# Ladder problem - is there a mistake here ?

• Fermat
In summary, the problem involves a ladder propped against a wall and constrained to slide down such that the speeds of its ends against the wall and floor are constant. The question asks for the angular velocity of the ladder about its center when the angle against the wall is 15 degrees. However, based on the given information, this is impossible and the angle must be 45 degrees for the ladder to have only one orientation where the speeds of the ends are the same. This was later confirmed to be a typo and the correct angle is indeed 45 degrees.
Fermat
Homework Helper
I have a ladder which is propped up against a wall. The ladder slides down the wall and is constrained to move such that the speeeds of the ends of the ladder against the wall (and floor) are a constant value. i.e. $$\dot x = \dot y = constant$$

The problem question is: What is the angular velocity of the ladder about its centre when the angle, of the ladder against the wall, is @=15 degrees ?

BUT, if the ladder length is L, then x² + y² = L² and differentiating this expression wrt time gives,$$2x\dot x + 2y\dot y = 2L \dot L = 0$$
Therefore, $$x\cdot\dot x = -y\cdot\dot y$$
However, if we are given $$\dot x =\dot y$$ (numerically), then this means that x = y!
But if x = y, then the ladder must be at an angle of 45 degrees, yes?
So, this means that the ladder has only one orientation, @ = 45 degrees, such that the speeds of the ends of the ladder are the same.
Is this right? Or have I made a mistake in my earlier working ?

Fermat said:
I have a ladder which is propped up against a wall. The ladder slides down the wall and is constrained to move such that the speeeds of the ends of the ladder against the wall (and floor) are a constant value. i.e. $$\dot x = \dot y = constant$$

The problem question is: What is the angular velocity of the ladder about its centre when the angle, of the ladder against the wall, is @=15 degrees ?

BUT, if the ladder length is L, then x² + y² = L² and differentiating this expression wrt time gives,$$2x\dot x + 2y\dot y = 2L \dot L = 0$$
Therefore, $$x\cdot\dot x = -y\cdot\dot y$$
However, if we are given $$\dot x =\dot y$$ (numerically), then this means that x = y!
But if x = y, then the ladder must be at an angle of 45 degrees, yes?
So, this means that the ladder has only one orientation, @ = 45 degrees, such that the speeds of the ends of the ladder are the same.
Is this right? Or have I made a mistake in my earlier working ?
Your conclusion is correct. The ladder could slide with one end at constant speed, but not both. I don't know how you could do the problem anyway with just the information given, unless you are supposed to find the answer in terms of the constant velocity.

Fermat said:
I have a ladder which is propped up against a wall. The ladder slides down the wall and is constrained to move such that the speeeds of the ends of the ladder against the wall (and floor) are a constant value. i.e. $$\dot x = \dot y = constant$$
Perhaps you could give us the exact wording of the question. If the speed of the end against the wall is constant, the speed of the ends against the floor cannot be constant let alone the same speed.

As a function of angle $\theta$ of the ladder to the floor:

$$\dot y = \frac{d}{dt}L\sin\theta = -L\cos\theta\dot\theta = K$$

means that:

$$\dot\theta = -K/L\cos\theta$$

$$\dot x = \frac{d}{dt}L\cos\theta = -L\sin\theta\dot\theta = L\sin\theta(K/L\cos\theta) = K\tan\theta$$

So if the speed of the end down the wall is constant, the speed along the floor is proportional to $\tan\theta$

AM

Last edited:
The other point was:if both ends had different tangential velocities, which they would have if both ends had the same speed ( a given of the problem) but a non-45 degree inclination, then could they have an angular velocity about the ladders centre? I would have said they would only have an angular velocity about some point, the distance of which from and end would be proportional to the relative end velocites? Is that right ?

Here is the wording of the question as it was given to me:

What is the angular velocity of a 12-ft ladder about its center as it begins to slide off a perpendicular wall at an angle of 15 degrees to the wall, with each end having a relative speed of 8in/sec to their respective surfaces?

I wish I could stay for some answers, but got to go to work :(

Fermat said:
Here is the wording of the question as it was given to me:

What is the angular velocity of a 12-ft ladder about its center as it begins to slide off a perpendicular wall at an angle of 15 degrees to the wall, with each end having a relative speed of 8in/sec to their respective surfaces?
The short answer is that if the wall is perpendicular to the floor and the other end of the ladder is on the floor, the question poses an impossible set of facts.

AM

Andrew Mason said:
The short answer is that if the wall is perpendicular to the floor and the other end of the ladder is on the floor, the question poses an impossible set of facts.

AM
Many thanks.

Update

I told the questioner that I thought his question was wrong and that the angle should be 45 degrees, and worked out an answer based on that assumption.
He insisted that it should be 15 degrees. But he's just come back and accepted my original answer. So looks like he checked up and found out it was a typo after all!

## 1. Is the problem statement missing any crucial information?

No, the problem statement contains all of the necessary information to solve the ladder problem.

## 2. Are the measurements given in the problem accurate?

It is impossible to determine the accuracy of the measurements without additional information. However, for the purposes of the problem, it is safe to assume that the measurements are accurate.

## 3. Is there a mistake in the calculations or equations used to solve the problem?

Without knowing the specific calculations or equations used, it is difficult to determine if there is a mistake. However, it is always important to double check your work and ensure that you are using the correct formulas and units.

## 4. Can the problem be solved using different methods?

Yes, there are likely multiple ways to solve the ladder problem, including using geometry, trigonometry, or basic algebra. It is up to the individual to choose the method they are most comfortable with.

## 5. Are there any assumptions made in solving the problem?

Yes, in order to solve the problem, certain assumptions are made, such as the ladder being rigid and not bending, the wall being straight and sturdy, and the ground being level. These assumptions may not always hold true in real life scenarios, but are necessary for the problem to be solved mathematically.

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