Work over time; humans versus automobiles

I've been doing some research lately on torque and horsepower to understand how they are connected to one another.

Take my car, a 2.0 TDI, outputs 390 Nm torque. 390 Nm alone without considering an engine or considering effect, isn't particularly much. You could create 390 Nm with your bare hands.

The engine creates those 390 Nm per rotation, and at peak flywheel torque (≈ 2000 RPM), it rotates about 33 times per second.

Considering work over time, that results to some 110 horsepower.

But if you have a bicycle with pedals, say 0.5m from center for calculation purposes, 390 Nm could be created by creating 780 newtons of force half a meter from centre. That is around the equivalent of my weight.

How is it then that the engine can push a 1.6 ton car in 230 km/t, but I can barely make a steep hill?

From my own understand, it all has to do with work over time. Maybe this also has something to do with when starting, the combustion can barely make the engine rotate, but once started it can pull a big car at high speeds?

You can create any amount of torque with your bare hands. Just pick a long enough lever. Archimedes famously said: "Give me a place to stand and with a lever I will move the whole world." (source: Wikiquote). Do note that Archimedes said nothing about the speed which he can move the whole world; he just said that he can in principle move the whole world.

However, the limitation here is power, not torque. The speed at which your hands or feet move is limited, and once you are operating near your maximum power, you may not be able to be as strong as you are when not moving at all.

Levers and gears can change the force or torque (the rotational equivalent of force), but none can change the power. The power is a fundamental thing that must come from an energy source.

There are two useful formulas: power is the rotational speed (in radians per second) times torque, or alternatively power is the linear speed times force. Levers work by modifying the speed and force in opposite directions; the end result is that their product is the same. Gears works similarly by modifying the rotational speed and torque in opposite directions; again, their product stays the same.

About torque levels: with a 170mm crank, a heavyweight strong bicyclist weighing 100 kg, pulling up with the rear shoe at 30 kg (and pushing up with the front shoe at an additional 30 kg), and pulling up from the handlebar at 25 kg, can reach 310 Nm: (100 kg + 30 kg + 30 kg + 25 kg) * 9.81 m/s^2 * 0.17 m = 308.52 Nm. Gears can multiply this: e.g. 22 tooth front gear and 32 tooth rear gear gives you 449 Nm. So, not only can you create torque more than 390 Nm with your bare hands, you can actually do so on a practical machine, a bicycle, with your feet.

But no way can a bicyclist produce 110 horsepower! A bicyclist is limited to about 1 horsepower in short bursts, and about one third to one half of a horsepower in longer durations.

So, how does a car engine differ from practical levers and bicycles powered by your hands or feet? It's the rotational rate! Car engines rotate so fast that no way could you produce such fast rotation with your hands or feet. The force and torque levels in car engines, however, are similar to what you can produce with your hands and feet in practical applications.

• Excellent and thorough answer. A followup question: when you start your car in the cold, it may struggle to start. The RPM is very, very low and the combustion event can barely rotate the engine. Once the engine is started, however, it revolves at an RPM around 900 rpm and can now easily move the vehicle without any additional throttle application than idle alone. I assume this all has to do with power? Power seems like an abstract concept; torque per revolution makes much more intuitive sense. Hard to imagine how time and frequency plays a big role – Erik Sep 14 '17 at 17:59

The engine creates those 390 Nm per rotation, and at peak flywheel torque (≈ 2000 RPM), it rotates about 33 times per second.

Considering work over time, that results to some 110 horsepower.

I concur:

``````Power = Torque × Angular Speed
= 390 Nm × ( 2,000 rev/min × 2π rad/rev × 1/60 min/s )
= 81680 W
= 110 hp
``````

But if you have a bicycle with pedals, say 0.5 m from center for calculation purposes, 390 Nm could be created by creating 780 newtons of force half a meter from centre. That is around the equivalent of my weight. How is it then that the engine can push a 1.6 ton car in 230 km/h, but I can barely make a steep hill?

There are a few parts to this:

• No human can self-power bicycle pedals to turn 33 times every second...

... let alone 3 revolutions per second, especially at that level of torque.

• 0.5 m is a bit of a stretch...

... even for a penny-farthing.

So you'd need even more force to sustain any given level of torque.

• Mass doesn't affect top speed...

... just acceleration.

This is a direct consequence of Newton's first and second laws. The limiting factors for a typical road vehicle are aerodynamic drag, road friction and drivetrain losses.

• Excellent answer! Could you please elaborate how mass doesn't effect top speed? Are we disregarding the mentioned limiting factors then? – Erik Sep 10 '17 at 19:19
• Sure, I'm assuming that the car's driving on a flat surface when I say that mass has no role to play (on an incline the story's very different). It's basically because at top speed acceleration is zero, so the motive force (sourced from the engine) is completely counteracted by the resistive forces acting on the vehicle. The only way mass would play a role here is if some of the resistive forces are proportional to the vehicle's mass. Such forces do exist in reality but their contribution is next to negligible, which is why I feel justified in saying that mass has no effect on top speed. – Zaid Sep 10 '17 at 19:35
• Very interesting. But to gain higher speed, you must accelerate. To accelerate, you need a force F = ma. Bigger mass -> bigger force. Doesn't that mean that mass affects top speed? – Erik Sep 10 '17 at 20:11
• It may take longer to reach top speed, but it will reach the same top speed – Zaid Sep 10 '17 at 20:18
• Is it reasonable to argue that if I can move an object with even the slightest acceleration (e.g 0.00001 m/s) over time, it will reach the same top speed regardless of weight? Because regardless of weight, you can always lower the acceleration indefinitely to ensure the forces required are below the maximum output of the engine? max_eng_output > mass_to_move * acceleration. Assuming the engine output and mass are constant, we can vary the acceleration to have the inequality be correct – Erik Sep 10 '17 at 21:19

I believe you have your calculations backwards. @0.5m from center, you'd need to cut the 390 in 1/2 ... down to 195Nm. In reality, the pedal is only about 0.2m from the center, which reduces it down to about 1/5th of the mystical 390Nm, or 78Nm in your example. Further in reality, if you Googled for how much horsepower a human can develop, you'd find the following information:

When considering human-powered equipment, a healthy human can produce about 1.2 hp briefly (see orders of magnitude) and sustain about 0.1 hp (74.57 W) indefinitely; trained athletes can manage up to about 2.5 hp (1.85 kW) briefly and 0.35 hp (260 W) for a period of several hours.

Considering even the slightest of cars can produce 60+ hp, they are far outworking us measly humans. Then you put gearing into the equation (with which automobiles is the great equalizer) you start seeing how an automobile can not only get itself going, but can keep itself at higher speeds for sustained periods of time, where as a human has a hard time moving a bicycle down the road for any appreciable amount of time.

• I believe the OP's calculations are fine. 780 N * 0.5 m = 390 Nm – Zaid Sep 10 '17 at 18:56
• @Zaid That is correct. My idea was that a car produces 390 Nm, so for a bicycle to do the same, we would need twice the amount of force at half the distance. Or, 5x the amount of force @ 0.2m distance – Erik Sep 10 '17 at 19:11
• @Erik - Gotcha. I would suggest that doesn't negate my point of how much HP can a human produce over a period of time versus a car. – Pᴀᴜʟsᴛᴇʀ2 Sep 10 '17 at 20:13
• Not at all. That is very true and sheds some important light to the question. – Erik Sep 10 '17 at 20:16

Not sure about the figures quoted already : the cyclist Chris Boardman maintained 450W for an hour see here: