Will a falling pencil hit the table? We do the maths! 

Would be nice to know why downward acceleration due to gravity is not a factor in that calculation.

"The bluetack doesn't affect the physics." *Pencil fails to move because of bluetack*

12:25 is the most perfect hand drawn Sinecurve I've ever seen.

Matt: Does it change the physics? Hugh: No. Pencil: Doesn't move

Whoever photoshopped and animated the scanned notes did a great job! It was much easier to follow along line by line rather than looking at a completely filled out paper full of formulas and drawings. And I didn't even notice that other lines/shapes would overlap what was already written until I saw the animation, rewound, and scrutinized the "past" version of the paper. Beautiful!

"why are all the pencils broken inside???"

I am deeeply disappointed for the pen length being 'a' and not 'b'.. "The pen is 2B long" would've been a perfect pun..

What a brilliant way of showing the working out on paper very clean! Must have took some editing skills, Brady would be proud lol

4:00 just some ducks drinking in the background

sin θ = 1/3 θ ⇒ θ = 0 :) Cool how the initial state is technically a situation in which one end of the pencil is "hitting" the edge of the table.

Finally another episode of "Matt and Hugh play with a thing and do some working out" I'm still waiting for a followup on the 3 sided coin video, though.

Matt's sheer elation at 1:39 is my favorite reaction to anything ever right now.

It never ceases to astonish me that you can scribble on a piece of paper some numbers and come up with a fairly accurate forecast of what happens in reality.

Matt Parker, answering the questions we never thought to ask.

3:22 "there's a center of mass somewhere in the very middle of the pencil" Clearly not as the topside is sharpened and given a sharpening angle of 75 degrees on a typical 18 cm long, 6mm thick pencil would result in the top half being ~13% lighter than the bottom half. As such the center of mass on such a pencil no longer coincides with the 9cm midpoint and is actually only ~7.8cm from the surface of the table.

I love these videos with Hugh. Hugh actually reminds me of Arthur Weasley (Ron's Dad) of HP.

That is the most beautiful hand-drawn sine graph I have ever seen

I cannot get enough of you two together! More Matt and Hugh please!

Loved the flock of drinking birds 😂

Sure, I can try. I have exactly 72 pencils lying around from a project I am yet to do :)

What amazes me is how simple of a reasoning you used to solve this. I was thinking parametric curves would be the way to go but this is much more efficient. Well done!

Somewhat counterintuitive result with a surprisingly intuitive explanation!

I just finished taking calculus-based physics last semester. It's interesting seeing a video on some topics we covered in class!

I love how the table has three legs, it is clearly made for that job (or similar).

"You wanna see a magic trick? I'm gonna make this pencil disappear."

The close enough graphic at the end is the best part

this one video has more information then one whole chapter of mechanics. thank you for this video.

I was actually in the mood for a standupmaths video today and I was rewarded!

Matt "Close Enough" Parker

Got humble pi for my b-day loving it!

I definitely prefer the animated scans. It conveys all the necessary information without an odd camera angle and unnecessary background details.

Really appreciated that moment of inertia moment.

his hand drawn sine wave is amazing

Really interesting topic. Keep it up Matt.

May I also say: God I love nerds so much! You guys are so fun. Thanks for making these videos and sharing your brilliance with all of us. =)

I find it awesome that there is a group of people on earth that find it fun to watch people do math. Its a good sign for the future of the earth.

1:48 Beautiful tower clock in the background! With gravity escapement! wow... a real jewel.

I would've clicked on this video 40 minutes ago had I know it was an episode of Matt and hugh play with a thing then do some working out , but alas that was absent from the title yet again

I really enjoyed the video, I never knew I needed to know the angle I need for my pencil to hit the edge of my table :D

Oh man, I do love a video where I can watch at Matt's face in 4K in slow motion!

It's always good to see 2 grown men playing with their pencils.

I prefer video like this one, when you can clearly see the solving process and not only the results, as in a lot of popular blogs about math.

Excellent - gotta love how things cancel out and that implies certain physical predictions. Maths is great!

The next step, now that we have a value for theta would be to calculate y = -a cos theta = 1/2 g t^2 and substitute in all the previous equations and end up with an formula for the impulse which would result in the pencil exactly hitting the corner of the table. And then go one step further and change it to an inequality: -a cos theta > 1/2 g t^2, and end up with what range of impulses will result in a table hit, and what range will have it miss entirely. Left as an exercise to the reader, naturally.

Timely. My physics class at school is testing momentum this week. Linear and angular versions. We look up moments of inertia in tables too.

What a wholesome way of explaining maths

Interesting that the experiment would work equally well on the moon or Jupiter. In space the pencil would just spin, but the tip would align with the edge of the table at the same angle. As to why 1/3ma^2 This is the inertia around the centre of mass since the formula is 1/12m m(2a)^2=1/3 ma^2

For people complaining about gravity not being factored in, it's in the assumption that the pencil just barely touches the table. Working from that, you can find that it then brings in the length and mass of the pencil in order to find out what impulse is required to make that situation happen. Having done that, the impulse required is: I=m*sqrt(ag(1+cos(θ))/2)

Awesome the derivation is just awesome sir

I love how Matt asked Hugh at the end to just hit as many pencils off the table as he could

Animated scans! More of that! 👍🏼

4:00 that flock of drinking birdies in the background lmao

Thats beautiful!!

Neat. Would like to see a follow up showing how the able changes as a function of gravitational acceleration.

Well, you're technically correct in what you did, but I was still a bit disappointed, since you didn't really answer the question that the title implies, which is "how hard you would have to hit it?". Instead you answered the question with "depends" and then asked "what would the angle be if it just barely hit?". I would also like to add that while I do understand how linear momentum and angular momentum are linked, you failed to mention that the given ratio heavily depends on WHERE you hit the pencil or how the mass is DISTRIBUTED in the pencil, which might lead to wrong conclusions for some viewers.

The moment of inertia is the variance of the mass from its center

I think the moment of inertia is 4/3ma^2 because the total length of the pencil is 2a at 5:17

I love how this guy is planting words so that we can think about something right before he says it just so we can feel smart too.

Add blue tac “Does it change the physics?” “No” Well strictly yes, ok I’m being pedantic.

And this was all just with the impulse parralel with the table. Id like to see different models with the impulse atracking the pencil from different directions.

Hugh Hunt: Excuse me while I do calculus by site. God I have forgot so much since I got out of school.

A very nice experiment! However it may leave the viewer with some questions: ▪Is the correct formula used for moment of inertia for a rod rotating about the centre of mass (which is I=ma²/3), or should we use that for rotation about an end point (which is I=4ma²/3)? [Note that a is the *half* length of the pencil] ▪What is the role of gravity? ▪What rotation speed do we need so that the tip does not touch the table? Let's do the physics a bit more rigorously, avoiding the formula issue altogether. We assume - the pencil to be an ideal rod (stiff, uniform mass distribution, no thickness) - it is hit at its very bottom point - no air resistance - gravity does play a role After it is hit, the motion of the pencil will move in such a way that 1. the center of mass (com) will move uniformly in the x-direction 2. the com is accelerated downward due to gravity 3. the pencil will uniformly rotate in the xy-plane When talking about angular momentum, we have to be very clear and careful about our reference point. In the video the com is just taken as a reference point, the motivation is not made very clear. Below I we will consider angular momentum with respect to the origin, which will be at the point of contact (which is at the table edge, where the pencil is kicked at its bottom). The positive x-axis is horizontal (away from the table), the positive y axis vertical, upward. Positive rotation will be anticlockwise. Angular momentum ( *L* ) of any particle at a location *r* and and a momentum *p* is defined as the vector product *r* × *p* . For an object we have to sum this quantity over its constituent part(icle)s. In our reference frame: The pencil at rest has *p* = *o* for all of its parts, so it must have *L* = *o* . The blow or impulse (= transfer of momentum) has *r* = *o* , so it must also have *L* = *o* . So, because of conservation of angular momentum, we know that the combined motion of the pencil must also have *L* = *o* . Immediately after impact: The com will be moving horizontally to the right. This gives rise to angular momentum of L=r×p=-amv (clockwise, so negative sign!). The rotation of the pencil itself will be anticlockwise with respect to its centre of mass, which constitutes an angular momentum of L=ωI. So we must have amv=ωI. We can also integrate total angular momentum for the rod. Using v=p/m-(y-a)ω, dm=ρdy, ρ=m/2a, and dp=vdm we write 0 = L = ∫ r×dp = ∫ r×vdm = ρ ∫ r×(p/m-(y-a)ω)dy Directly upon impact motion is all horizontal and r=y is vertical, so that the vector products line up. 2a = ρ ∫ y(p/m-(y-a)ω)dy = ρ (½(2a)²(v+aω) - (2a)³ω/3) = 0 which simplifies to v = aω/3. 0 Now aω is the rotation velocity of the tip about the com. This way we find the motion of the tip of the pencil: x(t) = vt - a sin(ωt) y(t) = a + a cos(ωt) - ½gt² Setting v = aω/3 and θ=ωt, indeed the x-coordinate of the tip of the pencil is 0 when θ/3 = sin(θ) or θ =2.2789. No matter how hard we hit the pencil, as long as the tip has y>=0 (otherwise it would have hit the table beforehand). Yes, we arrive at the same result. Back to Matts question @2:18: 'If you hit it so it just, just nips the end, what angle is the pencil on?', but later he says 'So we dont care how hard we hit it.' We should care, because it is fun. It just touches the table edge when y=0, too. What is the condition required for this? Set y(t) = a (1+ cos(θ)) - ½gt² = 0. We get t = √( 2a (1+ cos(θ))/g), and ω=θ/t As a realistic estimate: for a 0.15 m long pencil, t=73ms, and it needs to make at least 4.97 rps.

A yes. I remember this stuff. Didn't get anything to do with angular velocity, momentum and so on until University level stuff though. I actually once ran into a university version of the Physics textbook I used in High School. It was pretty much completely identical but the High School level one was missing about 1/3 of the pages and many entire chapters...

i personally think the sticky stuff makes the whole test inconclusive because when you strike the pencil you arent just overcoming gravity. You are overcoming the sticky stuff first. However weak or strong it is. Also you are having way too much fun. This reminds me of the 'magic' trick where you yank a table cloth out from under a set of plates and glasses fast enough and they dont come flying off the table

I like the constantly drinking ducks in the background. I left mine for months and they got horribly mouldy...!

here they are only considering the horizontal component of the motion, since their applied force is along horizontal direction, so it is not necessary to include the vertical gravitional force and the aceleration due to it that is gravitation aceleration.horizontal and vertical compnent of motion under gravitional field can be delt separately without any flaw in the mathematical results.

Think it’s worth noting why it only works in radians. Because it seems like an arbitrary restriction. Angular velocity, omega, can be thought of as the speed at which an angle is covered. This is (2 x pi)/T where T is the time period, the time taken to complete a full rotation of 2 x pi radians. By using this formula they have implicitly restricted the equation to only work in radians. It is convention that an angular velocity can only be denoted by omega, if it is in radians per second. However if we used a different angular velocity which we defined as 360/time period, then our later step involving theta would only work in degrees and not radians. So it isn’t that the formulas only work in radians, it’s that you must be consistent with degrees or radians throughout the method, and the method they used, implicitly took angular velocity in terms of radians not degrees.

At 2:27 Professional Mathematicians with a Professional Pencil Sharpener.

1:44 so excited about flying pencils☺️

By calculating for equal you get the angle for when it hits by a point, if you use an inequality like

Excellent !!!

Had Hugh Hunt as our lecturer, this was a question on the example paper

Now do keep in mind velocity is a vector here so while it may not be a matter of impulse once you assume the 1 case where it hits the table perfectly, it's still very much a matter of angle. If you shoot the pencil under an angle alpha from the centerline of the table with the same impulse, the pencil will still go the same velocity obviously because we know for sure the mass is a constant (unless you break the pencil ;) ). However, the useful parallel component to this centerline that's effectively moving away from the table will be shorter. This means you'll have to introduce a higher impulse to fire at a higher velocity so the pencil can still barely hit the table. That will increase Omega, so I'd expect that changes the angle Theta at which the pencil ends up hitting the table perfectly. When we consider an experiment in the 0

I’m curious to know if there’s a second valid solution where the pencil is hit just very slightly, so the bottom is just nudged off the edge of the table. The pencil would drop further and have an angle less than 90° when the tip clips the edge

Now do it for a pencil that does not have an even mass distribution and show the derivation of the moment of inertia.

Confusing to follow the explanation, because "momentum" in German is "Impuls" and "impulse" is "Moment"

Matt has the best face for slowmo

Masterpiece!

Fun follow up question: What is the minimum velocity that you would need to hit the pencil so that it does not hit the table?

Started with the extremes..so hard it does not hit the table, then so soft it doesn't leave the table

I like that the video length is the Golden Ratio

I liked the animated scan better than the GoPro

The pencil (his varicenter) describes a parabolic "falling" outside de table that depend of the initial impulse => horizontal force applyed to de pencil. That initial force is obstructed by the scotch tape at the base, because some of the force applyed get lose while breaking that friction and initiate movement. If you graph that parabolich equiation, dependant of the initial velocity/ force applyed, you can measure the minimun distance between the parabola and the edge of the table. Then, you can measure the time that toke thenpencil to get in that position, and calculating de amount of spinning in that period of time, check if the pencil hit the table or not.

Horrific pencil torture. A fallen pencil is a dead one.

I'm not @home but I'll try this when I get back. I have some broomsticks to try this with...

Would also depend on how high it is struck and how "stuck" to the table it is? Both of those would affect the angular velocity. Could you hit the pencil at the perfect height for there to be no angular movement?

Matt, I’d love to see a video on the math/physics of pencil fighting! I used to love it as a kid, but I was terrible at it. Also, seems like you might have a lot of useless pencils to work with now, so... 😊😊😊

What sort of geometrical shape, or distribution of mass, would allow: 1) the bottom of the shape to hit the table, (almost 360 degree rotation) or (different shape) 2) whacking the top causes the object to strike the table?

depends where you hit it relative to it's center of percussion.

Hey, I have a maths question. If you have a bottle (or cylinder for easier calculations), how much water to put in it to make it the most stable (so it returns to upright position from the largest angle)?

I'm also thinking that at some point, where the impulse is applied will come into play. For these calculations, it was assumed that the impulse was applied at distance 'a' from the center of mass (a-n where n=0), but as 'n' increases, and the impulse is applied closer to the center of mass, isn't the ratio of the linear and rotational momenta going to change? (With the extreme case of n = 2a, and you're whacking the top of the pencil instead of the bottom) If not, how am I thinking about this wrong?

When I saw the title, somehow I expected that you would end up calculating a probability of hitting the table, resp. determining ranges of angles and impulses (or whatsoever) for which the pencil does or does not hit the table... :-P

anyone else get entranced by the turkey/bird drinking bowl in the background

The center of maths😁

Now everyone has an excuse for flicking pencils across the classroom.

Finally, a reason to use radians in the real world! (I'm talking about myself here, I don't ever need radians in my normal life.)

Just paused the video to buy a cranked pencil sharpener after a flashback to school art lessons.

It has to fall, but it does not depend on the gravity! The same angle would take it on the moon! This is awesome!! Congrats

I'm pretty sure the blue track makes a huge difference

also you could do it just on a piece of paper and mark where you stand up the pencil then see where exactly the impact is