A 16.0-m uniform ladder weighing 490 N rests against a frictionless wall. The ladder makes a 63.0° angle with the horizontal.
(a) Find the horizontal and vertical forces the ground exerts on the base of the ladder when an 850-N firefighter has climbed 3.80 m along the ladder from the bottom.
Give the magnitude and direction of the vertical and horizontal components
1356
83 degrees
To find the horizontal and vertical forces exerted by the ground on the base of the ladder, you can use Newton's Second Law of Motion and consider the forces acting on the ladder in equilibrium.
First, let's calculate the weight of the ladder. The weight of an object is given by the formula:
Weight = mass * acceleration due to gravity
Given that the weight of the ladder is 490 N, we can use this formula to find the mass:
Mass = Weight / acceleration due to gravity
= 490 N / 9.8 m/s^2
≈ 50 kg
Next, let's consider the forces acting on the ladder. There are two forces acting on the ladder: the weight of the ladder acting downwards and the forces exerted by the ground acting upwards.
1. Vertical Forces:
The vertical forces acting on the ladder are the weight of the ladder (mg) acting downwards and the vertical component of the force exerted by the ground (Fv) acting upwards.
Taking the upward direction as positive, we can write the equation for vertical forces:
Fv - mg = 0
Substituting the values we have:
Fv - (50 kg * 9.8 m/s^2) = 0
Solving for Fv:
Fv = 50 kg * 9.8 m/s^2
≈ 490 N
So, the magnitude of the vertical force exerted by the ground is 490 N. Since the ladder is in equilibrium, the vertical force is equal to the weight of the ladder.
2. Horizontal Forces:
The horizontal forces acting on the ladder are the horizontal component of the force exerted by the ground (Fh) and the force applied by the firefighter (Fff) in the horizontal direction.
Since there is no horizontal acceleration, these forces must also be in equilibrium. Thus, we can write:
Fh = Fff
To find Fff, we need to consider the forces acting on the firefighter. The horizontal force applied by the firefighter is responsible for the horizontal component of the force on the ladder. The vertical force applied by the firefighter contributes to balancing the weight of the ladder.
Now, let's calculate the horizontal force applied by the firefighter (Fff):
Fff = Force_applied_by_firefighter * cos(angle)
Substituting the given values:
Fff = 850 N * cos(63°)
Solving for Fff:
Fff ≈ 850 N * 0.447
≈ 380.95 N
So, the magnitude of the horizontal force exerted by the firefighter is approximately 380.95 N, and since there is no other horizontal force acting on the ladder, the magnitude of the horizontal force exerted by the ground is also 380.95 N in the opposite direction.
To summarize:
Magnitude and direction of the vertical force exerted by the ground: 490 N (upwards)
Magnitude and direction of the horizontal force exerted by the ground: 380.95 N (opposite to the direction of the firefighter)