How high up on a ladder can he go without the ladder slipping?

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In summary, the ladder is 5m leaning on a wall, and it is 2.5 m away from the bottom of the wall, so the angle between the ground and the ladder is 60 degrees. The contact interaction between the ladder and the ground has a static friction force of no more than .4 mg, and mg is the weight of the person on the ladder. How high can he climb w/o the ladder slipping?
  • #1
rachie9
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The ladder is 5m leaning on a wall, and it is 2.5 m away from the bottom of the wall, so the angle between the ground and the ladder is 60 degrees. The contact interaction between the ladder and the ground has a static friction force of no more than .4 mg, and mg is the weight of the person on the ladder. How high can he climb w/o the ladder slipping?



The relevant equations are acceleration = 0 and torque = 0 because the ladder isn't moving.



I know that the static friction and normal force from the wall on the ladder must equal 0, and that the normal force from the ground plus the force of gravity must equal 0. Since the torque is 0, I think the total force x length of ladder x cos 60 must equal 0, but I'm not sure how to break up the individual forces and fit them into this equation.
 
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  • #2
The most important point to realize here is that the net torque on the ladder must be zero, as you said.

To start, try taking the torques about the upper end of the ladder.
 
  • #3
rachie9 said:
I think the total force x length of ladder x cos 60 must equal 0

Note: This statement is definitely incorrect -- it doesn't say net torque is zero. In fact, I don't think it is mathematically identified with anything physical about the problem.

rachie9 said:
but I'm not sure how to break up the individual forces and fit them into this equation.

Heres an outline of what you need to do in ANY statics problems (even when something like a distance is unknown, and variables have to be carried along):

1) Find all the bits of force and locate them along the length of the object with an orientation. some of these are related to gravitational force, some of them are related to other forces (like you say: friction, normal forces).
i.e. DRAW A FREE BODY DIAGRAM of the ladder!

2) Make lists of your forces in some coordinate system that has perpendicular axes: You can list either of:
a) Upwards forces vs. downwards forces
b) Forces parallel to the ladder versus forces perpendicular to the ladder
Note: you may have to break forces into components and use these components in your chosen listing.

You will use the above information to make a net force equals zero equation.

3) Chose a pivot point for referencing torques an identify the forces that cause torques when the system is oriented from the pivot.
edited to add: Hootenanny gives a wise choice for this pivot... why is the choice smart?

4) Find the components of those forces that are parallel to and perpendicular to the ladder.

5) Find all the moments of inertia/torques using ONLY the components of forces perpendicular to the length of the ladder -- using your CHOSEN pivot.
(I make separate lists of clockwise torques and counterclockwise torques).

You will use the step 3-5 information to make a net torque equals zero equation.

Can you start to fill in these steps?
 
  • #4
Thanks for your help.

I'm not sure how many components are in the net torque equation. I know there is a torque from the force of gravity, is there one for all the normal forces (acting on the ladder from the person, from the wall, and from the ground) as well? What about from friction?

Once I have my net torque = 0, how do I relate this to my net force = 0 equation?
 

1. How is the weight of the person on the ladder related to the ladder's stability?

The weight of the person on the ladder is directly related to the ladder's stability. The heavier the person is, the more likely the ladder is to slip. This is because the weight creates more downward force, causing the ladder to push against the ground with greater force, potentially causing it to slip.

2. What is the maximum weight that a ladder can hold before it becomes unsafe to use?

The maximum weight that a ladder can hold depends on the type and quality of the ladder. Before using a ladder, it is important to check the manufacturer's weight limit and make sure the ladder is in good condition. Additionally, the weight of the person on the ladder should also be taken into consideration.

3. How do the angle and placement of the ladder affect its stability?

The angle and placement of the ladder play a significant role in its stability. The ladder should be placed at a 75-degree angle from the ground and should be securely placed on a flat surface. If the angle is too steep or the ladder is on an uneven surface, it is more likely to slip.

4. Is it safe to use a ladder on a windy day?

It is not recommended to use a ladder on a windy day. Wind can cause instability and make it more likely for the ladder to slip. If you must use a ladder on a windy day, make sure to take extra precautions and have someone hold the ladder steady.

5. Are there any additional safety measures that should be taken when using a ladder?

Yes, there are additional safety measures that should be taken when using a ladder. These include making sure the ladder is in good condition, avoiding overreaching or standing on the top rung, and having someone hold the ladder steady if necessary. It is also important to follow proper ladder safety techniques, such as facing the ladder while climbing and maintaining three points of contact at all times.

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