The falling broom handle

Question

As the reaction to my previous physics puzzle was somewhat mixed, I'll try again. I found this in an online physics book.

You hold a broom as shown in the following picture:

The head of the broom is resting on the ground, and you are holding the handle end a couple of feet off the ground. You are also holding a bunch of keys at the same height as the tip of the broom handle.

You let go of everything at the same time. What happens?
A) The keys and the broom handle hit the ground at the same time.
B) The keys hit the ground first.
C) The broom handle hits the ground first.

Explain your answer.

Imho this should be migrated to the physics site for a definite answer. — Guntram Blohm, Oct 22 '19 at 16:25
As the starting angle of the broom has not been stated, there would appear to be more than one correct answer, or, the answers given are incomplete. — Weather Vane, Oct 22 '19 at 16:33
You accepted a wrong answer. In fact, the problem is ill-defined, and depends on the mass distribution and angle. — Jeffrey, Oct 22 '19 at 17:31
@WeatherVane In the question I say the end is two feet off the ground, and with a normal sized broom of about 5 feet you get an angle just below 25 degrees, very similar to what the picture in the question depicts. Broom handles almost always have a uniform mass distribution, and just like nearly every physics problem it is perfectly reasonable to assume uniform density without having to explicitly state so in the question. The mass of the head of the broom has virtually no effect since it is on the ground already. — Jaap Scherphuis, Oct 22 '19 at 18:22
Yes, the picture does show a broom angle of around 20°, rather less than the critical angle 35.3° given in the link from Conifers. I would say the effect of the broom head, being relatively massive, would be to make it a "hinged" rod and otherwise of no consequence. — Weather Vane, Oct 22 '19 at 18:27
Beyond not specifying the mass distribution and initial angle, it's not clear whether we assume there's a) no friction between the broom end and the ground, in which case it slides freely; b) total friction, in which case the end stays fixed like a hinge; or c) something in between. — xnor, Oct 22 '19 at 22:59
I tend to give this puzzle may not belongs here due to: 1) Insufficient criteria to solve this question by unknown angle of the broom. 2) If the angle could be calculated by the image shown(maybe you are measuring on your screen :D), then this puzzle should belongs to Physic Exchange due to involve specific physic formula/property to solve it. — Conifers, Oct 23 '19 at 01:36
@Conifers For (2), see here for my thoughts. As for (1), that could be remedied by providing additional specifications, such as what the OP did in a comment above — HTM, Oct 23 '19 at 03:04
@PiIsNot3 For(2) compared to your thoughts, this puzzle involved more specific physic property like Angular Momentum or Torque, which you will be learned in high school or university, should not so common...(maybe to think if this puzzle need calculus to solve, then it should belong to the PuzzleSE or MathSE? ) — Conifers, Oct 23 '19 at 04:40
@Conifers I did not intend for this to be a calculus problem. I asked for what happens and the reason why, and the answer I accepted was the first to explain the counterintuitive result. While one could calculate the effects of rotational inertia, the effect of the lateral movement due to the friction with the ground, and throw in friction with the air as well while you're at it, the fact remains that the effect of these is small enough to not change the answer in the situation as described and pictured in the question. The question is intended to be about the big picture, not the details. — Jaap Scherphuis, Oct 23 '19 at 06:38
@JaapScherphuis I agree that this puzzle is the big picture, not to detailed calculated the specific figures, however proven by xnor, neither 3 options are the best correct answer due to angle is undefined. Although you mentioned the off-ground height, but the broom length is unknown. I don't quite agree that broom should exist the common length to let you guaranteed the angle is always less than 47.9 proven by xnor. — Conifers, Oct 23 '19 at 07:21
@Conifers have you ever seen a broom that is $\frac{2\text{ ft}}{\sin 47.9}=2.7\text{ ft}$ or shorter in length? — Jaap Scherphuis, Oct 23 '19 at 07:26
@JaapScherphuis Yes the 2.69ft broom may be unusual, but if the cut-off angle is proven only 10 or 20?... — Conifers, Oct 23 '19 at 07:33
@Conifers But it isn't. That's reality for you. When I posed the question I knew the cut-off angle was somewhere in the 40s or 50s and set it up to be comfortably below that. — Jaap Scherphuis, Oct 23 '19 at 07:45
@JaapScherphuis Yes, but the cut-off angle is so common that could be the implicit premise for this puzzle...?(At least I don't know after the xnor's proof... :( ). Anyway this puzzle is good for the counter-intuitive clarification and fun in reality :) I may personally give xnor a bounty due to let me take a physic class :P — Conifers, Oct 23 '19 at 08:12
This is somewhat similar to a challenge "lie down faster than ball drops (from height of your head at the same time)". — Zizy Archer, Oct 23 '19 at 11:48
@WooShell that is what happens in the linked example posted by Conifers: the ball lands in a cup fixed to the beam. — Weather Vane, Oct 23 '19 at 13:40

xnor · Answer 1 · 2019-10-23T19:27:58.903

If the broom's angle to the ground is less than 47.9°, it will hit the ground first. Otherwise, the key will.

This assumes that the end of the broom on the ground doesn't slide, making it act like a hinge, and also that the broom handle is a uniform rod. The puzzle doesn't specify the properties of the broom, but I think these are a reasonable interpretation for a physical model.

Below are plots of the height, velocity, and acceleration of the top end of the broom and the key for when the broom starts at 47.9°. They hit the ground at the same time. Note that the broom falls slower at first but catches up at the end to tie. If it had started lower, it would win, and if it started higher, it would lose.

Derivation

I got these plots and estimated the cutoff angle 47.9° by numerically solving the following differential equation in the angle $\theta$, where $\ddot{\theta}$ represents its second derivative in time, e.g. the angular acceleration: $$ \ddot{\theta} = - \frac{3 g}{2 L}\cos \theta$$ Here, $g$ is the gravitational acceleration ($9.8 \thinspace m/s^2$) and $L$ is the length of the broom (I used 2 meters). Their values don't really matter, they just change the constant in $ \ddot{\theta} = - c \cos \theta$, which only affects the time scale.

This formula, also derived on this site, comes from considering the torque $\tau$ on the broom around its pivot on the ground via the gravitational force $mg$ acting on its center. This center is $\frac{L}{2} \cos \theta$ horizontal distance away, so $$\tau = \frac{mg L}{2} \cos \theta.$$ Then, we use that torque causes angular acceleration as $\tau = - I \ddot{\theta}$, where the moment of inertia for a uniform rod around it end is $ I = \frac{1}{3} M L^2$. This gives the formula above relating $\ddot{\theta}$ to $\cos \theta$. Since it's more intuitive to think about and plot height rather than angle, and we want to compare to the key, we relate the height $h$ of the top of the broom as $h = L \sin \theta$.

The site then solves for the angle $\theta$ where the initial downward acceleration $\ddot{h}$ matches the key's acceleration $-g$ due to gravity. This gives $\theta = 35.3^{\circ}$. But, this isn't the angle that makes them reach the ground at $h=0$ at the same time. The broom will accelerate faster during the smaller angles it passes through on its journey, so it will arrive first. Rather, we need to determine its whole trajectory $\theta(t)$ to find when it reaches zero.

Simulation

I doubt that such a second-order differential equation with $cos$ has an analytic solution, so I simulated it numerically. I used a time step of $0.2$ milliseconds, which I think is accurate enough. I compared it with a key starting from the corresponding initial height simulated the same way but under constant downward acceleration from gravity.

I searched for the initial angle $\theta(0)$ where the broom and key reach zero on the same simulation step, and found 47.9°. This might be a tiny bit off due to the discreteness of the simulation, but it should be very close.

Pro is here!, And this answer should be accepted! (I still can't figure out somewhere, physic is too hard to me :( ) — Conifers, Oct 23 '19 at 04:56
This puzzle seems easier than the site's(but both are hard for me :(, I can't even write an equation...) due to need consider more about the ball's normal force pressured on the rod, and this puzzle key & broom fall separately. — Conifers, Oct 23 '19 at 05:15
Thanks for the proper answer xnor :). Angular acceleration is beyond my knowledge set. — AndyT, Oct 23 '19 at 13:35

score 18 · Accepted Answer · answered Oct 22 '19 at 10:17

18

My instinct is that:

C. The broom handle hits first

This is based on:

The net effect of gravity acts through the centre of mass. As the centre of mass of the broom handle is half way along the handle, this is the part of the broom handle that will accelerate at 1g, i.e. 9.81m/s².
This means that the tip of the broom handle will accelerate at twice the middle of the broom, i.e. at 2g. Whereas the keys will just accelerate at 1g. Therefore the tip of the broom will hit the ground in half the time that the keys will.

answered Oct 22 '19 at 10:17

AndyT

1,298
9
18

1

You have good instincts, as this is the correct answer. The difference in speed is quite noticeable, and I highly recommend people try it out for real and see for themselves. A small amount of energy is taken up by the rotation of the broom, and if there is friction by the sideways movement of the broom, but at the kind of angle shown in the question those are small enough to make very little difference. – Jaap Scherphuis Oct 22 '19 at 13:54
Chalk one up for the Chartered Engineer getting a physics question right. Phew! – AndyT Oct 22 '19 at 13:57
I have tried it. When the broom is nearly vertical, it hits the ground a long time later. From an angle of about 60° to the horizontal, the broom still lands noticeably later. At lower angles I am unable to determine as they seem to land at the same time. If this answer (broom lands first) is correct, there must be a particular starting angle when they take the same time. – Weather Vane Oct 22 '19 at 15:32
@WeatherVane - I've now done a quick test with a short (15cm / 6in) ruler and a pen. I agree that when the broom/ruler is nearly vertical it lands a lot later. When the angle is below 45 degrees... it's all too fast for me to judge. Hmmmmm! – AndyT Oct 22 '19 at 15:39
2

So do you agree that there must be some starting angle when they both hit the ground at the same time? – Weather Vane Oct 22 '19 at 16:16
2

Maybe is it the answer? http://dev.physicslab.org/Document.aspx?doctype=3&filename=RotaryMotion_HingedBoard.xml – Conifers Oct 22 '19 at 16:44
23

This is incorrect. It depends on the broom angle. It is only true for broom close to the ground (low angle). If the broom was up at a angle of 89 degrees, visibly, the key would hit first. – Jeffrey Oct 22 '19 at 17:28
1

@Conifers page 2 of your linked example says the critical angle is 35.3° and although it says minimum angle I think it should say maximum angle as the broom plainly lands last when nearly vertical. – Weather Vane Oct 22 '19 at 17:40
I don't see how the linked problem demonstrates the angle at which both hit the floor at the same time. Their formula is finding the starting angle at which the initial angle of the end of the rod has the same downward acceleration as the ball (key). But, we want to know which one hits the ground first, which involves the acceleration over their whole paths. Presumably the equalizing angle is greater than 35.3°, so that the ball accelerates faster for the first part and slower for the rest. – xnor Oct 22 '19 at 22:56
@WeatherVane Yes, the linked problem should be maximum, and if the cup could catch ball, means the end of rod will fallen to ground faster then the ball, which seems equal to this puzzle's asking(rod->broom, ball->keys). So the answer is neither all of 3 options and should be the fourth: It depends on the brooms angle, key is faster when angle more than 35.3 and vise versa. – Conifers Oct 23 '19 at 00:57
Typo, not vice versa, should be slower when angle less than 35.3. – Conifers Oct 23 '19 at 01:03
@Conifers See my comment above, the 35.3 degrees isn't the right cutoff for this problem. I'm still not sure what they mean about the cup catching the ball, but the equation they solve is definitely just setting their initial accelerations equal at the starting angle. – xnor Oct 23 '19 at 03:32
@xnor if the cup could catch ball, then means cup falls faster. If not, that means ball should fall faster, however the rod is under the ball, so it might be they will accelerate commonly when angle is above 35.3(ball falls faster, and give the rod a normal force). Then if begin to below 35.3, they might motion seperately(bacause rod now falls faster than ball.) Anyway, I'm not major in Physics, need someone pro to validate :( – Conifers Oct 23 '19 at 04:52
1

@Conifers I just posted an answer, take a look. – xnor Oct 23 '19 at 04:54
1

This answer is incorrect. There are other vertical forces than gravity acting on the broom. Namely, the support force from the floor on the head of the broom. Therefore, the total acceleration of he centre-of-mass is not $g$. – JiK Oct 23 '19 at 04:54

score 9 · Answer 3 · answered Oct 22 '19 at 08:05

9

Reductio ad absurdum

Suppose the broom is very nearly upright (vertical) and the keys are level with its top.
It can be seen that the broom handle has to move sideways.
Even if the acceleration of the broom tip (sideways) were the same as the keys, it has a longer path to travel - nearly $\pi / 2 = 1.57$ times the distance travelled by the keys.
But to make it worse, the broom's acceleration is initially very small.

Now suppose the broom is very nearly flat (horizontal) and the keys are again level with its top.
This time, they will take almost the same time to fall, as both their acceleration, and distance to travel, are almost the same.
But the broom will still take longer. Because of the slight angle, it has to move a teensy bit sideways, and still travels in an arc and not in a straight line.

The only arrangement when they take the same time, will be when they are both already touching the floor! At all other times, the keys will arrive first.

answered Oct 22 '19 at 08:05

Weather Vane

14,420
1
22
54

3

Nice logic. Your reductio ad absurdum is very convincing. I'm not sure that it holds for lower angles though. – AndyT Oct 22 '19 at 10:20
1

Gravity is the only force pulling on the key; the broom gets pulled down by the same gravity, but also experiences a negating force where it touches the floor. So the sum of the forces on the broom is a smaller pull down which causes it to fall slower, meaning it will always reach the floor later. – Guntram Blohm Oct 22 '19 at 16:28
3

You can create another absurd case: the mass fully in the center of the broom. 0 inertia moment. The center of mass has half the distance to travel, the broom rotates as needed, without slowing down the broom. Then, the broom arrives first. Conclusion: the problem is ill-defined and depends on the inertia moment, length and angle – Jeffrey Oct 22 '19 at 17:33
@GuntramBlohm - But the upward force on the broom head will tend to rotate the broom handle tip downwards... – AndyT Oct 23 '19 at 13:34
"their acceleration ... are almost the same" is not true. You can't assume the tip of the broom has acceleration "g" since it's part of a rigid body. The net acceleration of the entire broom is less than g, but its center of mass starts lower. – aschepler Oct 24 '19 at 13:41

score 2 · Answer 4 · answered Oct 22 '19 at 17:16

Assuming the end of the broom doesn't slide along the floor and neglecting air friction, I believe that:

A is correct for an initial angle of the broom at 60 degrees. B is correct for initial angles greater than 60 degrees. C is correct for initial angles less than 60 degrees.

My reasoning was:

When the broom was almost upright it would start to fall very slowly allowing the keys to hit the ground first. When the broom was almost horizontal the end of the broom would hit the ground first because the center of the handle is the center of gravity and will fall at 9.81 (m/s^2). So the end broom will fall twice as fast.

A more technical explanation:

I equated the kinematic equations for the distance to fall and equated the time until impact. Combined the equations and solved for the angle of the broom. The exact angle calculation is too complex for Puzzling, but the question was good and thought provoking.

score 1 · Answer 5 · answered Oct 24 '19 at 01:44

So long as the starting angle is close to that shown, the broom end hits the ground first. The question isn't ill-defined so much as requiring a bit of understanding about real-world broom handle lengths and how being a couple of feet off the ground to start with will make that initial angle much less than 47 degrees.

Reason: The easiest way to think of it is that the centre of mass of the broom (roughly its middle) drops at "close enough to"* the same rate as the keys, but the middle is starting off at half the height of the keys, so the broom has an advantage. The end that started at 2ft off the ground will fall at roughly twice the rate of the middle.

*But against that advantage is the component of the force on the head of the broom (that touches the ground) that opposes the force of gravity to some extent - if the broom was perfectly vertical it would entirely cancel the gravitational force... if the broom were horizontal the force would only be rotating the broom; for angles encountered in this question the acceleration of the middle of broom is close enough to that of the keys for the double-rate advantage to let it win the race.

score -1 · Answer 6 · answered Oct 22 '19 at 20:26

For the moment, consider just the broom handle and the level at which it becomes horizontal: 1 - the tip of the handle and it's midpoint reach there at the same moment: the tip has travelled twice as far but at twice the speed. 2 - the keys must reach that level first.
The only force on the keys is it's weight and it's acceleration is g. On the broom, gravity pulls down, but the ground pushes up on the lower end. So the net force on the handle is less than it's weight and the acceleration of it's CoG must be less than g. The keys beat the CoG and hence the tip. 3 - At that point the keys are travelling faster than the handle CoG and are ahead of it.

For the extra distance required by the thickness of the head, it may be possible for the handle tip to catch up, as it is travelling aout twice as fast as the CoG when it passes horizontal. But if the handle started well above the horizontal, it's CoG is travelling much slower than the keys at this horizontal position, because it's acceleration has always been much less than g. I haven't done the analysis, but I suspect it requires a large broomhead and starting the handle nearly horizontal, if it is to be possible at all.

The falling broom handle

6 Answers6

Linked