Do you mean that their accelerations would gradually diminish until they reach terminal velocity (where the air resistance is now equal to the effect of gravity)?
What is a ‘fall rate’?
I taught college physics (and chemistry) for 7 years at a military academy prep school. Our lower-level track used a book called Conceptual Physics by Paul G. Hewitt. This book came with a ton of ancillary material, and I used a lot of it in my physics course as well. I thought it was a good idea to to encourage students to use conceptual thinking, rather than mindlessly “plugging-and-chugging” through the equations.
You guessed it: all three of the questions in the OP are word-for-word out of Hewitt. I used all three of the questions in my course as handouts or on overheads to stimulate classroom discussion.
Without looking at anyone else’s answers, the answers are:
- b) iron ball
- b) decreases
3a. Friction f is equal to P.
3b. Net force on the crate is equal to zero.
Treis, both parts of the third question are to be answered with the following key fact in mind: “It is pulled so that it moves with constant velocity across the floor.” Therefore the answer to #3a is: Friction f is equal to P, for the reason you stated for your answer to #3b.
Also, for some of you nitpickers, question #3 (in Hewett’s original form) was accompanied by one of his ubiquitous sketches. The sketch in this case indicated that P was a horizontal force, parallel to the floor.
In analytical (“rigid-body”) mechanics, acceleration of the center of mass can of course be treated as an “inertial force” when multiplied by the mass, as you suggest. But it doesn’t work in some other areas of mechanics. Hence the desire to preserve the conceptual difference between acceleration and force.
Isn’t a pound of feathers heavier than a pound of lead? I know that gold is measured in troy ounces, and feathers are measured in avoirdupois ounces. Would lead be measured by its troy weight, or avoirdupois weight? There are 12 troy ounces in a troy pound, and 16 avoirdupois ounces in an avoirdupois pound; but I don’t remember which ounce is heavier than the other, nor by how much. (Also, I heard it as ‘which is heavier; a pound of feathers, or a pound of gold?’.)
I only scanned the thread quickly, and I didn’t see this: Does Galileo’s alledged ‘falling balls’ experiment have any bearing? IIRC, he (alledgedly) used two balls of different weight that were also of different size. The balls are said to have hit the ground at the same time. If, from what I’ve seen in this thread, the heavier (same-sized) ball would fall faster, then would the larger ball in Galileo’s experiment not fall faster than the smaller one because it has greater resistance?
Also: They would fall at the same rate in a vacuum, right? (Remembering the ‘hammer and feather’ demonstration on the moon.)
Yes, I already admitted that in my first post I was too glib about that. However, what’s the problem in the post you quote? Do you deny that gravity produces an acceleration, that air resistance produces a separate acceleration, that the sum of these (the net acceleration) is downwards (thus answering your first post), or that the term giving the portion of the net acceleration for which air resistance is responsible increases upwards with increasing downwards velocity (which was my original point)?
No, the box was at rest and now is moving therefore at some point an unbalanced force was applied. Therefore we know at somepoint P was greater than F. Its a pretty badly worded question, it really should have been broken into two parts.
I deny that there are separate accelerations. You are doing fine in rigid or elastic dynamics as long as you simply multiply the acceleration by the mass to get a “force”, which you can consider as separate from other forces, but when you start talking about separate accelerations, then there are problems. Acceleration is an observable kinematic quantity: it is the second time derivative of the ball’s position. There is only one acceleration. Like any vector, acceleration can be considered a sum of components, but they are components of a single quantity. There is no conceptual basis for talking about “separate” accelerations that are combined into a “net” acceleration.
Mathematically, in rigid body mechanics or in elastodynamics, one may treat accelerations through the device of inertial “forces” in which case they may be identified separately if you like. But there is still only one actual acceleration.
This would all amount to conceptual hairsplitting, except that in the dynamics of inelastic materials, accelerations cannot be treated as inertial “forces.” That’s why, even in elastic or rigid body dynamics, we only add accelerations kinematically, in the sense of summing the components of an acceleration vector, never dynamically. Otherwise people would have to unlearn the “fact” that accelerations only matter insofar as they are force per unit mass.
I agree that in the observed world there is ultimately only one force or acceleration. However, the analytic/synthetic nature of the Newtonian model immediately leads to speaking in terms of different forces and different accelerations which are added to produce a net force.
As an example, do you insist that the gravitational pull of Alpha Centauri be factored into the fall of the parachutist? In the real world, yes it’s there. In the problem, however, the term representing that force is very small and – since forces add together as vectors in the Newtonian model – it can be taken as zero. Every time you include less than absolutely every force on a particle, you’re using this principle, so I don’t see how you can seriously fault my explanation of the air resistance term in isolation. This is expecially true, since the whole point of an analytic/synthetic theory is to promote comprehensibility, which is really the point of (I say again) a high school physics problem.
I also don’t see why you insist on using the term “inertial”. None of the forces into which I decompose the net force are the result of using an accelerated coordinate frame. That’s a more technical point, though, and would degenerate into self-righteous nitpicking of my own.
I think it’s OK to speak of more than one force, but not more than one acceleration. Mathematically, if one restricts to a theory in which acceleration only enters the equation of motion through the inertial term, one may get a correspondence between individual forces and their associated accelerations by multiplying by the mass, which is how I read your reply to RM Mentock. Then you can, if you want, think of the accelerations given rise to by the individual forces as separate effects.
But a one-to-one correspondence between forces and accelerations is not generally possible. For instance, if you stick a rod into a bucket of molten plastic and spin the rod, the plastic will climb up the rod. So the plastic, taken as a body, is accelerating upward. I don’t know of a way to get a correspondence between that acceleration and a specific force, using any theory consistent with the rod-climbing phenomenon.
Certainly, you are not going to make a practical mistake in thinking of separate accelerations when you’re talking about falling balls. But you can’t carry that kind of thinking into other situations without running into trouble. So I would argue that it’s best not to talk about separate accelerations at all. Because forces are dynamical concepts, I can always envision them as arising from distinct effects, but I can’t do that with acceleration, which is a purely kinematic quantity.
Can I ask what exactly you mean by the Newtonian model? That may be the source of our difference of opinion.
I’m not familiar with this effect, but I’m willing to bet (especially since you say “plastic” and imply that not just any viscous fluid will do) that it’s got something to do with fine enough structure that Newtonian physics fails to handle.
I mean that mechanics which follows from the three Newtonian “laws” as postulates. It’s a formal mathematical model which does a good enough job in everyday situations to be taken as “physics”. Get hard enough constraints on motions and it becomes easier to use Hamiltonian or Lagrangian mechanics as a model. Increase the characteristic velocities and a special relativistic model becomes more appropriate. Reduce the characteristic actions and quantum mechanics becomes more appropriate. Newton’s mechanics is what high school physics teaches.
On the other hand, if you’re correct that there are classical-scale accelerations which are attributable neither to actual forces nor to forces arising from an accelerated ccoordinate frame then I’ll grant that this is something to be pedagogically careful about lest it have to be unlearned. I’d have to see more about your example, though, to believe that there’s no force hiding in the microscopic interaction between the plactic and the rod.
Oh, and if it gets small enough I don’t mind that there’s no “force”, since force (in the Newtonian sense) goes out the window in QFT.
Maybe this is the source of the confusion. Where did you admit this?
Implicitly in my rewording. I don’t correct myself unless I was incorrect to begin with.
actually “incorrect”?! whoa, I would have settled for just “too glib” 
There’s nothing nonclassical going on, but you’re right, a linearly viscous, homogeneous fluid won’t do it. You need a viscoelastic fluid. The effect is described here. The interaction between the fluid and the rod isn’t what drives it, it’s the normal stress difference generated by the elasticity of the fluid. The fluid climbs the rod as long as there is a regular old no-slip boundary condition between the fluid and the rod.
Viscoelastic fluids are “non-Newtonian” in that they do not obey Newton’s law of viscosity, but they do obey classical mechanics, i.e. quantum or relativistic effects play no role.
There is a widespread misconception (especially in departments of physics) that classical, nonrelativistic mechanics consists only of Newton’s laws plus your choice of constitutive equation, like linear elasticity. I don’t know whether this is your view, but if it is, you are probably in illustrious company. But there is a whole literature of continuum mechanics that stands outside of (and complementary to) the atomistic theories. For example, physics textbooks typically develop the law of conservation of angular momentum for a body by treating the body as made up of point particles interacting through central forces and subject to Newton’s third law, even though everyone knows atoms don’t interact that way. The problem seems to be that the axiomatic development of classical mechanics stopped for a long time, and when it started up again, it started in departments of engineering and applied mathematics instead of physics.
I bring this up because if one sees classical or Newtonian physics as being point particles interacting through central forces, then there is no problem with talking about separate accelerations, because one can always trace them to the force divided by mass on a particular point particle. But that view is fatally flawed.
There are consistent, useful theories of classical continuum mechanics in which motions can be generated without any force at all, because the existence of torques that are not moments of forces is permitted.
Poorly worded. The example I gave was meant to illustrate the prevalence of poorly reasoned theories of extended bodies among physicists, not an example of modern continuum mechanics.
As I said, I’m talking about the Newtonian model, which is generally the extent of any high school course in physics.
This is, in my mind, a somewhat subtle point. As far as I’m concerned, torques are forces as well, and may enter on the ground floor depending on the configuration space of the system. Again, though, this goes far beyond a high-school course.