# Why do Horsepower and Torque Cross at 5,252 RPM?

There's an interesting phenomenon that occurs regarding horsepower and torque. As many of you know, this is that horsepower and torque always cross, and therefore equal each other at 5,252 RPM. But I've come to find out that many people may know this happens, but they don't know why this happens. You may know the equation and know why hp and torque equal each other at 5252 RPM, but how do you get that equation? After some random, nerdy, late-night physics browsing, I discovered that there are really no good explanations behind this. So, of course, I felt entitled to make one.

I will start off with the most common explanation behind this and will try to explain it the best I can. Then I'll give a real-world example that will hopefully help you understand if you still don't.

To begin, we should probably understand the importance of the gentleman named James Watt. He is the one who developed the entire idea of a "horsepower" and of course, the SI unit of power, the watt. Awhile back, Watt figured out that the average horse could do work at a rate 550 foot lbs per second, or 33,000 foot lbs per minute. This means that on average, a horse could move 33,000 lbs 1 foot in 1 minute. So 1 hp=33,000 foot lbs per minute. Makes sense.

Now, since we're talking about engines here, work is being applied to a rotating crankshaft. This means that we need to talk about torque. But what exactly does a ft-lb of torque mean? Well, it means that we have a 1 lb weight attached to a weightless bar 1 foot from the fulcrum. In the image below, "d" would equal 1 ft and "F" would equal 1 lb.

Now imagine that we rotate that bar an entire rotation, 360 degrees. The distance traveled is the circumference of the circle made, which is 2*radius*pi. In this case, it's 2*pi which equals 6.28319 ft. So the distance traveled in one rotation is 6.28319 ft and so 6.28319 ft-lbs of work (torque) was done.

Now let's go back to Watt's equations for hp. Watt figured out that 33,000 ft-lbs of work per minute equals 1 hp. If we divide 33,000 by 6.28319, we get 5,252.1. This means that for every 5,252 revolutions per minute of 1 ft-lb of torque, we get 1 hp. This can be modeled by saying:

1 hp= (1ft-lb)(5252 RPM)

Now this is where people usually get confused. Basically, this equation is saying that hp will always equal torque when at 5252 RPM. But this isn't our final equation. If we know that hp and torque must equal each other at 5252 RPM, then let's say that at 5252 RPM, a car has 300 hp and 300 ft-lbs torque. If we plug this into that equation, we get 300=(300 ft-lb)(5252 RPM). S0, 300=1,575,600. But wait, it doesn't take a physicist to realize that 300 does not actually equal 1,575,600. So somehow, we need to make the right side of that equation equal to 300 so our equation looks something like "300=300." To do this, we simply divide the right by 300 and get the magic number 5252 again. This means that the right side of that bold equation needs to be divided by 5252 in order for the equation to be true. So therefore,

hp=(ft-lb torque*RPM)/5252

Lo and behold, that is the equation to calculate hp. This is how dynos work, too. They don't actually measure hp. They measure torque and RPM and then calculate hp using this equation. So let's say that a dyno measures that a car puts out 450 ft-lbs of torque at 5252 rpm. Using this equation, you get (450*5252)/5252, the 5252 cancels and you have 450 hp. That is why horsepower equals torque at 5252 RPM. This leads me into my final way of explaining this phenomenon that should be easier to grasp.

We now know that 1 hp=33,000 ft-lbs of work per minute. That's a pretty extreme and unrealistic scenario that I personally find harder to understand. So let's instead say that 1 hp= 200 lbs, 165 ft in one minute. Keep in mind that this is the same as 33,000 ft-lbs (200 lbs*165 ft=33,000 ft-lbs). So we have a force of 200 lbs. Pretend that is being applied in a rotational manner so it is now 200 foot lbs of torque. Now we have already discovered that the circumference of a circle with a radius of 1 ft is 6.28319 ft. So if we are moving something 165 ft per minute, we can divide that by 6.28319 and get 26.26 RPM. So we now know that we have 200 ft-lbs of torque at 26.26 RPM. So we know that 1 hp=200 ft-lb*26.26 RPM.

Let's say we dyno test a car and we observe that the car produces 500 ft-lbs of torque at 4500 RPM. We can therefore say we have two equations:

1 hp=200 ft-lb*26.26 RPM

X hp=500 ft-lbs*4500 RPM

When we multiple those out, we get 1 hp=5252 ft-lbs RPM and X hp=2,250,000 ft-lbs RPM. Since we know what 1 hp should equal, all we have to do to find the X amount of hp is divide 2,250,000 by 5252. When you do this, you get 428 hp. So all we did to find hp was take the amount of torque measured at a certain RPM, multiply the torque and RPM, and then divide that by 5252. Or, hp=(ft-lbs torque*RPM)/5252.