What Is the Ground Wire For?

© Eugene Brennan

What Is a Ground Wire?

The hot and neutral cores in a power cable are used to supply current to the appliances in your home. So what about the green wire? Why is it needed? This is known as the "ground" and is an additional wire that is included for the safety of you and your home.

Note: This article was written for a US audience, so I refer to the protective conductor as "ground". However, it is also called "earth" in other countries. Another difference is that the term "hot" is used, which is also known as "live". The third difference is that a split-phase 120/240 volt supply is provided to most homes in the US, resulting in a 240-volt supply between the two hots in addition to a 120-volt supply. This 240-volt supply is for powering higher powered appliances.

What is the ground wire for? © Eugene Brennan

What Are the Wires in a Power Cable?

Hot

This wire is at a potential of 120 volts nominal in the US relative to ground. Current flows out through the hot wire to an appliance. Hot is also referred to as "live" in other countries and voltage can be either 110 or 230 volts AC nominal.

Neutral

The neutral wire is at a voltage close to or equal to ground. The current which flows to an appliance via the hot wire returns via the neutral core in a cable. (See note below).

Ground

This is a protective conductor, included to prevent shock and/or fire. Ground is also known as "earth" in some countries.

The supply from the transformer feeding your home is split-phase and in the U.S., 2 hots in addition to a neutral are provided. Lower power appliances are connected between either of the hots and neutral and this gives a 120 volt supply. The voltage between the two hots is 240 volt for supplying higher power appliances.

Note: The electricity supply in our homes is alternating current (AC). So while we tend to think of current flowing out through the hot wire to an appliance and returning via the neutral wire, current actually flows both ways. So for one-half of what is known as a "cycle", current flows out through hot and returns via the neutral wire. During the second half cycle, the process is reversed and current flows to the appliance via neutral and returns via hot.

AC and DC voltage waveforms. DC voltage is constant over time. AC voltage and current change direction at a rate given by the frequency. This is 60 times per second or 60 Hertz (Hz) in the US. Each repetition of the waveform is called a cycle. © Eugene Brennan

Voltage supplies throughout the world. Somusde, public domain image via Wikimedia Commons

Ground: The Protective Conductor

The flex or fixed wiring supplying metal cased appliances includes a ground conductor (colored green in the US or green/yellow in the EU) in addition to hot and neutral. Inside an appliance, the ground core of the cable is connected to the outer casing of the appliance. The connection may be made either using a screw terminal or a ring crimp and self-tapping screw/bolt. Spade crimps are generally not used to prevent a ground from being inadvertently removed instead of hot or neutral and not replaced. Fixed (e.g. storage heater, kitchen range) and portable i.e corded appliances with extraneous metal which can be touched in normal use must be grounded. Ground acts as a "bypass" for currents in the event of a fault.

The fault could be due to:

• Conductors (e.g. wires, terminals, components) at hot or near full mains potential breaking, bending or detaching and touching the casing of an appliance
• Breakdown of insulation. For instance, insulation on cores of the power flex could become damaged inside an appliance or insulating spacers could become dislodged. Also, metal parts such as screws or nuts which have come undone could bridge the gap between hot and the metal casing
• Making contact with a power cable when drilling through a wall

Inside a microwave oven, the ground core of the power flex is connected to the casing using a ring crimp and screw. Note the symbol for ground. © Eugene Brennan

What Happens During a Fault if an Appliance Isn't Grounded?

If a fault occurs, the external metal of an appliance will become live and the voltage with respect to ground a person is standing one will be anything up to 120 volts, depending on which part of the internal circuit touches the casing. If the metal isn't grounded and someone touches the appliance, current will travel through their body to ground.

If they are lucky and have rubber-soled shoes and are standing on a dry floor, they may just experience a tingling sensation. However, if conditions are damp, they have wet hands and are standing outdoors, they are more likely to experience a severe shock. If one hand touches the appliance and the other touches a grounded object (e.g pipework, poles, radiators or whatever), current will travel across their heart, a more dangerous scenario. If the person is unlucky or has a heart condition, this can kill.

Why Does Current Flow to Ground?

Understanding Electricity? What Are Volts, Amps, Watts, Ohms, AC and DC?

Why Is the Supply Transformer Grounded?

Grounding the neutral of the supply transformer is a safety measure taken to eliminate dangerous rises in potential (greater than the hot voltage) on the hot or neutral conductors entering a home. This could occur for instance if a very high voltage power line (possibly hundreds of kilovolts) breaks and lands on a "low" voltage (120 volt) line. Another scenario is the insulation between the primary and secondary of the transformer being breached. This could allow the primary voltage (>10kv) to appear on the secondary. Yet another possibility is a lightning strike on the lines. Static charge can also cause a buildup of voltage on lines.

Basically, grounding the neutral pulls down the voltage of the line so that neutral is close to the potential of the earth which we are standing on and the voltage on either of the hot lines doesn't greatly exceed 120 volts.

The path of fault current when someone touches a non-grounded faulty appliance (2nd hot in U.S. wiring systems not shown). © Eugene Brennan

How Does Grounding Of Appliances Make Them Safer?

Grounding provides a bypass, shunt or shortcut through which electricity can flow, instead of passing to earth through the person who touches an appliance. Wires called equipment grounding conductors (EGC) are run from the electrical panel through the fixed wiring to all socket outlets, fixed appliances such as ranges or water heaters, light switches and ceiling roses in your home. In the case of a portable appliance, this grounding path continues from the pin in the plug through the flex, to the metal body of the appliance. At the electrical panel, all of these conductors are joined at the main grounding terminal. A grounding electrode conductor (GEC) runs outside the premises to a grounding electrode embedded in the soil.

When a fault occurs, current flows via the grounding conductor back to the electrical panel. If a TNC or TNCS earthing system is in use, all neutrals are joined to ground at the panel (or the neutral and ground may be joined at the output of the supply meter see earthing systems schematic below), and so the hot to ground fault at the appliance effectively becomes a hot to neutral fault, practically a short circuit. A large over-current flows and this trips the MCB (miniature circuit breaker) and possibly also the GFCI (whichever acts first) for the circuit, cutting power and making everything safe.

Grounding, however, also has another important function. Even if the current is insufficient to trip a breaker (in the case of a TT grounding system), the neutral conductor breaks outside the home, or stray currents in the neutral cause a dangerous rise in potential, it reduces the touch voltage between the casing of the appliance and the area on the ground on which the person is standing to a safe level. Both EGCs and the ground rod and the bulk impedance from the ground rod to the supply transformer effectively behave as a potential divider. Since the impedance of an EGC and ground rod are a lot less than the equivalent impedance of the soil between the premises and the supply transformer, and since the two impedances are in series, a much smaller voltage is dropped across the EGC than the total supply voltage and so the hazard is reduced.

The low impedance ground conductor shunts current away from the person touching the appliance, reducing the touch voltage to a safe level. (In reality the conductor passes via the plug and flex of the appliance and electrical panel to the ground rod) © Eugene Brennan

Naming Convention in the USA and UK

Equipment grounding conductors (EGC) = Protective earths (PE) in the UK.

Main grounding terminal = Main earthing terminal in the UK.

Grounding electrode = Earthing electrode in the UK.

Double-Insulated and Non-Grounded Appliances

Appliances such as hair driers, TVs, handheld kitchen appliances, etc. generally have plastic casings. If a fault occurs inside the appliance (e.g. a wire or component touches the inside of the casing), there is no danger since the plastic body is an insulator. These appliances don't have a ground wire in the flex. Some appliances such as power tools are not grounded and instead are "doubly insulated". This means that although the external casing of the tool or appliance may be metal, sufficient separation and isolation of the external metal from internal high voltages is effected to prevent electric shock. These devices don't have a ground wire in the cord either.

Double insulated appliances can be extremely dangerous if they get wet. This is because the casing is not grounded and can become live if water breaches the separation between live parts and casing. Also, the MCB is unlikely to trip and the GFI may not operate either.

Double insulated symbol. © Eugene Brennan

GFCI

A safety device called a Ground Fault Circuit Interrupter (GFCI) also known as a Ground Fault Interrupter or Residual Current Device (RCD) is likely to be fitted in most modern installations. This device monitors the current flowing out through the hot conductor and back via neutral. Normally these currents are equal. If current leaks to ground, not all the current returns through the GFCI. Electronics in the device detects this imbalance, and it trips out, shutting off the power. The trip current for a GFCI is normally 30mA but can be higher or lower depending on conditions.

A GFCI handles situations such as someone touching a live conductor, such as a damaged power cord with exposed cores, or the connector of a kettle left in a pool of water on a sink. (It may also even trip if damp bread gets stuck in a toaster and touches the element!)

A GFCI also responds to faults as described above where hot makes contact with the grounded body of an appliance. The device cuts the power if the MCB doesn't "get there first".

Another function of the GFCI is to prevent fire. Consider the situation where a damaged and exposed conductor makes contact with damp timber or grounded material, e.g. conduit or piping. This could produce sparks and start a fire if there is any flammable material nearby, e.g. sawdust, wood shavings or insulation. The current may not be sufficient to trip a breaker, however, the small leakage current to ground is more likely to be detected by the GFCI, making it trip and shut off the power.

GFCIs can be installed at the electrical panel, they are available in the form of a GFCI socket outlet, and you can also buy a GFCI adapter that plugs into a socket. An appliance is then plugged into the adapter. This is a worthwhile safety accessory for an extension lead if you use power tools in the garden.

Three Types of Grounding Systems

TNCS or PME (Protective Multiple Earthing)

This system uses a combined ground/neutral back to the supply transformer. This is then split into separate ground and neutral conductors after the meter. A hot to ground fault effectively becomes a hot to neutral fault, and since the impedance back to the transformer is low, the large, short circuit current ensures that an MCB for the circuit will trip. The problem with this type of system is that full mains potential could appear on extraneous metalwork of an appliance if the neutral breaks outside the premises. This is why the ground electrode is so important. The bulk of the earth between the ground electrode at the premises and the point where the supply transformer is grounded acts like a potential divider.

If someone touches a grounded appliance, the touch voltage between their hand and feet is equal to the voltage between the point at which the electrode enters the ground and their feet. Since this distance is likely to be a fraction of the distance to the supply transformer, the voltage is reduced proportionately. The electricity supply company may install multiple earthing or ground points from the neutral line between transformer and premises to reduce the consequences and hazard of a broken neutral (especially if they are widely separated)

TNS

The TNS system is often used when a ground can be provided by the armor of the supply cable. If the armor becomes corroded causing bad ground, this system can be converted to TNCS.

TT

The TT system is used when the power comes in overhead. The system uses the bulk of the earth as the return path for fault currents. It doesn't have the risk of a broken neutral. If a home is distant from the supply transformer, the fault current during a hot to ground fault may be insufficient to trip a breaker because the resistance of the earth is too great. Since the development of GFCIs which can detect small leakage currents to the ground, this is less of an issue. TT systems may be converted to TNCS systems where the ground and neutral are neutralized or joined together at the exit point of the meter.

Un-Grounded and Grounded Socket Outlets: NEC Regulations

In the U.S., both ungrounded and grounded receptacles are used. Ungrounded outlets are prohibited in new buildings but in the situation where an equipment ground conductor is not present, NEC code exceptions allow these to be replaced by either another non-grounded receptacle, a GFCI receptacle or a grounding type receptacle fed by a GFCI as long as the receptacle is marked "No equipment ground" and "GFCI protected".

2-pin in-grounded receptacles can be upgraded to 3-pin grounded receptacles with the addition of new ground wiring.

Bonding

Metal services such as water and heating pipes and hot water heaters are grounded with a heavy gauge wire routed back to the electrical panel. This ensures that if a hot wire makes contact with these services, a large current will flow and trip the breaker. The heavy gauge wire is rated so that it can carry the current which may flow if a hot from a high current circuit makes contact with the service. Also, the heavy gauge keeps the resistance of the cable low. This ensures that as current flows through this resistance, the resulting voltage rise is kept below safe limits. This is vitally important in bathrooms where everything is damp and we may be in our bare feet and making relatively good electrical contact. Everything such as radiators, water pipes, wall heaters and the drain in the bath/shower are connected together by a bonding conductor. This "equipotential bonding" keeps everything at the same voltage and there is no difference in voltage between for instance a showerhead and the drain.

© Eugene Brennan

Electric Wiring Color Codes

This Wikipedia article gives lots of info about wiring and color codes used in various countries around the world.

References

Fish RM, Geddes LA. Conduction of electrical current to and through the human body: a review. Eplasty. 2009;9:e44. Published 2009 Oct 12.

Kuphaldt,Tony R., Lessons In Electric Circuits -- Volume I - Chapter 3 - ELECTRICAL SAFETY. Copyright (C) 2000-2020, Revised November 06, 2021.

Disclaimer

This article is accurate and true to the best of the author’s knowledge. Content is for informational or entertainment purposes only and does not substitute for personal counsel or professional advice in business, financial, legal, or technical matters.

© 2013 Eugene Brennan

Questions and Answers 

Cal Stowe from Houston, TX on August 12, 2019:

In my outside main breaker panel, the neutral bus is connected to:

1) the ground bus

2) earth electrode

3) conductor back to the transformer

4) 100 kva surge protector

This panel is connected to an inside panel with all the individual breakers.

The plumbing is connected to a separate earth electrode 4 feet away. Should the plumbing earth electrode be connected to the power earth electrode?

If they are separate:

1) Pipe ground fault voltage should be relatively high, current low, and breaker may not trip due to high impedance from pipe earth electrode to transformer earth electrode.

2) A high voltage surge would create high potential between connected devices and plumbing faucets, etc. There would be no potential plumbing to floor.

3) A lightning ground strike would create some potential between the two earth electrodes, and some potential between plumbing and floor due to voltage gradient.

4) Surge protector dump to earth will induce little voltage on plumbing.

If the earth electrodes are connected:

1) In the event of a ground fault, voltage will be low, and current high through neutral back to transformer. Breaker will trip.

2) A high voltage surge will not raise neutral and plumbing voltage?

3) A lightning ground strike will induce some voltage and current between the ground electrodes.

4) A surge protector dump to earth will induce some voltage on the plumbing?

Jim G on March 18, 2019:

Hello Eugene

I’m hoping that you can advise. I recently came into possession of an old film projector from the 1940’s. The projector is primarily made out of metal, has tubes, and a cord with no ground prong.

I plugged in the projector and turned it in and success the lights came on. But then I noticed a slight tingle when I touched the projector. Luckily I was wearing rubber soles. I opened up the projector and could not see any obvious wire touching the case, etc. I got my multi meter and ran a wire to the house ground which is close to my work bench. With my meter connected to the house ground I would read 120v from multiple locations on the projector.

I’m basically at my limit of electrical competence. Would replacing the power cord with a grounded cord solve my problem and make the projector safe to use.

Thank you for your advice.

Jim

Eugene Brennan (author) from Ireland on March 18, 2019:

Hi Jim, it sounds as though there's a hot to ground fault somewhere in the appliance. If you ground it by replacing the flex and plug, it would still trip a GFCI unless the fault is sorted. Another scenario with a three core grounded cord is that you could have a bootleg ground in one of your receptacles and if hot and neutral have been reversed, this would place live voltage on the casing of the appliance.

Another scenario is that there are EMI suppression filters used and with these, a terminal of the filter can be connected to chassis. The chassis in AV equipment is often ungrounded to prevent ground loops and hum. However, it's possible to experience a tingle when touching an aerial connector on a TV, because of a capacitor in the filter being connected to the chassis. However, EMI filters are unlikely to be used in vintage equipment such as your lamp.

Might be a good idea to take it somewhere and get it checked. I don't give advice online on these matters because something could be missed and there could be an unfortunate outcome!

Tyrone on September 11, 2017:

In the event of a fault it seems the circuit is not being completed via ground but via the neutral wire. Therefore the grounding on both the transformer and residence are stand alone and are only there to bring the neutral down to 0 volts?

Eugene Brennan (author) from Ireland on September 11, 2017:

A circuit is normally completed via neutral, but if a hot to ground (live to earth) fault occurs, there's also a parallel path for current to flow through via ground back to the transformer (and also the neutral conductor, if ground is connected to neutral at the panel).

Tyrone on September 11, 2017:

I'm thinking there wouldn't be much current travelling along the ground path because of the much higher resistance compared to the neutral wire?

Eugene Brennan (author) from Ireland on September 11, 2017:

It depends on the grounding system Tyrone. If TT grounding/earthing is used, the bulk of the ground may have a large resistance and so current would be lower than that in the neutral in a fault situation (possibly insufficient to trip an MCB, but when the installation is protected by a GFCI/RCD, this will trip). In a TN-S system, a large short circuit current will flow back via the separate ground to the supply transformer. In a TN-C system which has a common combined neutral/ground back to the transformer and appliance cases connected to neutral, a larger than normal current will flow on this conductor during a fault. A broken neutral conductor can cause a dangerous potential on the metalwork of appliances. A TN-CS system provides separate ground and neutral to appliances. If there's a fault, a large current will flow through the equipment ground (protective earth) back to the panel. This will be larger than the current flowing in the neutral core of the cable feeding the appliance, but will flow in the neutral conductor back to the transformer.

So it's really only TT systems where the resistance is high. The actual ground wire from the appliance back to the panel would have a low resistance.

Izhaan on August 28, 2017:

Plz tell me

How can i identify earth wire in household circuit.?

Eugene Brennan (author) from Ireland on August 28, 2017:

Hi Izhaan - This link shows the colours of earth wires in electrical installations:

https://en.wikipedia.org/wiki/Electrical_wiring#Co...

Earthing/grounding conductors may have a coloured sheath or be bare without a sheath in fixed wiring cables. The earth conductor in a flexible power cord supplying an appliance normally has an insulating sheath. In socket and lighting outlets and electrical panels, bare earth conductors are provided with a coloured sleeve to insulate them from inadvertent contact with live/hot parts and also to identify the conductors.

If you are considering making any electrical modifications, I highly recommend you consult a qualified electrician. A simple mistake can prove fatal!

Denise field on May 19, 2017:

Okay here goes my hairdryer lead (flex) has been chewed in half so I gave it to my neighbour (yes I'm rubbish!!)

He put on a new plug, the other one was fixed on the lead. But said its not earthed it only has 2 wires which were visible cutesy of the dog!! These visible wires are copper im not going to get electrocuted am I?

Eugene Brennan (author) from Ireland on May 19, 2017:

Hi Denise,

If the casing of the hair drier is plastic, it doesn't need to be earthed. I presume you mean the copper of the wires was visible? If there are still sections of the flex, after the plug was replaced, where the copper wires are exposed, the whole lead needs to be replaced (or possibly shortened if the bad bit is near the plug end).

Denise field on May 21, 2017:

Thank you so much Eugene. Youve really put my mind at ease. Unfortunately the lead has been shortened so no exposed wires!! But! It worked for about 1 minute then stopped. There was no smell indicating the motor burnet out no no bang no spark just stopped maybe the plug is old what do you think obviously without seeing it. Just glad to know I'm not going to be shocked etc.

Eugene Brennan (author) from Ireland on May 21, 2017:

Hi Denise,

Possibly there is invisible damage, i.e. broken wires, in the flex from the dog stretching it. Sometimes if the stranded copper cores in a flex are broken, the wires can be pulled until they snap and the lead then shortened. However the safest thing would be to completely replace the lead.

Ned Tallyho on April 10, 2017:

The green/yellow (EU) cable connecting copper pipes near my gas combi boiler is bent over to fit in but the bare wire can be seen where the cable has split due to being bent. Is this dangerous and should I get it changed?

Eugene Brennan (author) from Ireland on April 10, 2017:

Hi Ned,

The earth wire doesn't carry a dangerous voltage (at least it shouldn't assuming the earth rod is working properly and the touch voltage is below safe levels during a fault situation).

The covering on an earth wire is used for identification but can also give the inner core some protection from damage. It also acts as insulation if for instance the wire feeds into a socket outlet/light fitting. This prevents live parts/wires inadvertently touching the earth conductor when for instance socket plates are removed/replaced. Usually however live wires are insulated and live parts are shrouded, so this doesn't often happen.

I don't think it's a major issue if the cable insulation is split, in any case the pipes it connects to are uninsulated. I would be more worried that the inner core has been damaged and strands broken from the cable being repeatedly bent during installation/maintenance work. It may have just been chaffed when work was done, but it's no harm getting the cable checked by a qualified electrician.

MG Seltzer from South Portland, Maine on August 02, 2015:

This topic was just on my mind as our GCFI outlet in the bathroom isn't working, which means that the one near the kitchen door has kicked itself off. I had just been thinking, "I really need to understand how the current flows through the system." Also, years ago, a contractor told me the grounding plugs on adaptors were "useless," and could be pulled out with pliers to let a three-prong plug fit a standard outlet. And I remember thinking, "Now is this true?" I am bookmarking this Hub because I see lots of good detail that I think will answer my questions. Voted thumbs up, of course.

Eugene Brennan (author) from Ireland on August 03, 2015:

The older style NEMA 1-15 two pole socket was 15 amp rated and un-grounded. It is not permitted in new construction but there are probably lots of these still out there. If an appliance has a metal casing (e.g. a metal kettle), it needs to be grounded. The exception is double insulated appliances which have metal on their exterior but are not grounded. This is because the metal is separated from the inner workings to the extent that it is highly unlikely to become "live".

Ideally you should get your sockets upgraded to 3 pin grounded types...

Re-Discovering the Mill Stream Arch

One mill is marked on the OSI c.1837 first-edition six-inch map. Image courtesy the National Library of Scotland, licence CC-BY.

I hadn't seen it since around 1982 or '83. I thought it was covered with scrub and brambles over the decades, but not so. This is a good time of the year to investigate it before the brambles get going. My theory is that it could have been the opening to a culvert/mill race that diverted water to a second mill in the adjoining field beside the cemetery car park or it may have been the return feed to the stream. One mill is marked on the 6-inch, first-edition OSI map and was located just over the wall from where the bins used to be kept in the cemetery. The ruin is still there. There's an arched culvert under the Mile Mill road, south of the lodge, that fed water to this mill.
I think the arched opening into the stream is unlikely to be a latrine outlet or drain as it's too wide. I thought most of the arch was submerged below the waterline, but possibly the bed of the stream has dropped over the last four decades due to erosion. Anyway, I didn't have a camera with me. That's a shame because the arch was nicely lit up by the setting sun.
Will investigate further tomorrow and take some photos.

The map above is a ChatGPT-sharpened version of a screenshot of the c. 1837 first-edition six-inch map available on the National Library of Scotland website. The scans appear to be a bit clearer than those available on Tailte Éireann's Irish Townland and Historical Map viewer.
Map reproduction licence CC-BY, courtesy the National Library of Scotland.

YouTube Requirements and Broadband Speed

Connection speed required for YouTube. Image from Google Help

Unless you have lots of people in your home feeding off  your Internet service and sharing the bandwidth, ultra-high-speed connection promotions such as "Gigabit broadband" are just marketing hype. The table above shows the recommended connection speeds from Google for streaming video from YouTube. Vodafone keeps trying to sell me such a service using a fibre-optic connection that replaces copper wires to the cabinet and which would increase my speed from 100 Mbps to 1 Gbps, but I'm stubbornly refusing as the current speed is perfectly adequate. A faster connection probably increases upload speed (which I think is normally about one tenth of download speed), but most people aren't in a hurry when uploading, unless they can't wait to backup thousands of photos or put their latest video online. Eventually I'll have to give in, because copper lines, which Vodafone rent from Eir, will ultimately be replaced. The process is slow, because there are legacy systems that still require them. 
I ring Vodafone every year, three months before my contract is due for renewal and negotiate a loyalty discount, getting a six-month discount. It's only €5 per month for 6 months, but better than nothing.

A Child Can Push Harder Than a Lawn Mower Blade

AI image generated by Google Gemini.

Did you know that a child can probably produce more torque using their arm, than a lawn mower engine can generate to turn the blade? Surprisingly, the figure for the torque of a 5 hp engine used on a mower is 7 to 10 foot-pounds (ft·lbf) in the imperial system or 11.5 to 13.6 newton-meters (Nm). Applied torque is simply force multiplied by distance, so 1 pound-force applied over a distance of 1 foot from a pivot is 1 foot-pound. In the metric system, the unit of torque is the newton-metre (Nm). A torque of 1 Nm is equivalent to one newton—the SI unit of force—applied over a distance of one metre. The speed of a lawn mower blade is more important than torque, and blades are driven directly by the crankshaft without reduction or overdrive gearing. Blades typically spin at the engine speed of around 3000 RPM, giving a blade tip speed of around 200 mph, or 90 metres per second.

As we saw in the article Interesting Facts - What is Torque?

power = torque x angular rotation speed

So while a child may be able to exert more torque with their arm than a mower engine can exert on a blade, they wouldn't be able to match its power. Another piece of the jigsaw is that the quoted figure of 11 to 13 Nm is for the drive torque from the engine. Since a blade is spinning at around 3000 RPM, it has a lot of inertia. The technical term for this when a rotating body such as a flywheel or blade is involved is moment of inertia. When a blade slows down as it impacts grass, the change in angular momentum creates a force, just like the way a car or any linearly-moving object creates a force as it slows down due to friction, or when it hits something (think of the huge force produced by a hammer, typically many tonnes, when it hits a nail). So the force produced by a blade as it impacts and shears grass is much greater than the force that the engine itself could have generated.

Note: You may be wondering why "newton" in "newton-metres" above isn't capitalised.  The unit is named after the English scientist Isaac Newton. It's not a typo. In the International System of Units (SI), or Système international d'unités, units named after a person are not capitalised. However, the symbol is, N in this case, Units are normally lower case, as are prefixes. However, prefixes over one million are capitalised—so kilometres is km, but gigahertz is GHz.

Cantilevered Boardwalks For Motorway Bridges?

Bridge over M9 Kilcullen Bypass on Sunnyhill Road. Image courtesy Google.

When Kildare County Council, or whoever was responsible, produced the spec for this bridge, (and the one over the M7 at the Bundle of Sticks Roundabout outside Naas), they left out foothpaths. Maybe it was lack of common sense and an attitude of "Sure, only cars go out there". Or perhaps it was due to penny pinching and lack of foresight and a narrow bridge was cheaper than one a couple of metres wider? Anyway, I'm wondering could boardwalks/cantilevered foothpaths be fixed onto the sides, exterior to the parapets?
It would obviously be impractical to install foothpaths on the inside of the parapets due to to both lanes being reduced in width. As it is, cars can't pass pedestrians safely without partially moving into the other lane. An article by Paul MacDonald, Kildare National Roads Office, on the construction of the M7 Newbridge Bypass and M9 Kilcullen Bypass is available on the Kilcullen Diary here.

The section about the earthing of the steelwork of the underbridge for the River Liffey is interesting—High voltage pylons can induce voltages and currents in conductors, potentially leading to different and hazardous potentials between isolated elements of a structure. Bonding them together eliminates these voltages and earthing a structure pulls the voltage down to ground (similar to the equipotential bonding in bathrooms, where metal objects such as drains, radiators, towel rails, water pipes and metal baths are bonded together):

Bridge over M9 Kilcullen Bypass on Sunnyhill Road. Image coiurtesy Google.

How to Calculate Bolt Circle Diameter (BCD) for Chainrings and Bash Guards

Bolt circle diameter (BCD). © Eugene Brennan

What Is a Bolt Circle Diameter (BCD) or Pitch Circle Diameter (PCD)?

As you may be aware, if you've found this guide, a bolt circle is an imaginary circle that passes through the centre of the bolt or screw holes in a round pattern.

Thomas McGarry of McGarry's Lane

McGarry's Lane. © Eugene Brennan

I think I may have found the person who lent his name to the lane.

The Primary Valuation, also known as Griffith's Valuation was a mid-19th property valuation in Ireland. Published between 1847 and 1864, it was the first full-scale survey of its kind, and collected information about the occupiers of land and buildings, their landlords and the amount and value of property held. Griffith's Valuation is searchable here on the Ask About Ireland Website.

In the valuation, a Thomas McGarry appears, with an address in Nicholastown. The Nicholastown townland extends out as far as Yellowbogcommon townland to the south and New Abbey townland to the east. I was aware of the presence of a building, visible on the first edition, c. 1837 6" Ordnance Survey map. This was located across the road from the cottage in McGarry's Lane. By the time the c. 1900 last-edition 25" OSI map was drafted, this building had disappeared. It appears from the map and record below that Thomas McGarry was listed as an occupier of the 8 acres and 3 roods of land on which the building stood, with Robert Brereton as lessor (The owner of New Abbey House). 

Valuation book showing the entry for Thomas McGarry. Image courtesy Ask About Ireland.

Map showing corresponding parcel of land leased by Thomas McGarry (16). Map attribution: Ask About Ireland.

The Tithe Applotment Books, however, show no record of a Thomas McGarry. These books were records of tithes, a form of religious tax paid to the established church in Ireland—the Anglican Church of Ireland. Tithes were extremely unpopular, as the Roman Catholic majority also had to pay them, leading to the Tithe War of the 1830s and eventual repeal of the tax by The Irish Church Act 1869, which disestablished the Church of Ireland. In the records, there are three entries for a John McGarry, and he's recorded as having paid 3/3½  (three shillings, three and a half pence) in one of these records. No information is available on whether John was a brother or father of Thomas. I found a death record for Thomas McGarry on the Irish Genealogy website. He died on 14th March, 1889, with the location indicated as "New Abbey".
It's possible he's buried in New Abbey Cemetery.

Tithe payment by a John McGarry. Image attribution: The National Library of Ireland

The Tithe Applotment database search page is available here.  Image attribution: National library of Ireland

Death record for Thomas McGarry. Image attribution: Irish Genealogy.

The structure shown on the map was located behind this fence. No trace remains.  © Eugene Brennan

Cycle Paths, Moone High Cross and Poor Road Signage

© Eugene Brennan

Yesterday's Sunday cycle took me to the area around Moone, via a more roundabout but pleasant and traffic-free route up Brewel Hill, then through Colbinstown and Timolin. I returned home via Ballitore, Crookstown and Narraghmore Bog, a 30-mile round trip. I wanted to see the infamous Timolin-Moone cycle paths on the R448. As per usual, the path layout includes those ubiquitous grass verges. I'm not sure whether this feature is purely functional to reduce water run-off onto the carriageways or for growing wildflowers in the grass keep to the environmental lobby happy. I dropped in to see the High Cross of Moone. I hadn't been there for a few years. The cross has been protected by a corrugated-Perspex-roofed canopy for over 15 years. Such constructions help to protect carvings on heritage stonework from being eroded by the elements. Perhaps their use could be more widespread, as inevitably more detail will be lost from stonework over the coming centuries?

Unusually, I met lots of drivers on the narrow L-roads I normally travel on. Many of these are only the width of a car. It wasn't until a woman stopped her car and asked me for directions, while I was taking a cycling break at a bridge on one of these L-roads that branches off the Timolin-Baltinglass road, that I realised what was happening: Blockades on the M9 had forced drivers to resort to their sat-navs to find alternative routes. The woman was on her way to Dublin and the sat-nav had sent her down this boreen. Unfortunately, the lack of signage on many minor junctions, which is pretty typical in Kildare and Wicklow and presumably most other counties, meant that she was lost and had to resort to technology. Coupled with the obscuring of signs because landowners neglect to cut back vegetation, drivers who aren't familiar with roads must find it difficult. As a 9-year-old child, I was given the job of "navigator" by my father when we got lost on a Sunday drive near Avoca and I noticed we had taken a wrong turn on the journey home. My father told me to go to John Joe Dowling's newsagent and buy a Sheet 16 map of Kildare Wicklow to use for navigation (he was a navigation officer in the LDF during the Emergency, or perhaps later in the 50s as member of the FCA). From then on I kept track of our progress on drives, making sure we didn't get lost. We always had paper road maps in the car—something that people may no longer do, relying on technology to get them to where they want to go.

New Moone–Timolin cycle lanes. © Eugene Brennan

Moone High Cross. © Eugene Brennan

Carving detail on Moone High Cross. © Eugene Brennan

Carving detail on Moone High Cross. © Eugene Brennan

The L8286, off the Timolin–Baltinglass road. © Eugene Brennan

Obscured road sign near Boleybeg. © Eugene Brennan

Missing Clip on My Quick-Release Skewer

Quick-release axle skewer from a bike. © eugene Brennan 

The cam lever fell out of the housing when I took my front wheel off to change a puncture, half-way between Ballymore Eustace and Harristown last Sunday. I hadn't noticed that the retaining clip had disappeared, possibly rusting off over the last 29 years.

On the Couch With ChatGPT

Created by Grok

Eugene (transcript of me incoherently expressing myself in voice mode):

Sometimes when I'm on the verge of dozing and falling asleep at the table during the daytime, it's as if my mind goes back to maybe 40 years ago and I feel like I did then.

How to Find the Probability of an Event and Calculate Odds, Permutations and Combinations

Learn about probabilities, odds, permutations and combinations. blickpixel, public domain image via Pixabay

What Is Probability Theory?

Probability theory is an interesting area of statistics concerned with the odds or chances of an event happening in a trial, e.g., getting a six when a dice is thrown or drawing an ace of hearts from a pack of cards.

17th Century Dunlavin Headstones

A 17th century headstone in the old cemetery in Dunlavin village, Co. Wicklow. © Eugene Brennan

I visited the little park adjacent to the market house in Dunlavin on my Sunday cycle yesterday. I've cycled past it dozens of times over the last thirty years, but never actually went in. This was the location of a 17th-century church and graveyard, the graveyard now having been “recycled”, with the headstones stacked up against the back wall of the space. It’s a practice I’m not too fond of, and it's something that has occurred in several locations in Dublin—Cemeteries have been turned into parks, such as at the rear of St Mary’s Church (foundation stone laid in 1700) on Mary Street, and at St Kevin’s Church, Camden Row (behind the now-demolished Kevin Street College of Technology, my alma mater). It was a shock to discover on Street View that the college had gone, but it was an ugly building.
The cemetery in Dunlavin has a couple of old 17th-century headstones, something which isn’t commonplace, as the inscriptions on headstones older than the 18th century are normally eroded and illegible unless they’ve been sheltered from the elements.

Edit: Dunlavin local historian Chris Lawlor has kindly sent me a link to his thesis, The Establishment and Evolution of an Irish Village: The Case of Dunlavin, County Wicklow 1600 -1910, which includes some details about the cemetery on p. 44:

Map courtesy Tailte Éireann.

Another 17th century headstone in Dunlavin old cemetery. © Eugene Brennan 

The location of the cemetery in Dunlavin village. Image courtesy Tailte Éireann.

Chimney Stacks and Pots in Kilcullen

Chimney stacks, some topped with pots, on buildings along Kilcullen’s Upper Main Street. 

I've been reading the cover article from the April edition of The Bridge about the planned streetscape survey of Kilcullen's Upper Main Street. Zooming in on a c. 1900 image of the street, it appears that many of the chimney stacks were devoid of chimney pots—the same is true for Lower Main Street. A chimney pot serves several functions beyond mere aesthetics. It improves airflow by effectively raising the chimney's height without requiring the entire stack to be built taller. Since a chimney pot is narrower than the flue inside the stack (these chimney stacks were often wide and sometimes flue-less; in fact, children used to climb the stacks to sweep the chimneys), it creates a venturi effect, increasing airflow speed and improving suction.

Chimney pots became more common in the 18th century, with the peak of the fashion occurring in the 19th century.

Another feature of the chimney stacks is that many of them appear to have been constructed from brick, rather than rough stone. Over the intervening 130 years or so since the Lawrence Collection photo was taken, most of the stacks have been rendered with cement or lime mortar.

Images courtesy The National Library of Ireland.

Brick-constructed chimney stacks on buildings along Kilcullen's upper main street.

 

Lawrence Collection photo, c. 1900, of Kilcullen's Upper Main Street, 

 

How to Calculate Arc Length of a Circle, Segment and Sector Area

Circumference, diameter and radius. © Eugene Brennan

 

What You'll Learn

In this tutorial you'll learn about:

• names for different parts of a circle
• degrees and radians and how to convert between them
• chords, arcs and secants
• sine and cosine
• how to work out the length of an arc and chord
• how to calculate the area of sectors and segments
• the equation of a circle in the Cartesian coordinate system

What Is a Circle?

"A locus is a curve or other figure formed by all the points satisfying a particular equation."

A circle is a single-sided shape, but it can also be described as a locus of points where each point is equidistant (the same distance) from the centre.

Angle Formed by Two Rays Emanating From the Center of a Circle

An angle is formed when two lines or rays that are joined together at their endpoints, diverge or spread apart. Angles range from 0 to 360 degrees. We often "borrow" letters from the Greek alphabet to use in math and science. So for instance, we use the Greek letter "p" which is π (pi) and pronounced "pie" to represent the ratio of the circumference of a circle to the diameter. We also use the Greek letter θ (theta) and pronounced "the - ta", for representing angles.

An angle is formed by two rays diverging from the centre of a circle. This angle ranges from 0 to 360 degrees. © Eugene Brennan

360 degrees in a full circle. © Eugene Brennan

Parts of a Circle

• A sector is a portion of a circular disk enclosed by two rays and an arc.
• A segment is a portion of a circular disk enclosed by an arc and a chord.
• A semi-circle is a special case of a segment, formed when the chord equals the length of the diameter.

Arc, sector, segment, rays and chord. © Eugene Brennan

What Is Pi (π) ?

Pi represented by the Greek letter π is the ratio of the circumference to the diameter of a circle. It's a non-rational number which means that it can't be expressed as a fraction in the form a/b where a and b are integers.

Pi is equal to 3.1416 rounded to 4 decimal places.

What's the Length of the Circumference of a Circle?

If the diameter of a circle is D and the radius is R.

Then the circumference C = πD

But D = 2R

So in terms of the radius R

C = πD = 2πR

What's the Area of a Circle?

The area of a circle is A = πR ²

But R = D/2

So the area in terms of the radius R is

Scroll to Continue

A = πR ² = π (D/2)² = πD ²/4

What Are Degrees and Radians?

Angles are measured in degrees, but sometimes to make the mathematics simpler and elegant it's better to use radians which is another way of denoting an angle. A radian is the angle subtended by an arc of length equal to the radius of the circle. ( "Subtended" means produced by joining two lines from the end points of the arc to the center).

An arc of length R where R is the radius of a circle, corresponds to an angle of 1 radian.

So if the circumference of a circle is 2πR i.e 2π times R, the angle for a full circle will be 2π times 1 radian or 2π radians.

And 360 degrees = 2π radians.

How to Convert From Degrees to Radians

1. 360 degrees = 2π radians
2. Dividing both sides by 360 gives
3. 1 degree = 2π /360 radians
4. Then multiply both sides by θ
5. θ degrees = (2π/360) x θ = θ(π/180) radians
6. So to convert from degrees to radians, multiply by π/180

A radian is the angle subtended by an arc of length equal to the radius of a circle. © Eugene Brennan

How to Convert From Radians to Degrees

1. 2π radians = 360 degrees
2. Divide both sides by 2π giving
3. 1 radian = 360 / (2π) degrees
4. Multiply both sides by θ, so for an angle θ radians
5. θ radians = 360/(2π) x θ = (180/π)θ degrees
6. So to convert radians to degrees, multiply by 180/π

How to Find the Length of an Arc

You can work out the length of an arc by calculating what fraction the angle is of the 360 degrees for a full circle.

1. A full 360 degree angle has an associated arc length equal to the circumference C
2. So 360 degrees corresponds to an arc length C = 2πR
3. Divide by 360 to find the arc length for one degree:
4. 1 degree corresponds to an arc length 2πR/360
5. To find the arc length for an angle θ, multiply the result above by θ:
6. 1 x θ = θ corresponds to an arc length (2πR/360) x θ

So arc length s for an angle θ is:

s = (2πR/360) x θ = πRθ /180

The derivation is much simpler for radians:

By definition, 1 radian corresponds to an arc length R

So if the angle is θ radians, multiplying by θ gives:

Arc length s = R x θ = Rθ

Arc length is Rθ when θ is in radians. © Eugene Brennan

What Are Sine and Cosine?

A right-angled triangle has one angle measuring 90 degrees. The side opposite this angle is known as the hypotenuse and it is the longest side. Sine and cosine are trigonometric functions of an angle and are the ratios of the lengths of the other two sides to the hypotenuse of a right-angled triangle.

In the diagram below, one of the angles is represented by the Greek letter θ.

The side a is known as the "opposite" side and side b is the "adjacent" side to the angle θ.

sine θ = length of opposite side / length of hypotenuse

cosine θ = length of adjacent side / length of hypotenuse

Sine and cosine apply to an angle, not necessarily an angle in a triangle, so it's possible to just have two lines meeting at a point and to evaluate sine or cos for that angle. However sine and cos are derived from the sides of an imaginary right angled triangle superimposed on the lines. In the second diagram below, you can imagine a right angled triangle superimposed on the purple triangle, from which the opposite and adjacent sides and hypotenuse can be determined.

Over the range 0 to 90 degrees, sine ranges from 0 to 1 and cos ranges from 1 to 0

Remember sine and cosine only depend on the angle, not the size of the triangle. So if the length a changes in the diagram below when the triangle changes in size, the hypotenuse c also changes in size, but the ratio of a to c remains constant.

Sine and cosine are sometimes abbreviated to sin and cos.

Sine and cosine of angles. © Eugene Brennan

How to Calculate the Area of a Sector of a Circle

The total area of a circle is πR ² corresponding to an angle of 2π radians for the full circle.

If the angle is θ, then this is θ/2π the fraction of the full angle for a circle.

So the area of the sector is this fraction multiplied by the total area of the circle

or

(θ/2π) x (πR ²) = θR ²/2 with θ in radians.

Area of a sector of a circle knowing the angle θ in radians. © Eugene Brennan

How to Calculate the Length of a Chord Produced by an Angle

The length of a chord can be calculated using the Cosine Rule.
For the triangle XYZ in the diagram below, the side opposite the angle θ is the chord with length c.

From the Cosine Rule:

c ² = R ² + R ² -2RRcos θ

Simplifying:

c ² = R ² + R ² -2R ²cos θor c ² = 2R ² (1 - cos θ)

But from the half-angle formula (1- cos θ)/2 = sin ² (θ/2) or (1- cos θ) = 2sin ² (θ/2)

Substituting gives:

c² = 2R ² (1 - cos θ) = 2R ²2sin ² (θ/2) = 4R ²sin ² (θ/2)

Taking square roots of both sides gives:

c = 2Rsin(θ/2) with θ in radians.

A simpler derivation arrived at by splitting the triangle XYZ into 2 equal triangles and using the sine relationship between the opposite and hypotenuse, is shown in the calculation of segment area below.

The length of a chord. © Eugene Brennan

How to Calculate the Area of a Segment of a Circle

To calculate the area of a segment bounded by a chord and arc subtended by an angle θ , first work out the area of the triangle, then subtract this from the area of the sector, giving the area of the segment. (see diagrams below)

The triangle with angle θ can be bisected giving two right angled triangles with angles θ/2.

sin(θ/2) = a/R

So a = Rsin(θ/2) (cord length c = 2a = 2Rsin(θ/2)

cos(θ/2) = b/R

So b = Rcos(θ/2)

The area of the triangle XYZ is half the base by the perpendicular height so if the base is the chord XY, half the base is a and the perpendicular height is b. So the area is:

ab

Substituting for a and b gives:

Rsin(θ/2)Rcos(θ/2)

= R ²sin(θ/2)cos(θ/2)

But the double angle formula states that sin(2θ) = 2sin(θ)cos(θ)

Substituting gives:

Area of the triangle XYZ = R ²sin(θ/2)cos(θ/2) = R ² ((1/2)sin θ) = (1/2)R ²sin θ

Also, the area of the sector is:

R ²(θ/2)

And the area of the segment is the difference between the area of the sector and the triangle, so subtracting gives:

Area of segment = R ²(θ/2) - (1/2)R ²sin θ

= (R ²/2)( θ - sin θ ) with θ in radians.

To calculate the area of the segment, first calculate the area of the triangle XYZ and then subtract it from the sector.. © Eugene Brennan

Area of a segment of a circle knowing the angle. © Eugene Brennan

Equation of a Circle in Standard Form

If the centre of a circle is located at the origin, we can take any point on the circumference and superimpose a right angled triangle with the hypotenuse joining this point to the centre.
Then from Pythagoras's theorem, the square on the hypotenuse equals the sum of the squares on the other two sides. If the radius of a circle is r then this is the hypotenuse of the right angled triangle so we can write the equation as:

x ² + y ² = r ²

This is the equation of a circle in standard form in Cartesian coordinates.

If the circle is centred at the point (a,b), the equation of the circle is:

(x - a)² + (y - b)² = r ²

The equation of a circle with a centre at the origin is r² = x² + y². © Eugene Brennan

Equation of a Circle in Parametric Form

Another way of representing the coordinates of a circle is in parametric form. This expresses the values for the x and y coordinates in terms of a parameter. The parameter is chosen as the angle between the x-axis and the line joining the point (x,y) to the origin. If this angle is θ, then:

x = cos θ

y = sin θ

for 0 < θ < 360°

Summary of Equations for a Circle

Table 1. Circle formulas. θ is in radians.

Quantity	Equation
Circumference	πD
Area	πR²
Arc Length	Rθ
Chord Length	2Rsin(θ/2)
Sector Area	R²θ/2
Segment Area	(R²/2) (θ - sin(θ))
Perpendicular distance from circle centre to chord	Rcos(θ/2)
Angle subtended by arc	arc length / (Rθ)
Angle subtended by chord	2arcsin(chord length / (2R))

Example

Here's a practical example of using trigonometry with arcs and chords. A curved wall is built in front of a building. The wall is a section of a circle. It's necessary to work out the distance from points on the curve to the wall of the building (distance "B"), knowing the radius of curvature R, chord length L, distance from chord to wall S and distance from centre line to point on curve A. See if you can determine how the equations were derived. Hint: Use Pythagoras's Theorem.

© Eugene Brennan

References

Harris, J., & Stöcker Horst. (1998). Handbook of Mathematical Formulas and Computational Science. Springer.

This article is accurate and true to the best of the author’s knowledge. Content is for informational or entertainment purposes only and does not substitute for personal counsel or professional advice in business, financial, legal, or technical matters.

© 2018 Eugene Brennan

 

Q&A From Readers 

George Dimitriadis from Templestowe on May 18, 2018:

Hi.

A good introduction to the basics of circle properties.

Diagrams are clear and informative.

Just a couple of points.

You have So C = πD = πR/2, which should be C = πD = 2πR

and A = πR^2 = π (D/2)2 = πD^2/2

should be A = πR^2 = π (D/2)2 = πD^2/4

Eugene Brennan (author) from Ireland on May 18, 2018:

Thanks George, I should have proof read before publishing, instead of beta testing on the readers !!

Larry Rankin from Oklahoma on May 19, 2018:

Very educational.

Troy Sartain on March 19, 2019:

How about a similar article for ellipses? Just a thought. Obviously, another level of complexity, even if not rotated.

Eugene Brennan (author) from Ireland on March 19, 2019:

Thanks Troy, I'll keep it in mind. Parabolas will probably come first though.

Mazin G A on April 01, 2019:

Hi,

How can I calculate the angle at the center of an arc knowing radius and center, start, and end points? I know how to do that if I have the length of the arc, but in my case I don't have it.

Eugene Brennan (author) from Ireland on April 05, 2019:

If you mean you know the coordinates of the start and end points of the chord, you can work out the length of the chord using Pythagoras's theorem. Then use the equation for length of a chord (2Rsin(θ/2) to find θ.

Lakshay on September 19, 2019:

Good efforts

darrell on April 06, 2020:

how do i calculate the length of a segment of a circle

Eugene Brennan (author) from Ireland on April 07, 2020:

If you mean the chord length, it's 2Rsin(θ/2).

See the derivation above.

Austen SMITH on April 28, 2020:

Hi I have a simple but frustrating problem- I want to build a regular curved wall a set distance from a straight wall - the centre of the circle /arc of the wall falls within the building.

I need to work out distance from the straight wall to measure, at regular intervals, to create the perfect curve starting and ending on the chord (2nd) forming the distance from the straight wall.(1st chord)

Hope you can help.

Eugene Brennan (author) from Ireland on April 30, 2020:

Hi Austen, I spent hours trying to figure this out using angles, but it turned out that since the chord length is known between two ends of the curved wall (is this correct?), it can easily be worked out using Pythagoras's Theorem. I've drawn it up as an example at the bottom of the article, hope it helps.

Suggestion, you could put the values into a spreadsheet to do the calculations.

Austen on May 31, 2020:

Many thanks for solving the curved wall issue.

I’ racked my brains back 45 plus used the existing formulae on your website to crack it- but as always when someone who “knows” tells you “how” - it becomes so clear you wonder how you couldn't see it before.

Thanks for relighting the knowledge thirst.

A

Eugene Brennan (author) from Ireland on May 31, 2020:

Thanks Austen.

I worked out D in the diagram above knowing R and L/2. In reality that's probably not necessary because you may already know the distance from the centre of the arc to the inside of the wall. Adding this to S gives you D.