Dear Science,
My 9-year-old son went on a field trip to the Space Needle. The
kids’ task was to find certain objects on a list, from the city below.
When the pencil that my son and his friend were using broke, my son
encouraged his partner to throw it over the edge to the ground below.
(Surely doing a science experiment of his own.) When his second pencil
broke, they did it again, and this time they got caught. What velocity
can a pencil reach, and assuming the eraser end is the heaviest end, is
it even possible for the pencil to land point-down? What are the
chances it could harm someone standing on the sidewalk down
below?
Embarrassed But Curious
Kudos to your kid for his experimental nature. But please encourage
him to try less hazardous ones in the future. Still, don’t worry about
him killing anyone with a pencil flung off the Space Needle; at worst,
someone just lost an eye.
Let’s consider the pencil at the moment after it was flung from
the Space Needle. It’s going to start falling, thanks to the force
of gravity tugging it down. Gravity accelerates everything, regardless
of mass, at about 10 meters per second squared. The observation deck of
the Space Needle is about 160 meters above the ground. Calculus was
invented for this sort of problem. (Thanks, Newton!) A pencil dropped
from 160 meters will hit a velocity of about 60 meters per second just
before it hits the ground, assuming gravity is the only force acting on
the pencil. But it isn’t. We’ve forgotten about the air!
Two forces tug on the pencil in the vertical
axis. Gravity is pulling down; the pencil must move the air out
of the way as it falls, creating a drag force that pushes the pencil
up. The faster the pencil is going, the stronger this force
becomes. As the pencil accelerates, the upward drag force will increase
until it perfectly balances the downward force of gravity. When these
two opposing forces are in balance, the pencil has hit its terminal
velocity. Running through a bit of math (and a few guesses), Science
has figured the terminal velocity of a pencil dropped is about 15
meters per second, quite a bit less than when we weren’t accounting
for the air. The 160 meters is enough distance to fall for the pencil
to achieve its terminal velocity.
How much punch does a pencil
traveling at about 15 meters per
second have? About 1 joule of energy. A paintball has about 12 joules
of energy. It wouldn’t be fun to be below your son’s pencil, but
you’d probably survive.
Free-Body Diagrammingly Yours,
Science
Hmmm. I don’t think friction from the air actually pushes the pencil up. It slows the acceleration due to gravity, but pushes up? Not unless it’s magic air.
Right; the air doesn’t literally “push the pencil up.” Remembering back to my college physics class, the forces acting on the pencil are what make it move–in this case, downward.
what Golob was saying is that, in the Newtonian sense, the force exerted by gravity acts on the pencil “downward,” toward the center of the Earth, whereas the force of air resistance acts on the pencil in the opposite direction–that is, “up.”
All this means is that the force of air resistance is subtracted from, rather than added to, the force of gravity when calculating the total force exerted on the pencil.
Physically speaking it does. When you press against a wall the wall ‘pushes back’ with the same amount of force – according to physics books and professors.
and party smants? you’re a moron. go take a basic physics course and learn about vectors. gravity is a vector pushing down. the friction of the air is a vector pushing up. duh. i learned that in high school. why didn’t you?
Actually, the force of gravity is a vector pointing downward (in a reference frame with our normal experience). Friction is usually approximated as a vector pointing up, but is actually dissipating the energy provided by gravity into heat. Friction isn’t pushing up, it’s transforming the energy of falling into heat. Newton’s 3rd law only applies here by some back force from the compressibility of air.
Right, so my overwhelming need to report something ‘correct’ took over and I’m back again.
First, friction isn’t a vector. That’s just a really convenient way of expressing the loss of energy due to friction. They teach you that in physics 101 and 102 because it’s convenient and easy, and very few people care beyond that, but it’s incorrect.
Second, the dominant slowing force here may be friction (I don’t have the numbers in front of me), but we shouldn’t discount drag, or air resistance. This is a bit of a combination of friction and Newton’s 3rd law, of a body acting on the air below it. The air compresses, and acts like a spring to provide a small opposing force. How strong this opposing force is has to do with the shape of the body; more compact things compress a smaller area of air than more spread-out things. Hence why a crumpled-up ball of paper falls faster than a flat piece, despite being completely the same.
So Brandon’s right, and, while the air can be considered as ‘pushing up’ on the pencil, this isn’t 100% correct. Wind effects are ignored here, but you can imagine how those work.
Quoted from Joe M:
“First, friction isn’t a vector. That’s just a really convenient way of expressing the loss of energy due to friction.”
I’m not exactly sure what you mean by friction isn’t a vector. Forces in general are vector quantities (they have both magnitude and direction), and the frictional force is a force. I agree that there can be different representations for forces, but calling it a vector is still 100% correct (just not the entirety of what a “force” means).
Quote from Joe M:
“Second, the dominant slowing force here may be friction (I don’t have the numbers in front of me), but we shouldn’t discount drag, or air resistance.”
I’m not sure how long ago you took phys 101/102, but it seems you have some of the terms a bit confused. Air drag is just another term for the friction force due to a body moving through a fluid (both liquids and gasses are fluids). This friction force is usually termed drag since it depends on the relative velocity of the object and the medium through which it is moving, where as (approximately) the parallel component of the contact force between two objects (e.g. blocks) does not depend on the relative velocity between the two objects.
Quote from Joe M.
“This is a bit of a combination of friction and Newton’s 3rd law, of a body acting on the air below it.”
I’m not sure how you “combine” N3 with friction. N3 states that all forces arise from interactions between two objects. Thus for every force on a particular object (A) due to a second object (B) there exists a force on B due to A. Applied to this case, there is a friction (drag) force on the pencil by the air, as well as a friction (drag) force on the air by the pencil.
Quote from Joe M:
“That’s just a really convenient way of expressing the loss of energy due to friction.”
The analysis of the author of the article is perfect, and I think put in very nice terms that would be understandable by a majority of the population. Talking about the loss of energy due to the friction (drag) force is not needed in this discussion. An analysis of forces gives the terminal velocity of the system consisting of just the pencil (not the air) which can then be used to calculate the kinetic energy of the system of just the pencil. Even though the pencil system is loosing energy to the air through the drag interaction, it is irrelevant to the discussion.
Personally I would have compared the momentum of a paintball to the momentum of the falling pencil, as the change in momentum divided by the time over which that momentum changed gives the average force necessary to change the momentum. (Look up impulse-momentum theorem. I did a horrible job explaining it here) And it is this force that one “feels”.
And in case you think I’m crazy, you can see that I’m a PhD student at UW. I’ve been a head TA for about 2 years now, and have lots of experience teaching this level of physics.
the little bastards should have had their asses whipped for throwing shit off the tower.
the little assholes.
Once, I was hit in the head by a quarter that was thrown from the 300 level ramp on the outside of the Kingdome. It cut my scalp and hurt, but it didn’t kill me. I never saw the asshole who threw it. Some other asshole saw it happen and then scrambled for the coin that had just bounced off my head. My faith in humanity was sorely tested that night.
Also, Mythbusters did an episode about this, using a penny and the Empire State building for their calculations. They made a pneumatic gun that would shoot pennies at terminal velocity, and shot it at each other. That was cool.
Golly Isaac L, you should just ask Joe M out.
“Even though the pencil system is loosing energy to the air through the drag interaction, it is irrelevant to the discussion.”
You spelled ‘losing’ wrong.
My friend droped a bic lighter off the needle, and went down and found it. The lighter only had a small nic in the wheel, and worked fine for quite along time.
Tell your son at 25.00dollars a ticket to the top, he should of droped his bookbag!
Isaac Said:
“I’m not sure how long ago you took phys 101/102, but it seems you have some of the terms a bit confused. Air drag is just another term for the friction force due to a body moving through a fluid (both liquids and gasses are fluids). This friction force is usually termed drag since it depends on the relative velocity of the object and the medium through which it is moving, where as (approximately) the parallel component of the contact force between two objects (e.g. blocks) does not depend on the relative velocity between the two objects.”
Friction is not the only component of air drag. When the body is falling quickly through the air, there is a pressure differential – higher pressure below, lower pressure above. This creates a net upward force which is accurately described as a “push.” Air pressure below pushes up, air pressure above pushes down, but not as hard. This is called “pressure drag.” The mechanism by which a parachute slows a fall would be quite difficult to explain with friction drag alone.
In general, very small object are more influenced by friction drag while large objects are more influenced by pressure drag.
The reynolds number for a falling pencil would be in the thousands (V*L/kinviscos = 15m/s *.01m /(10^-6) Reynolds numbers in this range indicate that the pressure drag is likely to be a more important factor than the friction drag.
In summary: Air drag = Friction Drag + Pressure Drag.
Re 100 Ignore Pressure Drag
Halliday and Resnick do a terrible job of explaining this – but it is a survey of physics, not a complete treatment. Any decent fluid mechanics book should lay it out clearly.
And Isaac said:
“And in case you think I’m crazy, you can see that I’m a PhD student at UW. I’ve been a head TA for about 2 years now, and have lots of experience teaching this level of physics. “
That’s nice.
My friend and I were Discussing this article and wondering, assuming it was a sharpend unbroken pencil Would it (given enuf distance) Point down in it’s decent Versus tumble end over end?
Drag has two components – friction drag and pressure drag. For macroscopic object traveling at reasonably notice speeds, pressure drag is much more important. Friction drag is really not an issue with a pencil.
Pressure drag is the difference in air pressure between the front and back of a moving object. For a falling pencil, the air pushes from top and bottom both, but harder from the bottom than the top. So yes, the drag forces do “push” upwards.
As far as gravity, it is a mutual thing. The earth does not exert gravity on the pencil – they have an interaction which causes a gravitational attraction between the two. Since it is easier to accelerate a small object than a large one, we think of the earth pulling on the pencil, but really the force is mutual and they pull on each other with equal strength.
The minimum velocity to perforate skin (note: not eyeball, which would be markedly less) varies depending on the weight of the projectile, but can be as low as 101 meters/sec (331 ft/sec) for a 8.25 grain .177 airgun pellet (“BB”), 75 m/sec (245 ft/sec) for a .22 airgun pellet (16.5 grains), down to 58 m/sec (191 ft/sec) for a .38 caliber roundnose bullet. You can see that the requirement to perforate skin exceeds greatly the 15 meters/second terminal velocity a pencil would achieve.