevator system. In typical elevators, there is a counterweight attached to the other end of the pulley system equivalent to about the weight of the car at 50% loading. Counterweights don’t do all of the work in modern elevators, but they help stabilize the system and reduce the load on the elevator motor. Assuming an elevator car was loaded at 25% of its capacity, all the motor would need to do is brake the elevator car at the right floor (when going up). The counterweight in an elevator also means that if a motor brake were to fail, the car would be slowed from falling quite as fast as it would otherwise. Elevator systems as a whole are a little more complicated than explained here, but looking at counterweights, this is the essence of their function.
So, through a combination of pulleys and counterweights, we can design machines that give us mechanical advantage and make moving large masses fairly easy with a small input force.