Math of Nuts and Bolts

You might think I made a mistake in the blog title. I hear you saying "It ought to be the Nuts and Bolts of Math." Not today. As you can see, these are real bolts (no nuts in this photo, though).

NOTE - Bolts may also be called fasteners, screws or other terms.

Bolts are identified in a number of ways. All using NUMBERS. We can describe the:

• diameter of the bolt (thickness of the shaft)
• length of the bolt
• length of the thread portion
• form or shape of the thread
• angle of the thread
• pitch of the thread
• size and depth of the thread
• type of head on the bol
• handedness (rotation to tighten)
• material
• grade or strength

Only one bolt in this set is designed to screw into wood. It cuts its own threads as it is turned into a pre-drilled hole. All the other bolts are designed to screw into a matching set of threads.

Two of the bolts have fine threads while the rest have been cut with coarse threads.

Two of these bolts may be called socket cap bolts, or more commonly, Allen bolts. These are used in places where a regular wrench wouldn't easily fit.

Five bolts have a non-threaded portion at the top of the shank of the bolt, just below the head. That leaves space for a washer, or for the bolt to pass through and retain a separate piece of material.

Here are a few more. The color differences reveal the fact that some of them are of different material. One has no head, but has a collar in the middle. One has a conically-tapered head that fits a matching recess. Another has a domed cap and a square shape above the threads. Another has a lock washer, large washer and nut. Finally, one nut with a rounded bottom end is here too!

Here are some of the bolt heads. You can see that different tools must be used to turn them, and also some code markings on them. The markings reveal the strength of the bolts.

You can see that without a math education it would be very hard to distinguish between all these bolts!

They are everywhere - more than 200 billion are used every year in the USA.

In case you are wondering, No, Excel Math is not a hardware store. But our pal Joe accumulates lots of interesting bolts in his business, and he loaned me a few of them for today's blog. Thanks, Joe.

Check Wikipedia for more bolt facts and figures.

Can You Resist Playing Hookey?

Yes, I mean Playing Hookey, not playing hockey.

Hockey is a sport involving pucks and sticks. Hookey comes from a Dutch phrase that means hide and seek or an Olde English term Let's Hook it, meaning to run off and escape, or another phrase by hook or by crook.

Other similar terms are skipping, ditching or cutting. A stricter, more legal term is truancy. Truancy even sounds very serious, whereas playing hookey conveys a sense of youthful whimsy, rather than criminal evasion.

Playing Hookey implies you are skipping out of school or work to do something else less responsible, like going fishing! Do you know Mark Twain's characters Tom Sawyer and Huckleberry Finn? Or can you recall that other famous character called Huckleberry Hound™? (Click to see Huckleberry deal with truant kids).

You may be asking, where's the math in this? We find it in the aptly-named Play-Hookey Website. This site's purpose statement says:

While you're out of the classroom, there's no reason why you can't learn a little something and enjoy yourself while you're doing it.

This site is packed full of math, computer operation/repair/programming, electronics, physics, chemistry, and optics. The author explains resistors, capacitors, transistors, rectifiers, diodes, inductors, transformers, logic, etc. Don't run! It's not going to hurt you to learn something new!

Here comes our math. Resistors are simple components that have the ability of resisting the electrical current flowing through a circuit. This property is measured using in units called Ohms.

Here are some resistors. Because they are tiny, it's tough to print legible resistance values on them. So ages ago they began to be labeled with colored stripes.

Let's look at the 5-band resistor in the chart below. Using the first three stripes, we create a 3-digit number. In this case it's 237. The third stripe tells us how many zeros to add ( 000 ). That makes it 237,000 ohms.  The fifth stripe tells us how close to its specification this resistor is guaranteed to be. In this case the tolerance is 1%. [You may also find 4 or 6-band resistors.]

This resistor color code chart was created by Peter Veness at Bournemouth University in Dorset, England.

While we were trying to play hookey today, we learned about math, color codes and electricity.

Analog, or can you lend me a hand?

The last blog posting subject was the word DIGITAL. Today's is ANALOG (or analogue).

Analog is derived from the term analogous which means proportionate, or similar - in other words, something that is similar enough to another thing that we can use it to make a comparison.

Why is this math? We start teaching analog in the First Grade when we describe a clock that has hands to point out the time. Most do not have hands in the sense that this Mickey Mouse watch has hands, but they are similar to arms and hands. We call this an analog clock to distinguish it from a digital clock which has digits to indicate the time (and no hands).

Here are a few sample analog clocks and watches. We provide an analogous grouping in First Grade.

We often use the word analog to refer to things that are done in the old way rather than the new digital way. For example, a pencil might be analog and a computer keyboard digital. This is a messy way to use the words, but most people would probably understand what we mean. Shall we compare a few items?

DIGITAL	ANALOG
keyboard	pencil
solid state	vacuum tubes (valves)
switch	rotary knob
touch-tone	rotary dial
numerals	hands
digits	moving needle
on/off	variable
CD and DVD	vinyl records
looking through a window screen	looking through an open window

Here's a way to combine analog and digital to display the time:

Some clever Swedish software guys figured this out. You need an array of 6 externally-controlled analog clocks for each digit. And it takes some fancy thinking to work out the controls!

 CLICK TO SEE THE CURRENT TIME

How many clocks are needed to show 24 hours, minutes and seconds?
Do any of those analog clocks need to have 3 hands?

Digital Digital Digital (repeat 677,000,000 times)

Digital is a favorite word nowadays.

• My Google search turned up 677 million results.
• My Bing search turned up 424 million results.

What is digital? What does digital mean? 

• Does it mean anything with a battery? 
• Or anything that is programmed? 
• Or anything that older people don't understand?

The English word digital comes from a Latin word digitus meaning digit or finger. As in counting on your fingers rather than drawing with your finger in the sand.  As a part of speech, digital is an adjective. It modifies some other word. Here's one definition:  

A digital system is a data technology that uses discrete (discontinuous) values whereas non-digital (or analog) systems use a continuous range of values to represent information.

That doesn't help very much, does it? And since digital has been an English word for 400 years, it's certainly not the only definition. Here are some examples of digital communication that even old folks will understand:

• smoke signals
• Morse code - SOS
• semaphore (optical telegraph) signals
• Braille
• ship flags
• hand stuck out the window to signal a turn, etc.

Many newer forms of communication are generally described as digital, because

• they have an electronic, programmatic basis
• they indicate values using digits (numbers, not fingers!) rather than a needle or graph
• they can be read or manipulated or stored by a computer rather than the "naked eye"

Facebook is one of those new digital communication mechanisms, and we at Excel Math have a Facebook account in addition to this blog and our website. Come on over and see us sometime.

NOTE - When your first name is EXCEL and a large outfit in Redmond, Washington uses EXCEL for one of its favorite products, there are sometimes difficulties! Even if you have been around since the beginning of the digital era ...

You might think digital means lightweight, portable, inexpensive, etc. For users that's often true, not necessarily for the techies who have to make everything work back in the home office. Here's a nice little digital communication system ... for the captain of a ship who wants to talk to people on shore without everyone else listening in.

 



 

How do you rate?

Rate is a math term that implies one measured quantity is being placed in relationship to or judged by a second measured quantity.

For example, speed means distance traveled over a period of time. A common unit for rate of speed is mph or kph, which means miles or kilometers (measured by odometer) per hour (measured with a watch).

Since speed is a very common term, we usually don't bother to say rate of when we speak.

A "pulse rate" is the number of beats (counted) per minute (measured with a watch). Since measuring our heartbeat is very common, we usually don't bother to say "rate" when we speak, we just say pulse.

Doctors, nurses and emergency personnel need to measure heartbeat all the time. And they may not have a computer or calculator handy. In the field, they don't even usually wait one entire minute. They count for 15 seconds and multiply their result by 4 to get beats per minute, or count for 10 seconds and multiply by 6.

Another approach is to count a few beats, then look at a scale to tell what the rate would be per minute.

Here's a watch that makes it easy for emergency medical personnel. Notice that this watch has a 4-ended second hand. That way you never have to wait more than 15 seconds for a hand to come along! I've never seen another watch like this.

Instructions are indicated on the dial. When a second hand (in this case, the orange hand) comes by the pointer at the bottom (below 6 o'clock), you start counting respirations (breaths). When you have counted  5 breaths, the orange hand will be pointing at a number on the outer scale. That indicates the respirations per minute. This scale shows a respirations range of 10-60 a minute. Normal for an adult is about 15-20.  In this picture, my respiration rate was about 14.

You do the same thing to get a pulse, but using the top and right side of the watch. Put your finger on a spot where you can feel a pulse. When a hand passes the top pointer (above 12 o'clock) you start counting. In this case, you count 15 pulsations, then look at the moving hand. If it's where this hand is pointing, it means 70 beats per minute. This scale shows a pulsations range of 50-180 per minute.  Normal for an adult is about 60-100. My pulse rate was about 70.

As I said earlier, this watch is designed to save the wearer from having to do any math. It's easy enough to count 15-20 respirations in a minute, but much harder to count to 70 or 100 in a minute. So doing a 15 second pulse count and multiplying by 4 is the normal process.

Remember that we started with the term rate. Other common uses of the word are with birth rate, unemployment rate, interest rate, exchange rate, inflation rate, and so on. And of course at this time of the year many of us may be thinking about the tax rate!

Push Button, Cars Stop, Emergency Vehicles Go

Yesterday we discussed the button on a pole that pedestrians use at the crosswalk (zebra crossing). It makes the lights change sooner, and stops the cars a bit longer so pedestrians can walk across.

Today we'll talk about a different button. A button in a bus, fire truck, ambulance or police car. It's called the traffic signal preemption button.

For a bus, this magic device can HOLD a signal GREEN a bit longer, so buses move more quickly through heavy traffic. The bus system delivers a message to a sensor at the traffic signal. It's like shouting Hold that door! as you run towards an elevator.

A slightly-different device is in emergency vehicles to speed their response to an accident or fire. If the signals in front of the vehicle are GREEN, the emergency vehicle can go more quickly and safely. This type of device CHANGES a signal from RED to GREEN. 

Once the traffic signal has changed all other directions to RED, it gives the emergency vehicle a bright white light to show that all is clear. As soon as the traffic signal it loses its traffic preemption notice, the lights return to their normal pattern.

How does it work?

There are many systems. One uses bright strobe lights flashing at 10-14 cycles per second. Another depends upon an infra-red signal, like your TV remote. It's directed up and forwards towards the traffic signal. Another system uses a limited-range radio transmitter which isn't as susceptible to fog, rain or other atmospheric conditions that could block the visible or infra-red lights.

A more complex model has the vehicle communicating with the lights, and the lights warning those along the route that a vehicle is coming. That takes real intelligence! It's covered in patent 7116245, in case you care to read up on it ... here's one of the diagrams.

Yet another system relies on a GPS location signal emanating from the vehicle. It's covered under patent 7327280. With this approach, a central depot can see the vehicle moving along a screen and change the signals for the emergency vehicle. This system is also at the mercy of signals bouncing around through tall buildings, trees, etc.

Note - Both these systems were invented by Aaron Batchelder, of CalTech in Pasadena, California.

Traffic signal preemption is used at railway crossings too, to make sure all vehicles have cleared a set of railroad tracks and that no new vehicles are allowed to approach too closely. If possible, the signals allow nearby non-crossing traffic to keep moving, even "out of cycle" from the normal pattern.

I know pedestrians are wondering - if you are anxiously pushing a button at a crosswalk, do you have to wait for the emergency vehicle?  YES. It takes priority over your request, as well as priority over the cars.

You might be the impatient type who always wants a GREEN light, but I would advise you NOT to try to buy or build your own light-changing device. It is illegal for drivers or pedestrians to use them.