21
$\begingroup$

Suppose you fall from the top of a ladder straight down. You will hit the ground with an amount of force.

Now suppose that you fall over while holding onto the ladder, tipping over in an arc instead of falling straight down. You will hit the ground with another amount of force.

Neglecting the mass of the ladder and air resistance, which impact will have the most force? Falling or tipping?

This has been a bit of a debate between myself, my father, and my grandfather. I believe that they would fall with the same force because, relative to the ground, you start with the same amount of potential energy in both situations. My grandfather and father, however, guess that it would fall with around half the force because the force is dissipated somewhat by the forward motion and the support of the ladder. We would appreciate an answer from a third party to help us find the solution.

$\endgroup$
7
  • 3
    $\begingroup$ Hi user1583259, and welcome to Physics Stack Exchange! No worries about the tags, I fixed them up for you. $\endgroup$
    – David Z
    Aug 8, 2012 at 1:14
  • $\begingroup$ as you see here you've already got two opposite answers. So don't judge only by the words "yes/no", you should follow the calculations and more importantly the assumptions yourself. Are you capable of that? $\endgroup$
    – Yrogirg
    Aug 9, 2012 at 7:17
  • $\begingroup$ I think so. Both answers make sense in their own circumstances, but one of them factors in the mass of the ladder, something we were explicitly leaving out. $\endgroup$
    – Kenkron
    Aug 12, 2012 at 19:36
  • $\begingroup$ Both of the answers include maths, but only one, John Rennie's, handles it correctly. $\endgroup$
    – Thriveth
    Apr 30, 2014 at 21:23
  • $\begingroup$ I have unchecked the answer, out of respect for your confidence, but I am still unconvinced. John Rennie does not account for rotation. I'll have to think about whether that's okay... $\endgroup$
    – Kenkron
    May 2, 2014 at 16:23

1 Answer 1

24
$\begingroup$

Both you and your ancestors are wrong. But I bet you would never guess the real answer!

Assuming the base of the ladder doesn't slide you have a rotating system. Just like a freely falling man you convert potential energy to kinetic energy, but for a rotating system the kinetic energy is given by:

$$ T = \frac{1}{2}I\omega^2 $$

where $I$ is the angular momentum and $\omega$ is the angular velocity. Note that $v = r\omega$, where $r$ is the radius (i.e. the length of the ladder). We'll need this shortly.

Let's start by ignoring the mass of the ladder. In that case the moment of inertia of the system is just due to the man and assuming we treat the man as a point mass $I = ml^2$, where $m$ is the mass of the man and $l$ the length of the ladder. Setting the change in potential energy $mgl$ equal to the kinetic energy we get:

$$ mgl = \frac{1}{2}I\omega^2 = \frac{1}{2}ml^2\omega^2 = \frac{1}{2}mv^2 $$

where we get the last step by noting that $l\omega = v$. So:

$$ v^2 = 2gl $$

This is exactly the same result as we get for the man falling straight down, so you hit the ground with the same speed whether you fall straight down or whether you hold onto the ladder.

But now let's include the mass of the ladder, $m_L$. This adds to the potential energy because the centre of gravity of the ladder falls by $0.5l$, so:

$$ V = mgl + \frac{1}{2}m_Lgl $$

Now lets work out the kinetic energy. Since the man and ladder are rotating at the same angular velocity we get:

$$ T = \frac{1}{2}I\omega^2 + \frac{1}{2}I_L\omega^2 $$

For a rod of mass $m$ and length $l$ the moment of inertia is:

$$ I_L = \frac{1}{3}m_Ll^2 $$

So let's set the potential and kinetic energy equal, as as before we'll substitute for $I$ and $I_L$ and set $\omega = v/l$. We get:

$$ mgl + \frac{1}{2}m_Lgl = \frac{1}{2}mv^2 + \frac{1}{6}m_Lv^2$$

and rearranging this gives:

$$ v^2 = 2gl \frac{m + \frac{1}{2}m_L}{m + \frac{1}{3}m_L} $$

and if $m_L \gt 0$ the top of the fraction is greater than the bottom i.e. the velocity is greater than $2gl$. If you hold onto the ladder you actually hit the ground faster than if you let go!

This seems counterintuitive, but it's because left to itself the ladder would rotate faster than the combined system of you and the ladder. In effect the ladder is accelerating you as you and the ladder fall. That's why the final velocity is higher.

$\endgroup$
7
  • 1
    $\begingroup$ Thank you; I believe that you're right that the mass of the ladder will add to the impact, but the question said "Neglecting the mass of the ladder and air resistance", and I think what Prahar said about rotation is correct. $\endgroup$
    – Kenkron
    Aug 12, 2012 at 19:53
  • 1
    $\begingroup$ In fact, if you leave out the mass of the ladder, John's (correct) calculations show that both cases will just reduce to the simple free/fall case: $v^2 = 2gl$. So even so, Prahar's answer is wrong. $\endgroup$
    – Thriveth
    Apr 30, 2014 at 21:18
  • $\begingroup$ So a ladder with its mass evenly distributed falls faster than one with the mass concentrated at any end: I'm surprised at this. $\endgroup$ May 5, 2014 at 22:35
  • 2
    $\begingroup$ @LarryHarson: it's one of the classic problems you give students because their initial guess at the answer is always wrong :-) $\endgroup$ May 6, 2014 at 6:51
  • 1
    $\begingroup$ Yep, our professor at Classical Mechanics gave us the same problem (not a ladder, some other rod-shaped thingie). The gist of it is that it is the center-of-mass that falls with free-fall velocity, and so if the thing is rotating and the center-of-mass is not at the very end, the end will fall faster thhan this. $\endgroup$
    – Thriveth
    May 11, 2014 at 19:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.