A primer on Work, Energy, Power and other mechanical terms

If you are new to the science of mechanics, then you probably have seen a lot of unfamiliar words floating around. Some may sound familiar, but in mechanics their meaning may be something very different than their meaning in casual conversation. Other mechanical terms may be very new to you. Don't worry. Building a bot is a learning experience and most the terms are easy to grasp once you get the hang of it. In this tutorial, we'll examine some of the more common terms (or "units of measure") that you will likely encounter as you design and build your bot. We can break down these terms into several areas which include:

• Length or distance
• Mass
• Time
• Force
• Pressure
• Work and Energy
• Power

Length or Distance

This is the easiest unit of measure to grasp. Just about everyone has used a ruler to measure the length of something at one time or another. The metric system uses the "meter" as the base unit of measure. The meter is sometimes broken into smaller units called centimeters or even smaller units caller millimeters when its convenient to do so. Avoirdupoir units use the "foot" (or feet when plural) which in turn is broken down into inches when convenient. Use the following table to convert from one unit of length to another:

• 1 foot = 12 inches
• 1 inch = 25.4 millimeters
• 1 millimeter = 0.1 centimeter
• 1 millimeter = 0.0394 inches
• 1 centimeter = 10 millimeters
• 1 centimeter = 0.01 meter
• 1 meter = 39.37 inches
• 1 meter = 3.281 feet
• 1 meter = 100 centimeters
• 1 meter = 1000 millimeters

These units are often abbreviated as follows: "feet" as "ft", "inches" as "in", "meter" as "m", "centimeter" as "cm" and "millimeter" as "mm". You may also be interested to know that "length" is a "base unit of measure" which means it can be used to derive other units of measure (such as power which is discussed a little later). However, the reverse isn't true. Other base units include those of mass and time.

Mass

Mass is one of the most misunderstood terms. Most people perceive "weight" as "mass". So let's clarify the difference between weight and mass. Weight is actually a force. So when you stand on a scale to measure your weight, you're actually measuring the force of gravity the Earth is exerting against the mass of your body. Still confused? Ok, suppose you were to take your bathroom scale into space where there is no gravity, then measured your weight. You would find that you have 0 weight (you're weightless) even though you haven't gotten any thinner. Your mass hasn't changed, but without gravity there is no force pulling on you to generate the force we're calling "weight". This is an important point since not understanding the difference can render any calculations using mass totally useless.

So how do we measure mass? We use a scale. But wait... didn't we just say that we shouldn't use scales to measure mass because scales measure weight? Well, yes and no. While it's true that scales measure weight, we can still use them to measure mass. The reason we can do this is because the Earth's gravity is more or less constant no matter where you are (unless you're falling out a window). Given this fact, we can use one of Newton's equations to find the mass of an object if we know it's weight. Newton tells us that "F = m * a" which is read as "force is equal to the mass of the object times its acceleration". It turns out, we know what the acceleration is due to the Earth's gravity. It's 9.8 meters/second^2 or in avoirdupoir units it's 32.2 feet/second^2. If we rearrange Newton's equation to solve for mass we get "m = F / a" which says that all we need to do is divide the force shown on our scale by the acceleration generated by the Earth's gravity to get the mass. But wait! If you have a scale that uses the metric system, this has already been done for you. The scale has been calibrated to display the mass of an object directly. (It can do this because we know the Earth's gravity is constant.) It's when you use a scale that is calibrated in "pounds" that we run into trouble. Though we haven't discussed force yet, you should know that a "pound" is a measure of force, not mass. The avoirdupoir system measure of mass is called the "slug". To get the number of slugs, simply divide the weight (in pounds) by 32.2 ft/second^2 (which is Earth's gravity in avoirdupoir units). One more thing you should know, there is a pseudo-unit of mass in the avoirdupoir system called the pound-mass which is abbreviated as "lbm". This is an improper unit of measure, and it is what your bathroom scale is displaying if it's calibrated in pounds. However, since it's easily converted to slugs, we can use it with some restrictions applied.

So now it's time to summarize units of measure used to describe mass. The metric system uses the "gram" (or more often the kilogram which is 1000 grams) to measure mass. The avoirdupoir system uses "slug" or the "lbm". To convert between these, use the following table:

• 1 gram = 0.001 kilograms
• 1 kilogram = 1000 grams
• 1 lbm = 0.0311 slugs
• 1 slug = 32.2 lbm
• 1 kilogram = 2.2 lbm
• 1 lbm = 0.455 kilograms

Time

This is the easiest one of the bunch. Time is the same regardless of whether we use the metric system or the avoirdupoir system. For anything we're likely to do with our bot, you can safely assume that time will be measured in seconds (sometimes abbreviated as "sec"). For measuring things that are really fast, you may see the unit "ms" which is milliseconds or one thousandth of a second.

Force

We have all heard the term "force", but we need to define what a force is just to be clear. For our purposes, we can define a force as a push or pull. For example, when we try to pick up a heavy object, we feel the force of gravity trying to pull it back down. Most of us have experienced two magnets pushing or pulling towards each other. This is a magnetic force. Compressing a spring causes it to push back against us with a force. Now that we have an idea of what a force is, let's define a way to measure it.

Force is the first of our 'derived' units. We say it's derived because it's actually a combination of the base units we defined earlier. In particular, force is "mass * length / seconds^2". As you can see, the fundamental definition of force is rather awkward, so we create new units as a form of shorthand to make things a little easier. The units of force we will likely encounter with our bot are the "Newton" ("N") and the "pound" ("lb" or sometimes "#"). (Note: you may see the pound also abbreviated as "lbf" which is short for pound-force. This is to distinguish it from the pound-mass unit discussed earlier. You may be beginning to see why the avoirdupoir system of measure is frowned upon in the scientific community.) Use the following conversion table to convert between pounds and Newtons:

• 1 pound = 4.45 Newtons
• 1 Newton = 0.2247 pounds

Pressure

Mechanical pressure is not the same as the pressure we might find ourselves experiencing in competition with our bot. In mechanics, pressure is defined as a force per unit of surface area. For example, in the metric system we can describe pressure in units of Newtons per square meter ("N/m^2"). This same unit is also called a "Pascal" ("Pa"). Both terms mean the same thing. In the avoirdupoir system, we use pounds per square inch (psi). Now that we have defined pressure and the units it's measured in, what does it mean? It's simple. Let's get a better feel for pressure by restating our definition which says pressure is a force per unit area. What this means is that for a given surface area (say 1 square meter), we have a certain amount of force. So if the pressure is 100 N/m^2 (or equivalently, 100 Pascals) and a surface area of 1 square meter, then the push against the surface will be 100 Newtons. If we double the surface area, we now get 100 N/m^2 * 2 m^2 = 200 N. In generic terms terms, the force exerted against an area is equal to the size of the area times the pressure applied against it. (Beware! The units of measure must be consistent. Don't mix psi with square meters for example.) As a practical example, it's not uncommon for someone to ask how much force a pneumatic ram will exert for a given air pressure and piston diameter.  Let's assume we have a 50mm diameter piston and we are using a pressure of 1 million Pascals (abbreviated as 1 MPa where the capital M is short for 'million' and where MPa is pronounced as a "megapascal"). The first thing we need to do is make our units consistent. (We'll examine why dimensions are important in more detail later.) In this case, the air pressure is given as Newtons per square meter. Notice that the length dimension contained in our pressure unit of measure is in meters. But our piston is given in millimeters. We need to convert one or the other to the same units before we can get the right answer. For simplicity, we'll convert the piston diameter to meters. 50 millimeters converts to 0.05 meters (50 millimeters divided by 1000 millimeters per meter).

which tells us the surface area of a circle for a given diameter. Therefore

A = 3.1415 * (D / 2)^2
A = 3.1415 * (0.05 / 2)^2
A = 3.1415 * (0.025)^2
A = 3.1415 * 0.000625
A = 0.00196 m^2

F = pressure * area
F = 1,000,000N/m^2 * 0.00196 m^2
F = 1963 Newtons

Now for our table to convert pressure from one system to the other. We'll also introduce the unit of pressure called "atm" (short for 'atmosphere') or "bar" (a truncation of 'barometric' as in barometric pressure) both of which mean the same thing and refer to the atmospheric air pressure at sea level as a standard of measure:

• 1 Newton per square meter (N/m^2) = 1 Pascal (Pa)
• 1 N/m^2 = 0.000145 psi
• 1 psi = 6891 N/m^2
• 1 ATM = 1 bar
• 1 ATM = 14.7 psi
• 1 ATM = 101300 N/m^2

Work and energy

This is where the confusion really starts. Work, energy and power are the most misused terms in bot building. To clear things up, let me state the definitions of both 'work' and 'energy'. First we'll start with energy. Energy is defined as something that has the potential to do work. The 'something' can be a bottle of compressed gas, a charged battery, a stretched rubber band, etc.. Energy is something that we can convert into a force. The units of measure of energy that we'll likely use in our bot design are the metric Newton-meter (Nm) which is also called a "Joule" (J). The avoirdupoir measure is the "pound-foot" (lb-ft). Notice that both systems express energy as a force times a distance. This brings us to the concept of work. 'Work' is defined as a force applied over a distance and uses the same units as energy. This should be no surprise since it takes energy to do work. This brings up the question of "what's the difference?". The difference is that energy refers to the ability to do work, and work is the quantity of energy used when we do it.  It's a case of before we do something and after. Let's use an analogy to help us understand the difference. Suppose that money is a form of energy. We put this money in a bank for later use. This bank account represents stored energy. Now suppose we spend a portion of this money to pay a bill. The money that was paid out performs a function (it pays a bill) and depletes our bank account. The payment of our fictional bill represents work performed. But we can only pay so many bills before our bank account is drained, unless we put something back in. To do this, we go to 'work' to earn the money needed to recharge our bank account. Notice that all parts of our fictional bank transactions use the same units of measure such as 'dollars'. The concept of earning money, storing it in a bank, then spending it is very representative of the systems we find in a bot. For example, the batteries we use in our bot are likely rechargeable and perform the 'bank' function. We withdraw money in the form of electrical current to run our motors. After a match, we recharge our battery bank account using a battery charger in an effort to replace the energy spent (aka. work) by the motors. You may be asking why electrical work, such as produced by our batteries, is still measured as a force times a distance. The answer is that energy is energy regardless of its form and you can convert one form of energy to another. For example, the electric motor(s) in our bot converts electrical energy to mechanical energy which allows our bot to push (Pushing is force!) another bot over a distance. Get it? A force over a distance. The bot did work against the other bot. (There is one thing that you may not be aware of. Converting electrical energy to mechanical energy is not perfect. For example, if both bots are equally matched and a pushing contest erupts such that neither is covering any distance. We still burn energy even though no distance is covered. The energy is lost to heat produced in the motor windings because it's not turning freely while the bot is stuck against its opponent. Heat is another form of energy.)

As an example of work, let's take a second look at our imaginary pneumatic ram we proposed in our discussion of pressure. In addition to the pressure and piston size already given, let's assume it has a travel of 100mm. Given the information provided, how much work can it deliver if we allow it to travel its full length? This is very easy. Just multiply the force the piston is producing times the distance it travels. First convert the distance traveled to meters so that the units match. 100mm is 0.1 meters. Next, multiply this by the force we found in our pressure example which was 1963 Newtons. We get:

E = force * distance
E = 1963 Newtons * 0.1 meters
E = 196.3 Newton-meters (Nm)

• 1 Joule (J) = 1 Newton-meter (Nm)
• 1 NM = 0.7374 LB-ft
• 1 LB-ft = 1.356 NM

Power

I saved the most abused unit of measure for last. Power in casual conversion means something very different than when it's used in describing a mechanical system. In casual conversation, we equate power with the ability to do something. We tend to apply the term in much the same manner when we speak of machinery. But as we have just seen, the ability to do work is the definition of energy. Power is properly used to describe the rate at which we can produce or use energy. In other words, how fast we can do work. Formally, it's the amount of work performed per unit of time. There are lots of units in common use to describe power. These include horsepower (hp), Watts (W),  Newton-meters per second (Nm/sec), and lb-ft per second (lb-ft/sec).  Here's a conversion table:

1 hp = 746 Watts
1 hp = 550 lb-ft/sec
1 Watt = 1 Joule/sec
1 Watt = 0.00134 hp

Dimensions

As mentioned earlier, there is a something known as a "dimension". We've heard about the dimensions of an object. We usually think of dimensions in terms of width, height, and depth.  But in science, a dimension is defined as any unit of measure. For example, the units of measure used to describe power are dimensions. So are the units for  time, length, and mass. When we say that we need to use consistent units of measure, what we are really saying is that we need to use the same dimensions throughout our calculations, even if it means having to convert from one dimension to another (such as when we converted millimeters to meters in our piston example). Let's look at our example of solving for the work produced by our pneumatic ram. Remember that we said the piston was 50mm in diameter, that it traveled 100mm, and that we intended to use of pressure of 1MPa. To see what happens when we abuse dimensions, I'll 'forget' to convert the millimeters to meters and see what we get if we repeat our energy calculations:

First: find the piston area:

A = 3.1415 * (50 / 2)^2
A = 1963.5

Now multiply by the pressure to find the force:

F = 1963.5 * 1,000,000N/m^2
F = 1,963,495,409 N

Finally, find the work performed by multiplying F times the distance:

E = 1,963,495,409 * 100
E = 196,349,540,900 NM

Hmmm, that result is lot different from the correct result of 196Nm. The error is because we 'forgot' to convert millimeters to meters which in this case caused the decimal point to get placed in the wrong spot. The easiest way to see if we set up our equations correctly is to solve the equations using just the dimensions themselves (no numbers, just units). Let's try an example repeating our mistake from above and see what we get. To make things a little easier, I'll combine all three steps into single equation, first with the numbers provided so you can see how the equation was set up, then I'll remove the numbers and leave just the units:

E = pressure * area * distance
E = 1MPa * (3.1415*(50mm / 2)^2) * 100mm
E = 1MPa * (1963 mm^2 ) * 100mm

Remember that a MPa is actually 1 million N/m^2. Making this substitution and removing the numbers we get:

E = N/m^2 * mm^2 * mm

Combining the two "mm" terms and doing some minor rearranging, we finally get:

E = N-mm^3 / m^2

This is read as "Newton millimeter cubed per square meter". It should be simply "Newton meters". We can see that the dimensions came out to something very different from what we were expecting. This is illustrates how we can use dimensions as a quick way to spot errors when setting up our equations. If the dimensions come out wrong, then you need to look for errors in your equations because something is using the wrong units. Let's repeat our example using the proper units and conversion factors. Remember that 1mm is 0.001 meters or 0.001 meters/millimeter (m/mm). I'll leave the numbers in and algebraically combine the units of measure as we did before:

E = pressure * area * distance
E = 1MPa * [3.1415*(50mm * 0.001 m/mm / 2)^2] * (100mm * 0.001 m/mm)
E = 1MPa * [3.1415*(0.05m / 2) ^2] * (0.1m)
E = 1MPa * (0.001963m^2 ) * (0.1m)
E = 1,000,000N/m^2 * 0.001963m^2 * 0.1m
E = 196.3Nm

Amazing. The units are correct this time. As you can see, it's not a good idea to drop the units from your equations as it invites mistakes. But this method of catching errors is not perfect. Even if you get the proper units in your result, it doesn't guarantee that the numeric values are correct. You still need to be careful. (A missed decimal place can lead to serious errors for example, but it won't show up in a dimensional check.) There is also another trap in that some dimensions may actually be correct, but will still fail when checking dimensions. This usually happens when we use derived units of measure instead of the base units. For example, if we didn't know that 1 MPa was actually 1 million N/m^2, we'd get something like:

E = 0.0001963 MPa-m^3

which would be read as "megapascals meters cubed" which is actually correct, but very confusing. As you can see, it's a good idea to stay with the base units whenever possible in your calculations to avoid this particular trap. Just for fun I'll prove that 0.0001963 MPa-m^3 is really correct. Remember that 1MPa is actually 1 million Newtons per square meter or in other words "1 MPa = 1,000,000N/m^2". Let's take our last result and substitute the MPa unit with the 1,000,000N/m^2 equivalent:

E = 0.0001963 MPa-m^3
E = 0.0001963 * (1,000,000N/m^2) * m^3
E = 196.3 NM

No surprises here. The units are now as we expected since we replaced the MPa unit with the base unit.

Summary