Optimal Ladder Position for Painting: Friction and Wall Conditions Explained

[Myrddin]

Ladder Problem (urgent)

A light of negligible mass and length d is supported on a rough floor and leans against a smooth vertical wall makeing an angle (theta) with the floor. The coefficent of friction between the ladder and floor is M.

If a painter climbs the ladder up distance x, what value of x is when the ladder begins to slip? And how far can he climb is the floor is smooth and the wall is rough?

Attempt at solution:

Have a force diagram with mass of painter Mg downwards, with anti clockwise moment xMgcos(theta) taking base of the ladder as a pivot.

clockwise moment from the top off ladder reaction from the wall R2.
= R2cos(theta)d (i think)

R2 = Mmg (i think)

So both moments ---> Mmgcos(theta)d = mgcos(theta)x

so from this x = dM which doesn't seem right..

 

[SammyS]

A light ladder of negligible mass and length d is supported on a rough floor and leans against a smooth vertical wall making an angle θ (theta) with the floor. The coefficient of friction between the ladder and floor is M. (I'd rather use μ .)

If a painter climbs a distance x up the ladder, at what value of x does the ladder begin to slip? And how far can he climb is the floor is smooth and the wall is rough?

Attempt at solution:

Have a force diagram with mass of painter Mg downwards, with anti clockwise moment x·m·g·cos(θ) taking base of the ladder as a pivot.

clockwise moment from the top off ladder reaction from the wall R₂.
= R₂·cos(θ)·d (i think)

This should be R₂·sin(θ)·d .

R₂ = μ·m·g (i think)

So both moments ---> μ·m·g·cos(theta)d = m·g·cos(theta)x

so from this x = d·μ which doesn't seem right.

R₂ looks OK.

Just fix the anti-clockwise moment.

 

[Myrddin]

Anti clock wise moment should equal, dR2sin(theta)? so ---> dMmgsin(theta)
this gives anticlock wise = clockwise

So dMmgsin(theta) = xmgcos(theta)
so x = dMmgsin(theta) / mgcos(theta) = dMtan (theta)?

 

[SammyS]

x = d·μ·tan(θ) looks good to me.

x is proportional to d and μ. x increases as θ increases.

 

[rwisz]

I would first like to commend the urgency in finding the optimal ladder position for painting. It is important to consider the friction and wall conditions in order to ensure the safety and stability of the ladder while painting.

From the given information, we can see that there are two forces acting on the ladder - the weight of the painter (Mg) and the reaction force from the wall (R2). The coefficient of friction (M) between the ladder and the floor also plays a significant role in determining the optimal ladder position.

To find the value of x when the ladder begins to slip, we can use the principle of moments. The clockwise moment (R2cos(theta)d) must be equal to the anti-clockwise moment (xMgcos(theta)) in order for the ladder to be in equilibrium. Therefore, we can write the equation as follows:

R2cos(theta)d = xMgcos(theta)

Substituting R2 = Mmg (from the given information), we get:

Mmgcos(theta)d = xMgcos(theta)

Canceling out the common terms, we get:

d = x

Therefore, the optimal ladder position for painting would be when the distance x is equal to the length of the ladder (d). This means that the ladder should be placed at a 90-degree angle with the floor and the wall.

However, if the floor is smooth and the wall is rough, the coefficient of friction (M) would be different. In this case, we can use the same principle of moments to find the maximum height that the painter can climb before the ladder begins to slip. The only difference would be that the friction force (F) would be acting in the opposite direction of the motion of the ladder.

Therefore, the equation would be:

R2cos(theta)d - Fx = xMgcos(theta)

Solving for x, we get:

x = d(M + F/Mcos(theta))

This means that the maximum height the painter can climb would be less than d, as the friction force would limit the motion of the ladder. It is important to consider the coefficient of friction and the roughness of the wall in order to determine the safe and optimal ladder position for painting.