a fire fighter who weighs 800n climbs a uniform ladder and stops one third of the way up the ladder. the ladder is 5.0m long and weighs 180n. it rest against a vertical nail, making an angle of 53degree with the ground. find the normal and frictional forces on the ladder at its base
Fn=980N
Ff=267.5N
Solution for Ff please?!!!
To find the normal and frictional forces on the ladder at its base, we need to analyze the equilibrium conditions.
Let's start by drawing a free-body diagram of the ladder at the base:
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In this diagram:
- The weight of the firefighter is acting downward (800 N).
- The weight of the ladder is acting downward (180 N).
- The normal force is acting perpendicular to the ladder at its base (upward).
- The frictional force is acting parallel to the ground (opposite to the direction of motion).
First, let's calculate the force exerted by the firefighter on the ladder. Since the firefighter is one-third of the way up the ladder, the portion of their weight that acts on the ladder can be calculated as 1/3 * 800 N = 267 N. This forms an angle θ with the ladder, which we'll calculate next.
To find the angle θ, we'll use the given angle of 53 degrees and the ladder length. In a right-angled triangle, the adjacent side is parallel to the ground, the opposite side is the ladder, and the hypotenuse is the line formed by the ladder and the ground. Using trigonometry:
cos θ = adjacent / hypotenuse
cos 53° = 5.0 m / hypotenuse
Solving for the hypotenuse:
hypotenuse = 5.0 m / cos 53° ≈ 9.63 m
Now that we have the hypotenuse, we can calculate the angle θ using the cosine inverse (arccosine) function:
θ = arccos (5.0 m / 9.63 m) ≈ 58.3°
Next, let's resolve the forces vertically and horizontally. Since the ladder is in equilibrium, the vertical forces must balance each other, and the horizontal forces must also balance.
Vertically:
Sum of vertical forces = Normal force - Weight of the ladder - Weight of the firefighter = 0
Horizontally:
Sum of horizontal forces = Frictional force = 0
Now we can calculate the normal and frictional forces:
1. Normal force:
Since the sum of the vertical forces is zero, the normal force can be calculated as the sum of the ladder's weight and the firefighter's weight:
Normal force = Weight of the ladder + Weight of the firefighter
Normal force = 180 N + 800 N = 980 N
2. Frictional force:
Since the sum of the horizontal forces is zero, the frictional force is zero (no force opposing the motion in the horizontal direction).
Therefore, the normal force acting on the ladder at its base is 980 N, and the frictional force is 0 N.
da32
f1=11410.09 N
f2=12192.75 N