# How do I find the forces acting on a ladder?

• Eclair_de_XII
In summary: If they fix that, then they will get the same answer as you and I did for Fr. However, if they fix that, then they will get 134N, which is slightly different from your 133N. I suspect that the difference is due to round-off error in the book's calculation of the angle ##\theta=cos^{-1}(1/3)##.Ok, I'll have to look into that. Thanks for your help and for pointing that out!
Eclair_de_XII

## Homework Statement

"To get up on the roof, a person (mass 70.0 kg) places a 6.00-m aluminum ladder (mass 10.0 kg) against the house on a concrete pad with the base of the ladder 2.00 m from the house. The ladder rests against a plastic rain gutter, which we can assume to be friction-less. The center of mass of the ladder is 2.00 m from the bottom. The person is standing 3.00 m from the bottom. Find the normal reaction and friction forces on the ladder at its base."

## Homework Equations

##\theta=cos^{-1}(\frac{1}{3})=tan^{-1}(2\sqrt{2})##
##m_0=10kg##
##s_0=\frac{2 m}{tan\theta}##
##m_1=70kg##
##s_1=\frac{3 m}{tan\theta}##
##||s||=6m##

## The Attempt at a Solution

I'm choosing my lever arm to be the point at which the ladder touches the ground.

##∑|τ|=-g(m_0s_0+m_1s_1)+||s||cos\theta⋅F_r=0##
##∑F_x=-gcos\theta(m_0+m_1)-F_r+f_s=0##

##F_r=f_s-gcos\theta(m_0+m_1)##
##F_r⋅||s||⋅cos\theta=g(m_0s_0+m_1s_1)##
##F_r=\frac{g}{||s||}sec\theta(m_0s_0+m_1s_1)##
##f_s-gcos\theta(m_0+m_1)=\frac{g}{||s||}sec\theta(m_0s_0+m_1s_1)##
##f_s=gcos\theta(m_0+m_1)+\frac{g}{||s||}sec\theta(m_0s_0+m_1s_1)##

Can anyone tell me if I'm going in the right direction? Thanks.

Last edited:
You have the correct approach. But I noticed a few things:

(1) It is always helpful to define the symbols that you are using. For example, I had to try to decipher your equations to decide what angle you are calling ##\theta##. Likewise for the symbols ##s_1##, ##s##, and ##s_0##.

(2) The problem states that "the center of mass of the ladder is 2.00 m from the bottom". I think this probably means that the center of mass is 2.00 m along the ladder starting from the bottom of the ladder. If they meant it the way you interpreted it, I think they would have said that the center of mass is 2.00 m above the ground. Similarly for the position of the person.

(2) You wrote the equation ##∑|τ|=-g(m_0s_0+m_1s_1)+||s||cos\theta⋅F_r=0##. Check to make sure that the expression for the lever arm for ##F_r## is correct.

On more thing, it looks like you have assumed that the forces of gravity have a horizontal component. Is this right?

TSny said:
Check to make sure that the expression for the lever arm for ##F_r## is correct.

Oh, it's ##||s||sin\theta⋅F_r##, right? ##F_r## acts horizontally, so the lever arm is the vertical component of ##||s||##.

TSny said:
Is this right?

Forgive me. I think I should treat the ladder as a point mass for the summation of forces expression.

##∑F_x=f_s-F_r=0##
##∑F_y=-(m_0+m_1)g+F_N=0##

Eclair_de_XII said:
Oh, it's ##||s||sin\theta⋅F_r##, right? ##F_r## acts horizontally, so the lever arm is the vertical component of ##||s||##.

##∑F_x=f_s-F_r=0##
##∑F_y=-(m_0+m_1)g+F_N=0##
That all looks good.

##\theta=cos^{-1}(\frac{1}{3})##
##m_0=70 kg##
##s_0=3 m##
##m_1=10kg##
##s_1=32 m##

##∑|τ|=-gcos\theta(m_0s_0+m_1s_1)+F_r(||s||sin\theta)=0##
##F_r=(\frac{g}{||s||})(cot\theta)(m_0s_0+m_1s_1)=(\frac{9.8\frac{m}{s^2}}{6m})(\frac{1}{2\sqrt{2}})((70kg)(3m)+(10kg)(2m))=133 N##
##∑F_x=f_s-F_r=0##
##f_s=F_r=133N##

##∑F_y=F_N-g(m_0+m_1)=0##
##F_N=g(m_0+m_1)=784N##

I tried to do a quick check and it looks like your results for Fr and FN are correct.

Is that right? I probably need a new physics problem book because it says that ##F_N=784N## but ##f_s=376N##.

I just repeated the sum of the moment equation and once again calculated 132.8 N for Fr, which is equal to -fs, since those are the only 2 horizontal forces. So I think that is the correct answer unless you and I are both doing something fundamentally wrong.

It looks like the book incorrectly used (Fr)(||s||)(cosθ), instead of (Fr)(||s||)(sinθ) when calculating the moment from force Fr.

## 1. What are the main forces acting on a ladder?

The main forces acting on a ladder are gravitational force, normal force, and frictional force.

## 2. How does the weight of the ladder affect the forces?

The weight of the ladder contributes to the gravitational force acting on the ladder. The weight of the ladder is also balanced by the normal force from the surface it is resting on.

## 3. How does the angle of the ladder affect the forces?

The angle of the ladder affects the distribution of forces on the ladder. The steeper the angle, the more the weight of the ladder is shifted to the base, increasing the normal force. This also decreases the frictional force, making it easier to slide the ladder.

## 4. What other factors can affect the forces acting on a ladder?

Other factors that can affect the forces acting on a ladder include the material and surface of the ladder, the weight of the person on the ladder, and any external forces such as wind or objects pushing against the ladder.

## 5. How can I calculate the forces acting on a ladder?

To calculate the forces acting on a ladder, you will need to know the weight of the ladder, the angle at which it is leaning, and the coefficient of friction between the ladder and the surface it is resting on. With this information, you can use physics equations to determine the magnitude and direction of the forces acting on the ladder.

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