A ladder 13 feet long leans against a vertical frictionless wall with its lower end 5 feet from the wall. the ladder weighs 80 lbs and its center of gravity is at its center. Find the magnitude and direction of the force exerted on the lower end of the ladder.
To find the magnitude and direction of the force exerted on the lower end of the ladder, we can calculate the weight of the ladder and use trigonometry to determine the force components.
First, let's find the weight of the ladder. Given that the ladder weighs 80 lbs, the weight force (W) acting on the ladder is equal to the mass (m) of the ladder multiplied by the acceleration due to gravity (g). Assuming the acceleration due to gravity is 32 ft/s², we can calculate the weight as follows:
W = m * g
W = 80 lbs * 32 ft/s²
W = 2560 lb·ft/s²
Next, we need to break down the weight force into its components: one parallel to the wall, and one perpendicular to the wall.
The perpendicular component of the weight force, which acts at a 90-degree angle to the wall, does not contribute to the force exerted on the lower end of the ladder. The parallel component does contribute and is responsible for the force exerted.
Using trigonometry, we can determine the angle between the ladder and the ground. Let's call this angle θ.
From the given information, we can see that the ladder forms a right triangle with the wall and the ground. The length of the ladder forms the hypotenuse, and the distance from the wall to the lower end of the ladder is one of the legs. Therefore, we can use the trigonometric function tangent:
tan(θ) = opposite/adjacent
tan(θ) = 13 ft/5 ft
Solving for θ:
θ = arctan(13 ft/5 ft) ≈ 68.41 degrees
Now, we can determine the force exerted on the lower end of the ladder. Since the ladder is at rest, the net force acting on it must be zero.
The force exerted on the lower end of the ladder can be resolved into two components: the vertical component and the horizontal component. The vertical component must balance with the perpendicular component of the weight force, while the horizontal component must balance with the parallel component of the weight force.
The vertical component of the force exerted on the lower end of the ladder is equal in magnitude and opposite in direction to the perpendicular component of the weight. Therefore, it is:
F_vertical = 2560 lb·ft/s² * sin(θ)
The horizontal component of the force exerted on the lower end of the ladder is equal in magnitude and opposite in direction to the parallel component of the weight. Therefore, it is:
F_horizontal = 2560 lb·ft/s² * cos(θ)
Substituting the value of θ:
F_vertical = 2560 lb·ft/s² * sin(68.41 degrees)
F_horizontal = 2560 lb·ft/s² * cos(68.41 degrees)
Calculating these values will give us the specific magnitude and direction of the forces exerted on the lower end of the ladder.