Making Things Move: DIY Mechanisms for Inventors, Hobbyists, and Artists - Dustyn Roberts (2010)

Chapter 4. Forces, Friction, and Torque (Oh My!)

Have you ever tried to push open a door, only to realize you’re pushing on the wrong side—the one closer to the hinge—and it’s really hard to open? This happens because a door needs a certain amount of torque to open, and if you push too close to the hinge, you have to use a lot more force than if you push at the handle to create the same amount of torque. I suggest pushing on doors in the middle to avoid embarrassment.

In order to estimate torque, know where on the door to push, figure out if something will break, choose a motor, or pick a material for a project, it helps to think about the world around us in terms of numbers. So, before we get to examples of forces, friction, and torque, we need to review some math.

Torque Calculations

First, you need to understand the relationship between force and torque (also called moment). We talked about force in Chapter 1. Just as a force can be thought of as a push or pull, torque can be thought of as turning strength.

Torque is how hard something is rotated. More specifically, torque is force multiplied by the perpendicular distance to the axis of rotation. This distance is also called the lever arm or moment arm :

Torque = Force × Distance (⊥)

In the case of the unruly door, the hinge is the axis of rotation. You can see from Figure 4-1 and the equation that the greater the distance from the applied force to the hinge, the greater the torque. The force in this case is you pushing open the door. So when you accidentally push on the door very close to the hinge, you need to push with a lot of force to create the same torque as pushing with just a little force farther away from the hinge.

FIGURE 4-1 A picture definition of torque


You can feel torque in action with a simple exercise. Grab a can of soup from your pantry and hold it in your right hand, all the way out to the side so your arm is parallel to the floor. The strain you feel in your shoulder is your muscles creating the necessary torque to support the soup can. Your shoulder is acting as an axis of rotation, and the torque is the force of the can (its weight) multiplied by the distance the can is from your hand to your shoulder (see Figure 4-2). If the can weighs 1 lb, and your arm is 2 ft long (d1), the torque at your shoulder is 1 lb × 2 ft = 2 ft-lbs.

FIGURE 4-2 Shoulder torque when holding a can with your arm parallel to floor


Your shoulder will get tired after a while in this position, so lower the can about halfway (see Figure 4-3). Now the torque on your shoulder is less, even though your arm is still the same length and the can weighs the same. Why? Because the force of the can is still pointed down (gravity always is!), but the perpendicular distance to the axis—your shoulder—is smaller (d2).

Torque always has units of distance × force (sometimes written as force × distance). Unfortunately, there are many ways of measuring distance and force, so torque can be in foot-pounds, ounce-inches, millinewton-meters, and so on. You can go to and convert from any measurement you come across to one you prefer. There are even smart phone apps that do this for you. For example, if you find a motor that lists its torque in millinewton-meters, and foot-pounds make more sense to you, convert the motor torque to foot-pounds.

FIGURE 4-3 Shoulder torque while holding a can at an angle


The last bit of math we need to review before going through some examples is the basic geometry of triangles you probably learned in high school. Remember sine, cosine, and tangent (abbreviated sin, cos, and tan)? Those three trigonometric properties help you to estimate torque when, as in Figure 4-3, the line of force isn’t at a convenient right angle to what connects it to the axis of rotation.

A right triangle has one angle that’s 90° (indicated by the box in the corner of the triangle in Figure 4-4), and the side opposite the 90° angle, the longest side, is called the hypotenuse. The cool thing about right triangles is that you can figure out any one number you want—a side length or angle—just by knowing any two other numbers and using sine, cosine, or tangent. To remember the relationships, think SOHCAHTOA. It’s a mneumonic device to remember these formulas:

Sin (angle) = Opposite Side / Hypotenuse

Cos (angle) = Adjacent Side / Hypotenuse

Tan (angle) = Opposite Side / Adjacent Side

Use these relationships to solve for the distance of side X in Figure 4-4. All we know is that one angle is 45° and the hypotenuse is 2 ft. Since we want to solve for the side that is adjacent to the angle we know, we can use the cosine, like this:

cos (45) = X / 2

Now rearrange the equation to solve for X:

X = cos (45) × 2

When you type cos 45 into your calculator, the answer should be 0.707. Multiply that by 2 to get the distance of side X = 1.4 ft.

Now, did you realize you just solved for unknown distance d2 in Figure 4-3? If you assume the arm is at 45° from horizontal and 2 ft long, that’s exactly what you’ve done. In Figure 4-2, we already figured out that the shoulder torque when holding the can of soup straight out was 2 ft-lbs. Now figure out the torque when the can is held down at this 45° angle, as in Figure 4-3. The perpendicular distance we just solved for is 1.4 ft, multiplied by the weight of the can (1 lb), so the torque is 1.4 ft × 1 lb = 1.4 ft-lbs. That’s why it’s easier to hold the can lower than it is to hold it straight out.

FIGURE 4-4 Right triangle


Let’s take this example one step further and assume that you want to make a human-sized puppet that can raise and lower a can of soup. If you put a motor at its shoulder, how strong would the motor need to be?

In order to design mechanisms that move, you first need to understand how to estimate if something will break when it’s not moving, or static. Most of the time, you can isolate a static problem that represents the worst case of a dynamic (moving) one. In this case, it’s when the weight of the soup can is farthest from the motor shaft, thus requiring the greatest shoulder torque. Since the weight of the soup can doesn’t change, the highest torque will be needed when the distance from the can to the puppet shoulder is the highest. This happens when the puppet arm is parallel to the floor, as in Figure 4-2. Since we already solved for this maximum torque of 2 ft-lbs, you know to look for a motor that is at least that strong, and it will be able to handle all the other angles just fine.


Friction occurs everywhere two surfaces are in contact with each other. It’s what makes door hinges squeaky and mechanisms noisy. High friction is sometimes a good thing when your mechanism interacts with the environment, such as the way friction allows your car tires to grip the road. However, friction is usually your enemy when it comes to making things move. It can rob your mechanism of power and decrease efficiency. Low friction is what we strive for inside mechanisms to make things run smoothly. Low friction is what makes nonstick cookware slippery and causes you to slide on ice.

So what is friction, and how can you design projects and choose materials to minimize it?

Friction is actually a force, just like your weight is a force. In fact, the force of friction is a percentage of any object’s weight. Suppose you are trying to move a 50 lb box across a hardwood floor. At first, when the box is at rest, there are two forces acting on it, as shown in Figure 4-5:

FIGURE 4-5 Forces on a box at rest


1. Weight (W) The weight of the box is focused at the box’s center of gravity and points down toward the floor, in the direction of gravity.

2. Normal force (N) This means the floor is actually pushing up with the same amount of force as the box weighs in a direction normal (or perpendicular) to the floor. This might seem like a made-up force, but think of what would happen if you set the box down on a bed of marshmallows. Those marshmallows would squish until they were compressed enough to support the weight of the box. The floor doesn’t need to squish, because it’s already strong enough. (If your floor needs to squish to support a 50 lb box, you should get yourself a new floor!)

The normal force (N) equals the weight (W) of the box when it’s standing still. The fancy term for this is static equilibrium, which is when all the forces acting on an object cancel out so the object doesn’t move.

Since the box is heavy, you decide to sit on the floor and push it with your feet instead of trying to lift it. How hard do you need to push? You must push just hard enough to overcome the friction between the 50 lb box and the hardwood floor. This force is a fraction of the normal force. In equation form, the last sentence looks like this:

Force Due to Friction (Ff )) = Fraction (µ) × Normal Force (N)

The fraction, or coefficient of friction, commonly represented by the funny-looking Greek letter µ (pronounced miu), is less than 1. This means that you’ll need to push sideways with less than 50 lbs of force to move the box.

For example, let’s assume the coefficient of friction for our cardboard box on a hardwood floor is 0.4. Our equation looks like this:

Ff = 0.4 × 50 lbs

So the force due to friction that you’ll need to overcome is just 20 lbs. If you put the force due to friction and the force you’re pushing with into the diagram, it looks like Figure 4-6Friction always acts opposite the direction of movement.

So again, if all the forces cancel out, the box will be in static equilibrium, and it won’t move anywhere. You need to push with just slightly more force than the force due to friction in order to get the box out of equilibrium and moving.

FIGURE 4-6 Forces on a box when moving


In the previous example, we just assumed a coefficient of friction. But what if you don’t know the coefficient of friction and need to measure it? Let’s use the same 50 lb cardboard box, but this time on a piece of plywood. Put the box in the middle of the plywood, and then start lifting up one edge of the plywood until the box just starts to slip. Use a protractor or other angle-measurement tool. The coefficient of friction is just the tangent of the angle where the box starts to slip. In equation form, it looks like this:

µ = tan θ

The location of angle θ is shown in Figure 4-7. If the box starts to slide down the plywood at an angle of about 22°, that matches the coefficient of friction of 0.4 we assumed earlier.

FIGURE 4-7 Forces on a box tilted at an angle


All this talk of sliding boxes is great, but how does that help us design mechanisms? Think of friction as a force that is always working against you, so it’s something you need to compensate for when you are choosing a motor or other components for your project. Nothing is 100% efficient. This means you’ll never get out everything you put in. There are always losses, and many times these losses are because of friction.

In the preceding example, where the coefficient of friction was 0.4, 40% of the input force was lost to friction ! If it seems like a lot, that’s because it is—well, at least in this case. Friction is always relative. The combination of a box sliding on plywood might have a lot of friction, but a roller-skate wheel turning in its bearing might have a coefficient of friction of 0.05, for only a 5% loss.

The Coefficient of Friction

Some fancy math goes into making this simple equation to estimate the coefficient of friction. The force arrows in Figure 4-7 are not all at perfect right angles, like those in Figure 4-6. In order to cancel out the forces and calculate them, they need to be pointing in the same direction. To do this, we split up the weight force (W) into two components: one that is parallel to the plywood (opposite the force of friction) and one that is parallel to the normal force (N).

You can see that the angle at the top of this force triangle is the same as the angle the plywood is lifted off the floor (angle 1 on the left side of Figure 4-8). So we use some trigonometry and figure out that the component of the weight parallel to the plywood is = W × sin θ, and the component parallel to the normal force (N) is = W × cos θ. We can put these forces back in our diagram, as on the right side of Figure 4-8. Now we can start canceling out forces. We know these values:

N = W × cos θ and Ff = W × sin θ

But we also know that Ff = µ × N. If we substitute that into the equation on the right, we get this:

µ × N = W × sin θ

We can also substitute what we know about N into this equation:

µ × W × cos θ = W × sin θ

You can see here that the W values cancel out, so cross those out. To simplify the equation, realize that sin θ / cos θ = tan θ. So, the equation boils down to a simple µ = tan θ. Too easy!

FIGURE 4-8 Breaking the weight into components to solve for friction (left) and revised arrangement (right)


Project 4-1: Estimate the Coefficient of Friction

Let’s do a quick test to get a feel for the coefficient of friction between different materials, so you know what 5% and 40% losses look like.

Shopping List:

• Two small objects made of different materials (a 4 oz clay block and iPhone 2G with aluminum back are used here)

• Scrap wooden board

• Protractor

• Calculator


1. Position your protractor at the pivot point of the scrap wood board, as shown in Figure 4-9.

2. Place the first object (the clay block) on the board.

3. Increase the angle until the block just starts to slide.

4. Read the angle.

5. Repeat steps 2 and 3 three times, and take the average measurement. In this test, I found the clay slipping at 45°.

6. Repeat steps 2 though 5 with the second object (the iPhone). I found the iPhone slipping at just 12°.

FIGURE 4-9 Friction testing


7. Break out your calculator. To find the coefficient of friction between the wood and clay, take the tangent of 45° (remember µ = tan θ). You should get 1. This means that if you tried to push a clay block across the floor (as we did earlier with the box example), you would need to push with a force equal to its weight!

8. Do the same calculation with 12°. You should get 0.21. Since the iPhone is slipperier than the clay, it slides more easily. If you push an iPhone across a wooden floor, you need to push with a force equal only to 21% of its weight to get it to move.

Reducing Friction

Now that you know that friction is the enemy, let’s look at a couple of ways to decrease friction: clearance between parts and lubrication.


In Chapter 2, we talked about tolerances of materials and parts. So now you know that a 1/2 in shaft won’t fit in a 1/2 in hole very well if they are both 0.50000 in. You need to leave a little clearance between parts that move relative to each other.

Clearance is just a fancy word for space. You need to leave space around your 1/2 in shaft for it to move, so you may want to drill out a hole that’s around 0.515 in to give it some room to spin. There is no magic correct amount of clearance—it will depend on the size of your parts, their surface finish, and whether you want them to spin or stay put. For example, think of LEGOs. Some parts, like the axles, slide right through the holes in other parts. But the little gray stoppers, gears, and wheels you put on the axles are harder to slide on. Also, once you slide them to the right spot, they generally stay there. This is because there is less clearance between the axle and the gear than there is between the axle and the hole through the LEGO piece.

These differences in clearances between the parts allow you to constrain the motion of your LEGO parts just enough, but not too much. This follows the principle of minimum constraint design we talked about in Chapter 1. Using clearance appropriately is an excellent way to practice this principle.

Lubricants and Grease

It’s usually a good idea to lubricate things that move. It keeps friction lower, which increases efficiency by allowing more input power to transfer to the output. It also helps keep mechanisms quiet.

A lot of bearings, motor gearboxes, and other components come with grease already in them. For quick, multipurpose fixes, WD-40 is a good light lubricant; the company claims it has over 2,000 uses. Another well-known brand that’s a better lubricant is 3-IN-ONE. From bike chains to squeaky scissor hinges, a drop of this stuff will do the trick.

There are dozens of types of oils and greases available. Grease is thicker than oil and tends to stay where it’s put. Oil can be runny. If your chosen multipurpose lubricant doesn’t do the trick, try looking on McMaster for your application—for example, search for “gear grease”—to find a specific recommendation. Beeswax reportedly works well with moving wooden parts.

Free Body Diagrams and Graffiti Robots

One of the biggest problems my students have is figuring out how much motor torque they need for a certain project. It’s impossible to size a motor accurately without doing at least a little analysis of what you want the motor to do. Do you want your motor to drive a small mobile robot quickly? Or maybe you want to use your motor to lift a heavy weight or pull on a rubber band?

Being able to sketch free body diagrams is a handy skill when you’re trying to determine how to mount something or what kind of motor torque you need, and any other time you need to choose a component based on strength or torque. A free body diagram is like a simplified snapshot of all the forces and moments acting on a component. You can also use these diagrams to figure out the forces and moments you don’t know. The body referred to is just one object or component of a system.

Here is what is included in a free body diagram:

• A sketch of the body, free from any other objects, with only as much detail as necessary (most of the time, a dot is all that’s needed)

• All of the contact forces on the object:

• Friction always acts in the opposite direction of motion.

• The applied force is just what it sounds like—the force you apply to something. This could be a push or a pull (or a kick or a yank or a …). This can also be a force that something else applies to your object, such as the upward force that chains apply to the seat on a playground swing.

• The normal force acts perpendicular to the surface of contact. This is what stops you, or your chair, from falling through the floor and travelling through the center of the earth.

• Drag is the force that impedes an object when moving through air, water, or other fluids. Drag increases as an object moves faster. We generally ignore drag for things moving slowly in air.

• All of the noncontact forces on the object:

• Gravity acts on all objects on the earth, and pulls them toward the center of the earth. This shows up in free body diagrams as the weight of the object.

• All of the moments (torques) on the object

For the object to be in static equilibrium, three conditions must be met:

1. All sideways (horizontal) forces must cancel out.

2. All up and down (vertical) forces must cancel out.

3. All moments (torques) must cancel out.

If these three conditions are not true, you have a moving object.

Let’s walk through a real example to put this all in context. One of my former students built a graffiti-drawing robot that suspended a spray paint can from two motorized spools, as shown in Figure 4-10.

FIGURE 4-10 Graffbot original concept (credit: Mike Kelberman)


A motor and controller drive each spool, so working together, the spools can move the spray paint can in any shape possible on the wall below them. In order to figure out how hard the motors must work, let’s draw a free body diagram of the paint can platform, as on the left side of Figure 4-11.

The only forces in this free body diagram are the weight of the paint can platform (W) and a force pulling up from each rope (F1F2) to the motor spools. We can use what we know to find these forces so we can size the motor properly. So we need to use SOHCAHTOA to get all the forces pointing in straight lines.

On the right side of Figure 4-11, each rope forms a triangle with an imaginary horizontal line. If the can is hanging at 45°, part of the rope force is pulling sideways, and part is pulling up. By breaking the rope force into horizontal and vertical components, we can see that the two sideways forces (F1-OUT and F2-OUT) are equal and opposite, and therefore cancel each other out, satisfying our first condition for static equilibrium. That leaves us with just two up forces (F1-UP and F2-UP) and one down force (W). In order for the system to balance, the up forces need to equal the down forces to satisfy the second condition for static equilibrium:

FIGURE 4-11 Free body diagram of paint can platform in Graffbot (left) and rope forces broken up into components (right)


F1-UP + F2-UP = Weight

So if the paint can platform weighs 10 lbs, each FUP force must be 5 lbs. There are no torques on this paint platform, so we can ignore the third condition for static equilibrium.

Now we’re getting closer. We know the FOUT forces cancel, and each FUP force is 5 lbs. But we don’t know the actual forces, F1 and F 2, on the ropes that go to the spools. For the triangle on the right in Figure 4-11, recall from our earlier conversation that the sine of an angle equals the opposite side over the hypotenuse (the SOH part of SOHCAHTOA):

sin 45° = F1-UP / F1

This gives us F1 = 7 lbs. So we learned that it takes more force to pull something up at an angle like this than it takes to pull something up straight. If the angle gets smaller, so that the paint can platform starts closer to the motor spools, the force on the ropes will increase.

FIGURE 4-12 Free body diagram of Graffbot spool (left) and original Graffbot spool drawing (right) (credit: Mike Kelberman)


In order to choose the correct motor, we need to take this one step farther and draw a free body diagram of the spool, as shown in Figure 4-12.

We can neglect the weight of the spool and motor, because the holding force of the screws that mount them to the wall cancel it out. That leaves just the force from the rope and the torque from the motor that is in line with the spool shaft to resist the spool from being unwound. Remember that torque is just force times distance? The distance here is from the edge of the spool to the center of the spool. If the diameter of the spool is 4 in, then the radius is 2 in, and we can solve for the unknown motor torque:

Torque (T) = 7 lbs × 2 in = 14 in-lb

We now know that we need a motor that can turn with at least 14 in-lbs of torque to get this Graffbot moving. Each motor needs to be this strong to control the spray paint can platform.

How to Measure Force and Torque

You can measure your weight (the force you exert on the ground) by standing on a common bathroom floor scale. But what if the object you need to weigh doesn’t fit on a scale, or you need to measure the pulling force from something like a rope? And how do you measure torque?

Measuring Force

The simplest way to measure force, if you’re trying to weigh something, is to use a scale. Some scales are mechanical, using weights and springs to turn or balance a dial; some are electrical. Tools for measuring force come in all different shapes, sizes, and price ranges. Throughout this book, we’ll use force-measurement tools that are readily available and affordable.

Mechanical Options

The most affordable option is the standard bathroom scale. This is a smaller version of the scale you stand on at the doctor’s office. The kitchen scale, its smaller cousin, is used to measure lighter objects like ingredients for recipes and is more accurate. These are mechanically based scales, which are easy to use. They typically have a needle that comes to rest on a dial to indicate the weight. With these scales, the object pushes on a base to measure a force.

To measure pulling force, you can use a luggage scale or spring scale. You can also find these at sporting good stores sold as fish scales. Mechanical luggage scales go for under $10 and look kind of like the scales at grocery stores to weigh produce. You can purchase spring scales, which are literally just a spring attached to a hook, for even less. Most spring scales have a housing that indicates the pulling force based on how much the spring stretches. These generally work only for a small range of forces, like 5 to 20 lbs. So you need to have a good idea of what you’re measuring before you choose how to measure it.

Electrical Options

Bathroom and kitchen scales also come in electrical versions. Instead of a system of springs and levers underneath the platform, these use sensors to detect weight and display it digitally on a screen. You can use these kinds of sensors directly if you need to integrate them into a project, but they are not as plug-and-play as the mechanical options.

• Force-sensitive resistors (FSRs) are used to measure low forces. Their accuracy is not great (±5%–25%, depending on the application), so they are more useful for measuring relative weights or as a sensor to indicate whether something is being squeezed or sat on. An example is the SparkFun () sensor SEN-09375, which goes up to 22 lbs.

• Flexiforce pressure sensors are more accurate —about ±2.5%—but are about twice as expensive as FSRs. However, at around $20, they’re still on the low end of the price scale for force measurement options. An example is the SparkFun sensor SEN-08685.

• Luggage/fish scales also come in digital versions. A company called Balanzza makes popular versions that start at around $15. MakerBot Industries used a digital fish scale to measure the pull force of its plastruder motor for the CupCake CNC machine, as shown in Figure 4-13. See Project 4-2 for how it works and to learn how to make your own version.

FIGURE 4-13 MakerBot Industries used The Rack to measure the pull force of its plastruder motor.


The next step up on the price scale is a big one. Higher-end force measurement tools use fancy electronics for precise measurements. They generally come in two varieties: handheld and button types that can be integrated into projects.

• Digital force gauges, like McMaster’s 1903T51, start at around $370. You can attach a hook or a plate to the measurement end to measure force when pulling or pushing.

• Load cells, like the MLP-100 from Transducer Techniques (, start at around $300. Unless you’re a whiz at electrical engineering, you’ll also need the $400 display to read the force. However, if you need accuracy and a sensor you can integrate into a project, these are ideal.

Measuring Torque

Measuring an unknown torque directly can get expensive. You can set a torque wrench to a certain number and tighten a bolt just the right amount, but torque wrenches are not really made for measuring an unknown torque. Torque wrenches start at around $100 and climb up steeply from there.

You can clamp a torque gauge onto a motor shaft or a screw head and read the resulting torque in real time. However, at $580 for a gauge like McMaster’s 83395A29, this is not a very accessible option either.

Luckily, we can use the fact that torque = force × distance to our advantage here. All we need to do is measure the force and the distance from the axis to the point where it is being applied, and we get torque!

Project 4-2: Measure Motor Torque

When you buy a motor, it will usually come with a list of specifications to tell you everything you want to know about it. However, sometimes you’re stuck with a motor that doesn’t have a data sheet. Here, we will use an adaptation of MakerBot’s Rack (Figure 4-13) to measure motor torque indirectly by measuring motor force.

Shopping List:

• DC motor (Solarbotics GM9 shown)

• Shaft collar, gear, or other component that fits the shaft and has a set screw hub; replace the set screw with a regular long screw

• Luggage scale, spring scale, or fish scale

• Two C-clamps to hold the motor and scale

• Ruler

• Optionally, epoxy putty and small hook (like one for hanging pictures)


1. Fix the screw and shaft collar or gear to your motor shaft.

2. Fix the motor to the edge of your work table with a clamp (see Figure 4-14).

3. If necessary, use the epoxy putty and small hook to create an attachment on the scale that can hook around the screw.

4. Fix the luggage scale to your work table with a clamp.

FIGURE 4-14 Measuring motor torque


5. Turn the luggage scale on if it’s digital.

6. Connect the end of a strap or hook to the screw as close to the screw head as possible.

7. Power on your motor.

NOTE If you’re testing a motor for which you don’t have a data sheet, it’s best to use a benchtop power supply so you can find out the acceptable voltage range for your motor (see Chapter 5). Alternatively, you can try batteries that add up to the voltage your motor needs. Six volts would be a good guess for most small DC motors.

8. Your motor will try to turn, but the connection to the luggage scale will stop it. Read the luggage scale as the motor is stalled like this, and record the number. Do this quickly, before your motor heats up from working too hard!

9. Turn off the power supply.

10. Measure the distance from the center of the motor shaft to the location on the screw that your luggage scale was attached.

11. Multiply the distance you found in step 10 by the force reading you got in step 8, and voila, you have the torque of the motor for a given voltage.