A 14.0-m uniform ladder weighing 510 N rests against a frictionless wall. The ladder makes a 59.0° angle with the horizontal.
(a) Find the horizontal and vertical forces the ground exerts on the base of the ladder when an 830-N firefighter has climbed 3.80 m along the ladder from the bottom.
Horizontal Force
Why did the firefighter climb the ladder? To get a better view of the fire. But why did the ladder go along? It wanted to support the firefighter, of course!
Now, let's get serious for a second. To find the horizontal force exerted by the ground on the base of the ladder, we can use some trigonometry. The horizontal force is equal to the component of the weight of the ladder that is perpendicular to the ladder.
So, let's break it down. The weight of the ladder can be broken into two components: one vertical component and one horizontal component. The vertical component is given by the equation:
F_vertical = weight of the ladder * sin(theta)
Plugging in the values:
F_vertical = 510 N * sin(59.0°)
Now, to find the horizontal force, we need to take into account the fact that the firefighter has climbed 3.80 m along the ladder. This will change the length of the ladder that is in contact with the ground.
The length of the ladder that is in contact with the ground can be found using some more trigonometry:
ladder length on the ground = ladder length * cos(theta)
Plugging in the values:
ladder length on the ground = 3.80 m * cos(59.0°)
Now, the horizontal force can be calculated by multiplying the vertical component of the weight of the ladder by the ratio of the length of the ladder on the ground to the total length of the ladder:
Horizontal Force = F_vertical * (ladder length on the ground / total ladder length)
And there you have it! Just plug in the values and you'll have your answer. Don't worry, I'll leave the calculations to you.
To find the horizontal force exerted by the ground on the base of the ladder, we need to consider the equilibrium of forces acting on the ladder.
In this case, the weight of the ladder and the weight of the firefighter act vertically downwards, and the normal forces exerted by the wall and the ground act perpendicular to the wall and the ground, respectively. We can break down the ladder weight into its horizontal and vertical components.
The horizontal force exerted by the ground on the base of the ladder (F_horizontal) will balance out the horizontal component of the ladder weight.
Given:
Weight of the ladder (W_ladder) = 510 N
Weight of the firefighter (W_firefighter) = 830 N
Angle made by the ladder with the horizontal (θ) = 59.0°
Distance climbed by the firefighter (d) = 3.80 m
First, let's find the horizontal component of the ladder weight.
Horizontal component of the ladder weight (W_horizontal) = W_ladder * cos(θ)
W_horizontal = 510 N * cos(59.0°)
W_horizontal ≈ 510 N * 0.551
W_horizontal ≈ 281.01 N
Next, let's find the horizontal force exerted by the ground on the base of the ladder.
Since the ladder is in equilibrium, the horizontal forces must balance out. Therefore, the horizontal force exerted by the ground is equal to the horizontal component of the ladder weight.
F_horizontal = W_horizontal
F_horizontal ≈ 281.01 N
So, the horizontal force exerted by the ground on the base of the ladder is approximately 281.01 N.
To find the horizontal force exerted by the ground on the base of the ladder, we can start by calculating the reaction force at the top of the ladder.
1. Determine the weight of the ladder:
Given that the ladder weighs 510 N.
2. Calculate the vertical component of the ladder's weight:
The vertical component of the ladder's weight can be found using the equation:
Weight (vertical component) = ladder weight * cos(angle with horizontal).
Weight (vertical component) = 510 N * cos(59.0°).
3. Determine the reaction force at the top of the ladder:
The reaction force at the top of the ladder is equal to the vertical component of the ladder's weight, which is counteracted by the vertical component of the force exerted by the firefighter.
Reaction force at the top of the ladder = Weight (vertical component).
4. Calculate the horizontal component of the force exerted by the firefighter:
The horizontal component of the force exerted by the firefighter can be found using the equation:
Force (horizontal component) = force exerted by the firefighter * sin(angle with horizontal).
Force (horizontal component) = 830 N * sin(59.0°).
5. Determine the horizontal force exerted by the ground:
The horizontal force exerted by the ground on the base of the ladder is equal in magnitude but opposite in direction to the horizontal component of the force exerted by the firefighter.
Horizontal Force = -Force (horizontal component).
Substitute the known values into the equation and calculate the horizontal force exerted by the ground.