# Solving Static Equilibrium: 15.5 m Ladder & 500 N Weight

• xdevinx
In summary, a uniform ladder weighing 500 N and resting against a frictionless wall at a 56.0° angle with the horizontal has an 810 N firefighter standing 4.10 m from the bottom. To find the horizontal and vertical forces exerted by the ground on the base of the ladder, the sum of forces in the x and y direction must be equal to 0. In part (b), if the firefighter is 9.20 m up, the ladder is just on the verge of slipping and the coefficient of static friction between the ladder and ground can be calculated.

## Homework Statement

A 15.5 m uniform ladder weighing 500 N rests against a frictionless wall. The ladder makes a 56.0° angle with the horizontal.

(a) Find the horizontal and vertical forces the ground exerts on the base of the ladder when an 810 N firefighter is 4.10 m from the bottom.

Magnitude of the horizontal forceMagnitude of the vertical force(b) If the ladder is just on the verge of slipping when the firefighter is 9.20 m up, what is the coefficient of static friction between ladder and ground?

## Homework Equations

I have no idea how to do this /:
All I know is that the sum of the forces in the x and y direction has to equal 0.
Other than that...

## The Attempt at a Solution

xdevinx said:
All I know is that the sum of the forces in the x and y direction has to equal 0.
Good--that describes translational equilibrium. What's the condition for rotational equilibrium?

I would approach this problem by first identifying the key variables and equations that are relevant to the situation. In this case, the key variables are the length and weight of the ladder, the angle it makes with the horizontal, the weight of the firefighter, and the distance from the bottom of the ladder to the firefighter. The key equations that can be used to solve this problem are the equations for static equilibrium, which state that the sum of the forces in the x and y direction must equal 0.

To solve for the horizontal and vertical forces exerted by the ground on the base of the ladder, we can use the following equations:

ΣFx = 0, which means that the horizontal forces must balance out to 0

ΣFy = 0, which means that the vertical forces must balance out to 0

We can then set up equations using trigonometry to solve for the unknown forces. For example, to solve for the horizontal force, we can use the equation:

cos(56.0°) = Fx/810 N

This will give us the magnitude of the horizontal force exerted by the ground on the base of the ladder. Similarly, we can use the equation:

sin(56.0°) = Fy/810 N

to solve for the magnitude of the vertical force.

To solve for the coefficient of static friction, we can use the equation:

μs = (Ff/Fn)

where Ff is the maximum force of friction and Fn is the normal force exerted by the ground on the base of the ladder. We can use the same approach as before to solve for these forces, using the equations for static equilibrium.

In conclusion, as a scientist, I would approach this problem by first identifying the key variables and equations, and then using mathematical calculations and principles of static equilibrium to solve for the unknown forces and coefficients.

## 1. How do you determine the forces acting on the ladder and weight in static equilibrium?

To determine the forces acting on the system, we must first draw a free body diagram and identify all the external forces acting on the ladder and weight. These forces include the weight of the ladder and weight, the normal force from the ground, and any other applied forces such as friction or tension.

## 2. What is the equation for solving static equilibrium?

The equation for solving static equilibrium is ΣF = 0, where ΣF represents the sum of all the external forces acting on the system and must equal zero for the system to be in equilibrium. This equation is based on Newton's First Law of Motion, which states that an object will remain at rest or in motion at a constant velocity unless acted upon by an external force.

## 3. How do you find the unknown forces in a static equilibrium problem?

To find the unknown forces, we must set up and solve a system of equations using the ΣF = 0 equation. This involves setting up equations for the horizontal and vertical components of the forces and solving for the unknown forces using algebraic methods.

## 4. What is the role of torque in solving static equilibrium problems?

Torque is a measure of the rotational force acting on an object. In static equilibrium problems, torque must also be balanced for the system to be in equilibrium. This means that the sum of the clockwise torques must equal the sum of the counterclockwise torques. Torque can be calculated using the equation τ = r x F, where r is the distance from the pivot point to the line of action of the force, and F is the magnitude of the force.

## 5. How do you know if a system is in static equilibrium?

A system is in static equilibrium if the sum of all the external forces and torques acting on the system is equal to zero. This means that the system is not accelerating or rotating, and all forces and torques are balanced. If any of the forces or torques are not balanced, the system is not in equilibrium and will either accelerate or rotate.

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