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Oil and oil ratings
Mineral oil is refined from natural sources and made up of naturaly occuring minerals left behind by the dinosours.
Synthetic oil is made by man out of artificial man made chemicals.

Because refining mineral oils is not a perfect science, the molecules are not 100% uniform in shape or size and the oil is not always as pure as you might believe.
Mineral oils also contains paraffin, which thickens the oil when its cold when you really need it to be thin and fast flowing for starting the engine.

Fully synthetic oil is far cleaner as its not refined from mined minerals and the molecules are a more uniform rounder shape and are smaller. This means the oil is more slippery and performs better.
Thats good right? Well yes, but the downside of this is that it gets through small gaps easier than mineral oil. So if you have a small oil leak using mineral oil you will have a bigger one if you swap to fully synthetic.

For newer engines with higher tolerences, fully synth is the better option, but for older engines that may be a bit worn or have lower tolerences mineral oil can be a better option.
Whats the difference between mineral oil and synthetic oil
10w-40, 10w-50 or 15w-40. SAE30 or SAE20.
What do they mean?
The Society of Automobile Engineers laid out a standard to measure viscosity (thickness) of oil by measuring the volume of oil that passes through a meter called a viscometer at 100 degrees Centigrade (210 degrees F) which is the average temerature of oil around the big end amd crankshaft.
The figure this gives is called the SAE rating and gives us a known factor by which we can measure the performance of the oil in the engine.

On the following chart you can see that when a given volume of oil flows through a viscometer in more than 5.6 seconds, but less than 9.3 will have an SAE rating of SAE20 but greater than 9.3 and less than 12.5 will give it a rating of SAE30.

Multi grade oils have two ratings that are given in a format like 10W-30. The first of the two numbers is the W, or Winter rating and the second is the SAE rating.

You can see on this chart that when a given volume of cold oil flows through the viscometer in more than 3.8 seconds but less than 4.1 will have a W rating of 5W.

What this means is a 10W-30 oil will have the viscosity of an SAE30 oil at operating temperature but will have the viscosity of an SAE10 oil when it is cold.

If you have an engine that is supposed to have 10W-40, is it ok to use a 5w-50 oil?

Yes it is. You will not cause any harm.
The higher rated the second number, the higher the viscosity at 100 degrees which is good because it means it is providing good protection.
But bear in mnd it is possible to get too high a viscosity.

If the viscosity gets too high, the polymers in the oil will start to form chains as they are put under load. These chains can 'wrap' around the crankshaft causing resistance and therefore friction and power loss.
In addition to this, if the viscosity is too high the engine wont recieve the required flow of oil to lubricate it properly and the filter may not be able to pass the thickened oil through at the required rate.
So going slightly over the required rating wont cause problems, but going too far over it can be detrimental.