# Ladder sliding down with no friction

1. Nov 6, 2016

### Dustgil

1. The problem statement, all variables and given/known data
A uniform ladder leans against a smooth vertical wall. If the floor is also smooth, and the initial angle between the floor and the ladder is theta(0), show that the ladder, in sliding down, will lose contact with the wall when the angle between the floor and the ladder is

$$sin^{-1}(\frac {2} {3} sin \theta_{0})$$

2. Relevant equations

$$\frac {dL} {dt}=N$$

where L is the angular momentum and N is the net external torque

3. The attempt at a solution

I understand why the ladder loses contact with the wall at some point. There is a normal force on the ladder exerted by the wall that is not balanced by anything. So as the ladder falls, it sorta gains length in the x direction while at the same time being pushed away from the wall. Eventually there comes a point where the normal force exerted by the wall is zero and then negative, which will be at the angle I'm looking for.

Let's say that the angle that the ladder makes with the floor is theta. The translational motion equations are

$$m \ddot y=F_{n} - mg$$
$$m \ddot x=F_{w}$$

I start to get a bit confused when trying to define the rotational motion. The first question is where's the best place to define the rotation at? For now I choose the point where the ladder makes contact with the ground. This is where I run into my second problem. For whatever reason I'm having trouble confidently expressing the torques exerted from both gravity and the normal force from the wall. I'm just kinda getting mixed up by it. I make a guess:

$$\frac {dL} {dt}= N$$
$$I \ddot {\theta} = mgcos(\theta)(\frac {l} {2}) - F_{w}sin(\theta)(\frac {l} {2})$$

Are these the correct equations? It took me a long while to get to this point so It'd be cool to be confident before I soldier on.

2. Nov 6, 2016

### kuruman

Try the corner of the wall and look at the center of mass. First understand the path it follows if both ends of the ladder are constrained to be always in contact, then figure out what must be true for the CM to deviate from this path.

3. Nov 6, 2016

### Dustgil

So the CM moves in a circle. Won't the CM stop accelerating in the x direction when it loses contact?

4. Nov 6, 2016

### Dustgil

That makes the translation motion

$$m \ddot {r} = r \dot { \theta }$$

but i'm still unsure as to how to define the rotational motion. maybe I need to brush up on my trig..

5. Nov 6, 2016

### kuruman

Correct. What force provides the necessary centripetal acceleration? When that force goes to zero, the CM leaves the circle. Hint: This is essentially the same problem as the small block that slides down on the outside of a hemispherical frictionless surface.

6. Nov 6, 2016

### Dustgil

For the hint it would be the contact force the block makes with the surface, so the normal force on the wall, right?

7. Nov 6, 2016

### kuruman

And the floor.

8. Nov 6, 2016

### haruspex

Not sure that is going to work... Not obvious to me, anyway. The particle on the sphere has no moment of inertia.
The force "providing centripetal acceleration" is not one applied force, it is the radial component of the net force. Even though the normal force from the wall becomes zero, there is still gravity and the normal force from the floor. And this net component does not go to zero, it just becomes insufficient to maintain circular motion.

9. Nov 6, 2016

### kuruman

I am not sure what "insufficient" means. The condition for the CM to go around in a circle is $$F_r -mg\cos \theta = - \frac{mv^2}{r}$$ where $F_r$ is the radial component of the contact force resultant. From this $$F_r = m\left(g\cos \theta - \frac{v^2}{r} \right)$$As long as $F_r$ is positive, circular motion of the CM is maintained. What is the criterion for "insufficient" other than $F_r = 0$?

10. Nov 6, 2016

### haruspex

You wrote that the force necessary for centripetal acceleration goes to zero. That is not Fr; it is mg cos(θ)-Fr.

11. Nov 6, 2016

### kuruman

Yes, thank you for the correction. I made the mistake of trying to do it in my head without writing anything down - I should know better.

12. Nov 7, 2016

### ehild

What are x and y? Are they the coordinates of the CoM? how are the related to the angle theta?
Yes, you can solve the problem by writing the equation of motion of the center of mass, and also the equation of rotation about the CoM.

One other approach can be using conservation of energy. The kinetic energy is the sum of the KE of the CoM, and the rotational energy about the CoM. As you have found, the CoM moves around a circle. That gives you a relation between θ and $\dotθ$.
At the end, when the ladder loses contact with the wall, $\ddot x$=0. You can write it also in terms of theta and derivatives.