The very first textbook chapter I edited (way back in 2008!) was on work and machines, so I have a soft spot for the topic. It’s also the focus of the “Science Playground” exhibit at the science museum I work for (and no, I will never, ever tire of the phrase “the science museum I work for”.)
There are lots of things to push and pull, and it’s a ton of fun interacting with guests who are lifting 550 pounds for the first time. It’s made me nostalgic for my textbook days, so I thought I’d summarize some of the key concepts and formulas here.
Simple machines- such as levers, pulleys, inclined planes (generally known as ramps), wheel/axle combinations, etc.- are devices that make work easier. In a departure from the linked discussion, I’m going to introduce a formula.
work = force x distance (Let’s assume that the relevant force is always in the same direction as the motion of the object, allowing us to skip the trig functions.)
This formula is necessary in order to drive home a key concept: simple machines do NOT reduce the amount of work necessary to complete a task. For example, in order to lift a 10 N object off the ground to a height of 3 m, you would have to exert a 10 N force over a distance of 3 m, resulting in 30 joules (J) of work. If you used a lever, pulley, or ramp to lift the object, the task would be easier, but not because you could do less than 30 J of work. You could, however, exert a smaller force over a longer distance. A 6 N force applied over a distance of 5 m still equals the required 30 J of work.
From here on out, we’ll refer to the force, distance, and work that YOU put in as the “input”, and the force, distance, and work that actually end up applied to the object as the “output”. Let’s visualize the previous situation using the simplest of the simple machines, the ramp.
From everyday experience, we know that pushing something up a ramp is usually easier than lifting it straight up. It feels easier because we don’t have to push as hard; in exchange, we have to push for a longer distance to get the object to the same point.
Since physicists can’t resist quantifying how much easier something “feels”, there’s a measurement called ideal mechanical advantage (IMA). In an ideal, perfect (read: frictionless) world, the IMA tells you how much easier a task becomes when you use a machine.
IMA = input distance/output distance
Since friction is everywhere, the IMA is exactly what it says on the tin: an unattainable ideal. Some of the input force you apply goes into overcoming friction instead of actually doing work to lift the object. In our imperfect world, there’s also a measurement called actual mechanical advantage (AMA) that tells you how much easier your job actually becomes when you use a machine.
AMA = output force/input distance
Efficiency is a measure of how “good” your machine is- since we know some of our input force will be wasted due to friction, what percent of our input work goes towards the intended task?
efficiency = (output work/input work) x 100
(Note the “x 100”; we’re so used to turning decimals into percents automatically that we forget a formal step is required!) Because of friction, you can never get more work OUT of a machine than you put into it, and efficiency can never be over 100%.
Exercise for the reader: review the formulas presented here. How can efficiency be expressed in terms of IMA and AMA? Hint- to divide by a fraction, flip it over and multiply!