The energy spent when pushing a non-moving object versus pushing a moving object with the same effort

I have been wondering about some aspects of energy transfer and have become a bit confused - to illustrate my confusion, I have come up with this example. Hopefully you can clear up which of my premises or conclusions that are wrong.

Situation 1.

I am trying to push a box, but I fail, because there is a very heavy object on the other side of the box. I thus do no work on the box, and all the energy $$E$$ that I spend is used in my muscle cells.

Situation 2.

The next day, the heavy object has been moved, and I decide to push the box again, exerting myself exactly the same as yesterday. I assume this would mean applying the same force to the box. This time, the box moves, and I do work on the box, but with my level of exertion being the same as yesterday, I still spend the energy $$E$$, and for this level of exertion, my muscles need all of that energy (same as yesterday) to function. Thus there is no energy left for the box.

So, what is going on here?

• Do I need less energy for my muscles, when the box is actually moving, and the excess energy can be transferred to the box?

• Do I somehow spend different amounts of energy in the two situations, even though the force applied is the same?

If you (like me) are a bit uneasy with the volatility of a human being in this setup, it might be better to replace it with a robot that applies the same force to the box in situation 1 and 2. Perhaps the robot has a caterpillar track that is going at a constant rate, so just skidding on the ground in situation 1, but moving with the box in situation 2.

As you suggest, the thought experiment can be stated in a clear way when all the components in the setup are mechanical parts.

Heavy machinery usually as a type of clutch that is called 'torque convertor'. Let's say we have a machine with a a powertrain such that if the load increases the power output increases to meet the load, so that the drive axis is always kept going.

When a bulldozer comes up against an immovable object then initially the power output of the engine will rise. But the object is immovable, and at some point the torque converter will start to slip. The torque convertor is designed to do that; the torque converter helps to prevent the machine from destroying itself.

The power output that would otherwise go into moving stuff around has to go somewhere. When the torque converter is slipping all of that power output transforms to heat. (Heavy machinery has coolant lines close to the torque convertor so that heat can be withdrawn from the torque converter.)

If we disregard the various friction losses in other parts of the bulldozer:
For the power output of the engine itself it makes no difference whether the torque converter is just below the point where it starts slipping, or just past that point.

Power output is power output. If stuff is being moved around then that is where the power output is going; if the torque converter is slipping then the power output is transformed to heat.

Of course, it would be pointless to keep increasing power output beyond the point where the torque converter starts slipping.

Muscle contraction is, when you go all the way down to molecular level, a process of elongated molecular structures actively sliding along each other.

(Think of it as millipedes climbing in opposite directions, using the body of the other to pull themselves a bit further along.)

Imagine a weight handed to your outstretched arm. If you have just enough muscle strength to lift it the weight goes up. If the weight is just beyond your muscle strength then despite all your effort your arm sags down under the weight.

Despite your effort, the muscle fibers slide in the direction oppostite to your intent. That opposite-to-your-intent slding is the physiologic counterpart of a slipping torque converter.

If you are pushing something, but it is immovable, then the motor neuron connections to your muscles are instructing the muscle to contract, and that attempt to contract means that on molecular level your muscles are outputting power. But the object is immovable. But power output is power output. So all of the power output turns in to heat.

(Note: muscles do not turn all the energy that is expended into motion. I don't know any number, but I think the efficiency is below 50%. When the object that you are pushing is perfectly movable you will still generate heat in your muscles, simply because a sizable percentage of the energy expended ends up as heat anyway.)

• Thanks, this makes sense to me. And let me try to apply it to the case of a human being: I would in situation 1 need to be making a walking motion in order to get a setup similar to me walking when moving the box in situation 2. And the walking motion in situation 1 would amount to my shoes slipping on the ground, thus producing heat energy, as opposed to situation 2, where there is no slipping, and the energy goes to the box. Apr 14 at 14:25

Situation 1:

• The energy spent is zero, because the force you apply does not generate a motion.
• Stated otherwise : The reaction exerted by the object to be moved is equal and opposite to the force you apply. Therefore the total is 0. Therefore no energy is spent.

Situation 2:

• The object moves, because the friction is less, because the total weight of the obstacle is less.
• In this case the energy spent is F * L Where F is the force you apply and L is the distance traveled.
• I don't understand your answer. In situation 1, I apply a force to the box, and the box applies an opposite force on me, so since these two forces act on two different objects, they will not cancel out. And even though I don't move the box, I still spent energy trying to move it. Same as if I hold an object at a constant height in my out-stretched arm - my muscles are spending energy and eventually I get tired. Apr 14 at 8:19
• The 2 forces apply to both you and the box (action / reaction). Nothing is moving / Nothing is accelerated / Nothing is set in motion. In physics energy is defined as the the work done by a force to create some actual physical change. This change can be: a change in the trajectory of an objet: when a socker player kicks a ball, for instance, or when you bend a metal rod, or when you climb a flight of stairs. If nothing’s moving, then the acceleration of the box (and yours, which is equal and opposite) is zero ,so is the sum of forces and the energy spent. Apr 14 at 15:24
• ... more technically : You apply a force to an object, then only one of two things can happen in the presence of friction: either this force is large enough to set the object in motion (1) or it isn’t (2). Case (1) : the force applied is not moving, the energy spent is F * X = 0, since X = 0, since the object is not moving. Case (2) the energy spent is not zero, since X is not zero. Apr 14 at 15:32