From wikipedia: “In physics, acceleration is the rate at which the velocity of a body changes with time.”
How would you describe Newton’s first law of motion of force by definition is an acceleration?
What Chrono’s writes: “once the monkey is in motion no additional force is required, only the base, balanced amount of force needed to counteract gravity” is exactly Newton’s first law of motion. If the monkey’s velocity isn’t changing, either because it’s hanging still or moving upwards at constant velocity, the net force is zero and the pull exerted needs only be equal to G.
Please rephrase that for a world where any force, not only a net force, is an acceleration.
This is wrong, and I’m not sure how you are arriving at this. Look at Naita’s post 78 (for clarity, I’d add a fifth line “Elevator stands still again: 1G”). Do you disagree with that post?
There’s no way to get the elevator to stop without the “Elevator slows down: <1G” line.
Where do you think the force is coming from? The gravitational force is completely constant regardless of motion (F=ma; a and m are both constant). The only other source must be that from an acceleration. Starting and stopping is an acceleration, but constant motion is not. No additional acceleration means no additional force.
A real world scenario involving muscles is unlikely to involve constant force either. The point was that you appeared to quibble with Newton’s first law of motion, not with whether a real world situation would at any point involve constant velocity.
It’s not really critical. The force averages out to just the gravitational force. If it exceeds gravity for some short period, then it must be less than gravity for some other period. If it didn’t average out to be the same, then the monkey would be continually accelerating the whole way.
No matter what, the bucket (assuming frictionless/massless rope/pulley) exactly matches the monkey’s movement.
iiiieeeeeeeee… nnnooooooooooooo. Acceleration is not a force. Applying a force to an object will cause it to change speed. That is an acceleration. Forces cause acceleration, not vice-versa.
And, what nobody is taking into account is that as the monkey climbs, he is increasing his potential energy. That takes “work”, i.e. the application of a force, which doesn’t have to cause acceleration. If it just offsets the pull of gravity, you get a constant motion in the opposite direction. For example, terminal velocity. The speed at which the force of air resistance is exactly the same (but opposite) that of the pull of gravity. The object still has a lot of forces acting on it, but no acceleration just constant speed.
I’m not sure what the bucket might think of all this. Probably something along the lines of “what, hey, why am I hanging here? I hope that monkey doesn’t try to bite me! How come this rope is so light, kind of like it has no mass. And the pulley isn’t squeaking, must be frictionless! aarrgghhhh… dear Liza help, I’ve got a hole in me”.
It’s critical when discussing how many times the monkey will accelerate when climbing the rope. Averaging over time isn’t how the real world operates.
But I will say, after viewing some monkeys climbing ropes on youtube, it’s definitely smoother than a human climbing which is what I was picturing, but the ones I saw still had distinct upward motions as they switched arms.
Frictionless pulleys also aren’t how the real world operates. If we’re allowed to assume a frictionless pulley, then why not also assume a smooth-climbing monkey?
For the people (Chronos, et. al.) who think there is a single acceleration - this seems to mean that you think that if you are climbing a rope, you only need to pull on it once and you’ll fly to the top of the rope.
That’s obviously false - or else you could jump to the moon with one jump. So, every time the monkey pulls, he is applying additional force to the system. Which moves the bucket.
No, we don’t. If the monkey is just hanging on the rope, he has to apply a force just to stay in place. What we are saying is that for a portion of the climb where the monkey is moving upwards with a constant velocity, he will only have to apply the same force. Not more, as many people seem to believe.
Force and acceleration are essentially the same thing, aside from a proportionality constant of mass. Is gravity a force with a magnitude proportional to mass, or is it an acceleration? No one knows, and it doesn’t matter.
Work is force times distance. If there is no acceleration somewhere along the way, there is no movement (assuming there was no motion to start with). A monkey hanging applies a force (indeed, the same exact force as one climbing at a constant rate), but doesn’t produce work since it doesn’t apply the force through a distance.
I can guarantee that if you properly account for every force in the system, and do the average with the proper basis, then a[sub]ave[/sub] = F[sub]ave[/sub]/m, and v[sub]final[/sub] = v[sub]initial[/sub] + F[sub]ave[/sub]t/m. [sup]1[/sup]
It works both ways, of course. Hang on a rope. Raise and lower yourself, sometimes gently and sometimes quickly, and with varying duration. If you were stationary both before and after the exercise, then I can say with complete confidence that your average force is simply your mass times 9.8 m/s^2 (no matter what your final height).
For a very simple example, try this (theoretically): pull on the rope for one second at twice your weight. You will accelerate upwards at 1G. Now let go of the rope for one second. You will decelerate at 1G, but continue gaining height until your velocity reaches zero after one second and you grab the rope again. Clearly, your average acceleration (above that of gravity) is exactly zero. And yet you’ve gained 9.8 meters in height.
[sup]1[/sup] Okay, I’m assuming a non-relativistic monkey.
In hindsight yes, but then you’d have been much better off just saying “A real world monkey won’t have constant speed during the climb”. As usual with these discussions a major issue is people arguing from different, unstated, sets of premisses.
I see several posts here concerned about the monkey gaining potential energy which can’t come from nowhere. Of course it doesn’t come from nowhere. The increased gravitational potential energy is offset by a loss of chemical potential energy in the monkey’s muscles.
Martin Hogbin summed it up rather well. In the make-believe world of frictionless pulleys and massless ropes, every time the monkey pulled on the rope trying to lift itself 40 cm, the bucket would rise 20 cm and the monkey would rise 20 cm. In the real world where pulleys have friction, a smart monkey could climb slowly and the bucket would not move at all.
Here’s an easy way to see why that would work: Imagine a very heavy rope on a large pulley with a mouse holding on to one end of the rope and at the other end is a tiny bucket of sand which weighs the same as the mouse. When the mouse climbs the rope, do you expect the pulley to move? Of course not, the friction far exceeds the mouse’s pulling ability. The same could work for the monkey if the monkey intentionally limited its acceleration and climbed slowly.
You mean like when I said this:
“Well, yes, I’m describing how rope climbing actually happens (doesn’t have to be a complete stop, but there is a distinct increase and decrease in speed typically as arms are swapped back and forth).”
Sure that can happen and it’s ok, but it’s not my default assumption so it would have to be made explicit for me to understand that is what is meant, like ZenBeam did.
ZenBeam’s posts expanded on the area of confusion and explained how it would work in both scenarios, confirming that both you and I were correct but making different assumptions.