For thousands of years mankind has been perfecting the art of lifting heavy things, manipulating the laws of physics using nothing more than a few simple tools and a bit of ingenuity to allow huge loads to be moved very easily. This is only possible because cleverly designed machines like this pulley system are able to amplify forces in some way. The amplification of a force is called mechanical advantage, and we can quantify the amount of mechanical advantage a specific system can generate as the force that is output by the system divided by the input force.
To illustrate this consider a mass of 100 kg. To lift it off the ground you need to apply a force of around 1 kN. But if you use a device with a mechanical advantage of 2, you only need to input a 0.
5 kN force. To explain how this is even possible, this video will take a look at three devices - levers, pulleys and gears - that are able to do just that. I've also covered a fourth way of generating mechanical advantage, using hydraulic systems, in a companion video that you can watch right now over on Nebula.
The best way to get access to Nebula is by signing up to our bundle deal with this video's sponsor CuriosityStream, but more about that later. Let's start with the lever. Consisting of just a beam pivoting about a fixed hinge, called the fulcrum, it's one of the simplest ways of generating mechanical advantage.
Consider the amount of force we need to balance a mass located at one end of the beam. When discussing levers it’s common to talk about a load, which in this case is the mass, and an effort, which is the force that needs to be applied to lift the load. It’s quite obvious that this system with the fulcrum in the middle of the beam can only be balanced if the effort is equal to the load.
But what if we move the fulcrum away from the centre of the beam, towards the mass? We now only need a much smaller effort to balance the lever. But why?
We can explain this by considering the rotational stability of the lever. Both the load and the effort have the effect of rotating the lever, which means that they are generating moments about the fulcrum. A moment is a quantity that determines how much a force causes something to rotate about a specific point.
It is calculated as the magnitude of the force multiplied by the perpendicular distance from the line of action of the force to the point of interest. For the lever to be in rotational equilibrium the moments acting about the fulcrum must balance each other. We can then easily calculate how large the effort needs to be to balance the load, based on the distances a and b.
Earlier we defined mechanical advantage as the ratio of the output to the input force, so we can see that for a lever it is calculated as the ratio of the two distances. By moving the fulcrum we can change the mechanical advantage. Putting it very close to the load means we will be able to lift the mass with a much smaller force, which is incredibly useful.
It might seem like the lever is somehow breaking the laws of physics, by turning a small force into a much larger one. But it’s important to remember that levers don’t actually create energy. The amount of energy used to lift a mass is equal to the magnitude of the applied force multiplied by the distance over which the force acts.
So although the force applied to lift the mass when using mechanical advantage is smaller, it needs to be applied over a longer distance. The amount of energy used is always the same, with or without mechanical advantage. All devices that generate mechanical advantage work on this same principle - they convert a small force applied over a long distance into a large force applied over a short distance.
The pulley, which is essentially just a wheel containing a groove through which a rope or cable can run, is a bit more difficult to understand than the lever, but opens up even more possibilities for generating mechanical advantage. Let’s look at a simple configuration where a rope is passed through a single pulley and is attached to a mass. If a large enough force is applied to the end of the rope, the mass can be lifted.
This system allows us to re-direct the force needed to lift the mass, so that we can pull on the rope at an angle instead of having to lift the mass vertically off the ground, but it doesn’t actually provide any mechanical advantage - if the mass has a weight of 1 kN, a 1 kN force has to be applied to the end of the rope to lift it. Because the rope is being pulled on by the applied force at one end, and by the weight of the mass at the other, it's subjected to a tensile load of 1 kN along its entire length. In reality a slightly larger force will need to be applied to compensate for friction at the pulley, but to keep things simple we'll assume that the pulley is both frictionless and massless.
Let’s look at a different setup now, where the pulley is flipped upside down and is attached to the mass instead of the ceiling. This small modification completely changes how the system works - it might seem impossible but we now only need to apply half the force we previously did to lift the mass. This system has a mechanical advantage of 2.
The reason is that the mass is now supported by two segments of rope, instead of just one. To better understand what's happening let's take a look at the forces acting on the pulley. There is one downwards force, the weight of 1 kN, and two upwards forces caused by the two parts of the same rope that are supporting the pulley.
The tension in the rope is constant along its length, so the two upwards forces must be equal to each other. And for the system to be in equilibrium they must balance the 1 kN load. We can conclude that both upwards forces are equal to 0.
5 kN. Since half of the load is taken by the ceiling, the load can be lifted by applying a force of only 0. 5 kN to the end of the rope.
To make this system slightly easier to use, we can add a second pulley so that instead of pulling up on the free end of the rope we can pull on it at an angle. Like with the lever, the cost of dividing the force needed to lift the mass by 2 is that we have to apply that force over a double the distance. The additional distance comes from the fact that two segments of the rope need to be shortened to raise the mass, instead of just one.
Adding even more pulleys allows us to achieve greater mechanical advantage. This system has a mechanical advantage of four, for example. The mass is supported by 4 rope segments so producing an output force of 1 kN only requires an input force of 0.
25 kN. Because using this many pulleys takes up a lot of space, the individual pulleys are usually combined into two blocks, one movable block that moves with the mass, and one standing block that is fixed to the ceiling. Gears are another type of simple machine that generates mechanical advantage.
The teeth of one gear interlock with identical teeth on another to form gear trains that can be used to transmit and multiply forces. In each set of gears we can identify a driver gear that’s connected to a motor or a rotating shaft and transmits force to a follower gear, causing it to rotate in the opposite direction. In this case the follower gear, which has 20 teeth, is smaller than the driver gear, which has 40 teeth.
Since the circumference of the larger gear is twice that of the smaller gear, the smaller gear must rotate twice for every one rotation of the large gear. This configuration is called a speed multiplier - the follower gear is rotating twice as fast as the driver gear. To understand how gears can generate mechanical advantage let’s look at how force is transmitted through the system.
This motor provides a torque of 100 Nm to the driver gear. Torque is a moment that acts about a longitudinal axis, and is calculated as the magnitude of the force multiplied by the distance to the axis of rotation. The torque of 100 Nm is equivalent to someone turning the shaft with a force of 100 N at a distance of 1 m, or a force of 200 N at a distance of half a metre, for example.
The input torque causes the driver gear to rotate, generating a tangential contact force between the two gears. We can calculate the magnitude of the contact force as the input torque divided by the distance between the force and the axis of rotation, which is just the radius of the driver gear. The contact force causes the follower gear to rotate, generating an output torque that can be calculated as the product of the contact force and the radius of the follower gear.
This tells us that the mechanical advantage of the system is given by the ratio of the radii of the two gears. This is the same as the number of teeth on the follower gear divided by the number of teeth on the driver gear. With 40 teeth on the driver gear and 20 on the follower gear, this particular system has a mechanical advantage of 0.
5. Doubling the speed of the follower gear comes at the cost of halving the output torque. But if the smaller gear is the driver gear, the system will have a mechanical advantage of 2 - the output speed is halved but the output torque is double the input torque.
Even when a system has more than two gears the mechanical advantage only depends on the number of teeth on the first and last gears. This means a third gear, called an idler gear, can be added to make the driver and follower gears turn in the same direction, without affecting the mechanical advantage of the system. Using a small driver gear to turn a large follower gear and generate large torques is how a hand winch works.
A drum around which a cable is wound is attached to the large follower gear, and the small gear is driven by turning a handle. The larger number of teeth on the follower gear means that the input torque from the handle is amplified, allowing heavy objects to be moved by the winch. Levers, pulleys and gears aren't the only devices that can amplify forces.
Hydraulic systems, like you would find in a hydraulic press for example, use the incompressibility of liquids to generate mechanical advantage, which makes them useful for applications like the deployment of landing gear in aircraft flight control systems. If you're interested in learning more I've just published a video on Nebula that goes into the detail of how exactly hydraulic systems are able to generate mechanical advantage, and the different components you need to build a functioning one. Nebula is a creator-built streaming site that’s home to a growing collection of exclusive or extended Efficient Engineer videos, but also loads of ad-free content from over 100 other educational creators on the platform.
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