What is the mathematical relation between horsepower and torque?

Question

HP and torque are used with abandon and interchangably by automotive engine writers, but it seems like they don't have a clue what they're talking about.

HP is simply the measured by lifting a weight a specified height in a specified time, so what is torque?

Answer 1

Both torque and power can be observed "directly". Think of slowing a free-spinning tire with your hand. Feel the tug on your palm and the tension in your arm? That's a measure of torque, the torque the tire experiences as a result of your palm slowing it down. Feel the heat build up from friction? That's a measure of power.

Incidentally, water brake dynamometers get a direct measurement of power by measuring the increase in the temperature of water flowing past a propeller spun by the engine under test. You can solve for torque if you know engine RPM.

What an engine naturally produces, however, is torque. Think about one piston in a gasoline engine. When the gasoline ignites, it pushes on the piston, and the piston exerts pressure on the crankshaft, causing it to turn. The crankshaft feels some number of foot-pounds of torque in the process. There are three variables that affect torque:

The size of the piston face
The amount of pressure that the ignited fuel applies to the face of the piston
The distance the piston travels on each stroke (therefore the diameter of the crankshaft). The bigger the diameter of the crankshaft, the bigger the lever arm and therefore the more torque.
There is a direct relationship between horsepower and torque. You can convert torque to horsepower with the following equation:

HP = Torque * RPM / 5,252
That 5,252 number, by the way, comes from dividing 33,000 by (2 * pi). Imagine taking 33,000 foot-pounds and walking it around in a circle rather than a straight line. For example, if you took a 10 foot pole and attached it to a vertical axle, the circumference of its circle is:
circumference = 10 * 2 * pi = 62.8 feet
If one horse is pushing on the pole with 100 pounds of force (1,000 foot-pounds of torque), it can move the pole at 5.25 RPM. Torque and horsepower are directly related to each other.

Answer 2

Torque is energy to produce rotation. Formally it's the product of a tangential force and the radial distance the force is from a pivot point. Torque units typically are foot pounds or newton meters, the same unit coincidentally for work.

Horsepower is a RATE for doing work, that is work per unit time. Generally the greater the torque available the higher the horsepower potential.

Speaking loosely you gotta have enough torque to turn the wheels; the more horsepower available the faster you can get the wheels to turn.

Answer 3

Mathematical Relation
Power (Horsepower) = (Torque x RPM)/5252

The math is easy but as you note, grasping the concept for internal combustion engines is not. The following notes may help.

- Power is the rate that energy is transferred from one form to another. An internal combustion engine converts the chemical energy of fuel to motion plus heat loss . Power is needed to get any vehicle moving quickly and cover the energy losses. The energy of a moving vehicle is proportional to it's mass and velocity (squared). So to get any heavy vehicle to high speeds quickly requires one thing...power. Anyone who has seen a space shuttle launch can get a feel for the importance of power.

-Torque is a rotational force and represents a mechanical advantage. The common analogy is a wrench turning a bolt. Torque does not indicate how fast the bolt turns. Torque can still exist with no motion. The locked rotor of an electic motor produces torque but there is no motion.

-Torque is like the voltage of a battery and shaft speed is like electrical current. Power the battery delivers is a function of both current and voltage. You need both to energize an electric motor or energize a light. An electric car needs electric power. One small, high voltage battery is not enough if there is no current to produce the power needed. Likewise, you need both mechanical torque and shaft rotational speed to generate power to a vehicle. Torque alone is not enough.

-You can add all sorts of gearing to increase or decrease the ouput torque of an engine at the wheels. But because of energy conservation, there is nothing you can do with gears, transmissions or hydraulics to increase the power output. The advantage of gearing is that it allows a car engine to operate in the range of maximum power at various speeds during acceleration. More power means more energy converted to speed more quickly.

- Imagine a 1 foot radius pulley attached to the shaft of an engine. A rope is tied to a weight and wrapped around the pulley. With a 1 foot torque moment from pulley wheel radius, the engine torque is the amount of weight the engine can lift at any given RPM. Power is a function of the weight AND the rate the weight is lifted. And since the rate lifts at a rate propotional to shaft RPM, Power is proportional to shaft RPM.

-When things get complicated, energy and the rate engergy is used can simply things. Motion requires energy. Getting an auto moving quickly requires that energy be expended quickly (more power.) Adjusting mechanical torque (gears and transmission) insures an engine is providing maximum power at different speeds.

Answer 4

In Enginnering terms

Hp - horsepower
K - conversion factor
Hp =K X Torque x Rpm
Torgue is that twist force trying to rotate the pulley your useing to lift that something. while Horsepower is how fast that twist force lifts it .

Answer 5

Torque is radial force multiplied by the distance from the center of the circle. Think loosely as turning force on a wrench. It's usually measured in foot-lbs .

Answer 6

horsepower = 2 x pi x rpm x torque(lb.feet) / 33000

Answer 7

(X foot-pounds * 2 * PI * REVOLUTIONS)/MINUTE
Where torque is foot-pounds, and the result is HP.

HP and torque are not interchangable, nor does anyone that knows anything about them do so. Torque is the mesure of rotational force.

Answer 8

In racing terms, Torque is what launches you from the start, hp is what keeps you moving.

Answer 9

Twisting power, as in axle.