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Hoo boy, you want to start an argument with car people, this is how you do it! XD
Let's keep this as simple as possible. Torque is a unit of work - force multiplied by a distance, hence the units of foot-pounds and newton-metres. In other words, it's the amount of energy that an engine is producing, in an instantaneous snapshot.
Horsepower is a measure of power, which is, put in simple terms, how much work an engine can do in a specific period of time. Horsepower is basically, "how much torque did this engine generate in a second?" - foot-pounds per second, basically.
Torque is useful in that it basically shows where an engine is doing work most efficiently, and is directly tied to the engine's optimal airflow characteristics, but that starts to get incredibly complicated, dealing with stuff like volumetric efficiency, airflow resonances and a pile of other factors!
Torque is force
HP is how much Torque you can generate in a certain amount of time
Torque = force times distance
it's the force applied by the piston on the crankshaft.
Power = work
It's the amount of work done in an amount of time.
Power is what really matters, doens't matter what engine you have, if it's generating a specific number of horsepower, it will lift/pull a certain weight in a certain time.
Now comes rpm's and flow efficiency.
The engine varies it's capacity for sucking air. If it's too slow, it can't generate enough vaccum, if it's too fast, there's not enough time until the valve closes. So, the peak torque is where all the parts have the better efficiency in letting air inside the cilinder, it's where the bang is bigger, hence, a bigger number on the graphic.
Rpm is a form of frequency, just Like the 60/50hz of your house's electrical supply. In the engine, we measure in revolution per minute. 1.000rpm means that the engine is making 1.000 revolutions at every minute. So, at every revolution we got one or more expansion cycles, it varies on how much cilinders the engine have. More rpm's means more bangs per minute.
I'm trying not to make a too long comment. But the greater torque you have, the greater is the power, because more force is applied. Also, the greater the the rpm, the greater the power, because, the frequency is higher.
Example: a 4 cylinder 2.0 engine
Stock, aspirated: 140hp@6k rpm
Turbocharged: 200hp@6k rpm (the tunning increased the torque, no higher rpm)
Aspirated tuned: 200hp@9k rpm (the tunning increased the rpm, no higher torque)
Think about a semi a and a race car. The semi needs large amount of power in low rpm's to pull weight. A race car needs large amount of power in the lightest engine possible, hence small high-revving.
I hope this helps.
A car with 250NM of torque has 200hp at 5500RPM
Torque= Force
HP= How much Torque you create at a certain amount of time
so giving it at perspective
A truck with 470hp and 2200nm of torque can pull 30 tons of weight
But a Car with 470hp and 520nm of torque can't
If the truck had the car engine, it could still technically have the same top speed as the original engine, because it has the same hp, but it would also not be able to even run the truck properly, because it lacks TORQUE so it doesn't have the strenght to pull that much weight.
"the engine with the crankshaft that is 1 unit long would achieve the same torque at twice the RPM then the engine with the crankshaft that's 2 units long?"
Not necessarily. In the question you gave a example of changing the stroke. You need to account this regarding to diameter. But, increasing the stroke, you increase the levering, and the result is more torque. With the same diameter, and the same everything on the engine, the 2 units crank would have more torque than the 1 unit. The rpm the peak torque occurs will vary a lot from what else you have on the engine. Also, a heavier crankshaft tends to rev lower.
if you really want the math it's this
Horsepower = Torque x RPM / 5,252
So, breaking it down
Horsepower is equal to torque multiplied by where it's the peak power located at the RPM range divided by 5,252
5,252 is a general consense of where the Torque band crosses the HP band, so you divide by it, so let's have an example
HP = 950 LB.FT. x 5200RPM/5,252
HP = 941
This engine has an amazing 941HP.