I would like to know whether there is any difference in the energy consumed in climbing a flight of stairs, if the steps are taken one at a time vs taking them 2 at a time
[Physics] Energy spent climbing a flight of stairs
energy
Related Solutions
The center of mass KE is equal to the KE of the total mass of the system moving at the center of mass speed. The total KE in a stationary frame of reference is the sum of this center of mass KE and the KE of different masses relative to the center of mass frame of reference. In mathematical terms,
\begin{equation} KE_{lab}^{total} = KE_{lab}^{CM} + \sum\limits_{i=1}^n KE_{CM}^i \end{equation}
Here the subscript refers to the frame of reference while the superscript is the KE of that species. Hence the term $K_{int}$ refers to the KE of individual masses of the system relative to the center of mass, while $K_{CM}$ is the kinetic energy of center of mass as observed from a stationary frame. The kinetic energy lost during the explosion is the difference between the initial and final $KE_{lab}^{total}$. The center of mass kinetic energy can be calculated by knowing the total mass and the velocity of center of mass in the stationary frame. To calculate the internal kinetic energy, one needs to calculate the velocities of individual masses relative to the center of mass i.e $v_{lab}^{i} - v_{CM}$ (vector subtraction).
Suppose that you are standing stationary outside your house holding a 25kg sack of rice. Now walk 1km to your friend's house to give him the rice. He is not home so you return with the rice. You are now back where you started: stationary again with the sack of rice. Have you done any work? In a day to day sense, you have but you have neither gained not lost kinetic nor potential energy so you have done no work. You may have expended lots of energy but this has all gone to various inefficiencies.
Clarification. What I was trying to address was the distinction between work in its common day to day sense and its sense in physics. In my scenario, you have burned some food to make the trip and conservation of energy will not have been violated. So, the energy has gone somewhere. You and the sack of rice have no net gain in kinetic or potential energy so it has gone elsewhere. Most of it has become heat.
So did you do work? You probably felt that you had since you will be tired after the trip. Your wife who asked you to deliver the rice may think that you have not as the sack of rice is where it started. In physics, work has been done and the conservation of energy has not been violated, it is just that you might not notice or care where the work was done.
Back to your example, you might or might not have done more work by running up the stairs. You would need to determine the energy that went elsewhere than your kinetic or potential energy. This is possibly more a question of biology than physics. In which scenario is your body more efficient?
Best Answer
It shouldn't make any difference. The only force you're acting against is gravity if you don't take friction into consideration, and gravity is a conservative force, which means that the work it does (and hence the work that you do in this situation) doesn't depend on the path you take. So, the amount of energy you consume wouldn't depend on taking the steps one or two at a time.
Approximately, the energy you consume would be $mgh$, where $m$ is your mass, $g$ is the gravitational field near the Earth's surface, and $h$ is the total height you climb.
This is all ideal, of course; it assumes you are a point particle, for example. There is no accounting for the discomfort your leg would feel upon having to stretch and contract an additional amount, so in a real-life situation, there would probably be a deviation from what I stated above, and maybe it would require extra energy for you to climb two steps at a time. From a simplistic point of view, this is because since your leg has to reach up farther and contract more when it pulls you up (when you're climbing two steps at a time), there will be more energy lost due to friction between your joints and the difference in how your muscles contract.
From a different point of view, climbing two steps at a time can probably get you upstairs faster, and you will therefore spend less time climbing the stairs. If you spend less time climbing, you will spend less energy on things like maintaining body temperature, and breathing (since you are climbing for less time). So while climbing the stairs, you may spend less energy on those processes as a result of going up two steps at a time.
The human body is a pretty complicated system, and as you can see from the above speculations, it's not easy to see how exactly its processes would change when you change the way you climb stairs. However, you can approximately say that you spend the same amount of energy with both paths (let's pretend we're particles, it makes everything more simple :D).