A 15.0 m, 500 N uniform ladder rests against a frictionless wall, making an angle of 60 0 with the horizontal, as shown in Figure Q3(a). Draw the free body diagram and calculate the normal force FN exerted by the wall, horizontal FBI and vertical Fv forces exerted on the base of the ladder when an 800 N fire fighter is x I l m from the bottom
http://www.cpp.edu/~skboddeker/131/131hw/ch12h.htm
To calculate the normal force (FN), horizontal force (FBI), and vertical force (FV), we need to start by drawing a free body diagram.
The free body diagram would consist of the ladder and the forces acting on it. The forces acting on the ladder include:
- The weight of the ladder acting downward (W)
- The normal force exerted by the wall, perpendicular to the wall (FN)
- The horizontal force exerted by the wall, parallel to the ground (FBI)
- The vertical force exerted on the base of the ladder (FV)
- The applied force by the firefighter (F)
Since the ladder is in equilibrium, the sum of all the forces in the vertical direction and the horizontal direction will be equal to zero.
Now, let's calculate the values step-by-step:
Step 1: Calculate the weight of the ladder (W):
Given that the weight is 500 N, it can be calculated using the formula:
W = mass × gravity
Assuming the acceleration due to gravity is 9.8 m/s^2:
W = 500 N
Step 2: Calculate the angle between the ladder and the horizontal:
Given that the angle is 60 degrees, we can convert it to radians:
θ = 60° × (π/180°)
Step 3: Calculate the applied force by the firefighter (F):
Given that the firefighter exerts an 800 N force, it can be denoted as:
F = 800 N
Step 4: Determine the forces acting on the ladder:
Using the equilibrium condition, we can write the following equations:
ΣFy = FV - FN - Wsin(θ) = 0
ΣFx = FBI - Wcos(θ) = 0
Step 5: Calculate the normal force (FN):
Using the equation ΣFy = FV - FN - Wsin(θ) = 0, we can rearrange the equation to solve for FN:
FN = FV - Wsin(θ)
Step 6: Calculate the horizontal force (FBI):
Using the equation ΣFx = FBI - Wcos(θ) = 0, we can rearrange the equation to solve for FBI:
FBI = Wcos(θ)
Now, plug in the given values and perform the calculations:
- Weight of the ladder (W) = 500 N
- Angle between the ladder and the horizontal (θ) = 60° × (π/180°)
- Applied force by the firefighter (F) = 800 N
- Normal force (FN) = FV - Wsin(θ)
= 0 - 500 N × sin(60° × (π/180°))
- Horizontal force (FBI) = Wcos(θ)
= 500 N × cos(60° × (π/180°))
Performing the calculations, we get:
- Normal force (FN) = 500 N × sin(60° × (π/180°))
≈ 433.01 N
- Horizontal force (FBI) = 500 N × cos(60° × (π/180°))
≈ 250 N
Therefore, the normal force (FN) exerted by the wall is approximately 433.01 N, the horizontal force (FBI) is approximately 250 N, and the vertical force (FV) is zero (since it cancels out with the weight of the ladder in equilibrium).
To solve this problem, we can start by drawing the free body diagram of the ladder and analyzing the forces acting on it.
1. Draw a vertical line to represent the ladder, and label its length as 15.0 m.
---------------------------
| |
| |
| |
| Ladder |
| (angle of 60 degrees) |
| |
| |
| |
| |
--------------------------
| |
| Wall |
---------------------------
| |
| Ground |
---------------------------
2. Identify the forces acting on the ladder:
- The weight of the ladder (500 N) acts downward, represented by a downward arrow labeled "W."
- The normal force exerted by the wall (FN) acts perpendicular to the wall, represented by an arrow.
- The horizontal force exerted by the ground (FBI) acts parallel to the ground, represented by an arrow.
- The vertical force exerted by the ground (Fv) acts perpendicular to the ground, represented by an arrow.
3. The ladder is in equilibrium, meaning the net force and net torque acting on it are both zero.
Now, let's calculate the values of FN, FBI, and Fv when the firefighter is x meters from the bottom.
To find these values, we can use the following equilibrium equations:
Net Force in the x-direction:
ΣFx = 0
FBI = 0 since there is no horizontal force acting on the ladder (assuming no external forces).
Net Force in the y-direction:
ΣFy = 0
FN + Fv - W = 0
FN + Fv = W
Net Torque about any point:
Στ = 0
τFBI + τFN + τFv - τW = 0
τFN - τW = 0
FN * d - W * (15.0 - x) = 0
FN * d = W * (15.0 - x)
Using trigonometry, we can find the value of d, which is the horizontal distance from the bottom of the ladder to the point where the firefighter is standing:
d = cos(60°) * x = 0.5 * x
Now, substitute d into the equation FN * d = W * (15.0 - x) to solve for FN:
FN * (0.5 * x) = 500 N * (15.0 - x)
Simplifying the equation:
0.5 * x * FN = 500 N * (15.0 - x)
0.5 * FN = 500 N * (15.0 - x) / x
Solve for FN:
FN = 1000 N * (15.0 - x) / x
Finally, substitute the given x value (the position of the firefighter) to find the numerical value of FN at that location.