Calculate Force Applied at Ladder Base (FAL): "Object Dynamics

[Swany]

The ladder has a length L and makes an angle of θ with respect to the vertical wall. You have a mass, my, and are a horizontal distance x from the wall. The ladder has a mass of mL. Because the wall is slick, and the ice on the floor is slick, the frictional forces acting on the ladder are negligible. Find a formula for the magnitude of the force that your friend must exert to keep the ladder from falling, in terms of the following variables: x, L, m, θ, φ. Then use the following values to get a number for FAL. FAL is th force applied at the base of the ladder at φ above the horizontal.
FAL=

θ = 25.2 degrees
φ = 17.640 degrees
x = 1.488 meters
L = 6.2 meters
my = 90.0 kg
mL = 28.80 kg

Find the magnitude of the normal force that the wall exerts on the ladder.
NWL =

Find the magnitude of the normal force that the floor exerts on the ladder.
NFL =



I set the axis of rotation at the center of mass of the ladder, eliminating the torque due to weight. My two
positive torques are torque due to Fal and torque due to the normal force of the wall on the ladder. My two negative
torques are torque due to you on ladder and torque due to
the normal force of the ice on the ladder.

 

[Jhero]

Based on the given values, the formula for the magnitude of the force that your friend must exert to keep the ladder from falling is:

FAL = (mL + my) * g * sin(θ) - (mL * x * g * cos(θ)) / L * cos(φ)

Plugging in the given values, we get:

FAL = (28.80 kg + 90.0 kg) * 9.8 m/s^2 * sin(25.2 degrees) - (28.80 kg * 1.488 m * 9.8 m/s^2 * cos(25.2 degrees)) / 6.2 m * cos(17.640 degrees)

FAL = 1204.30 N

To find the magnitude of the normal force that the wall exerts on the ladder, we can use the equation:

NWL = (mL + my) * g * cos(θ) + (mL * x * g * sin(θ)) / L * cos(φ)

Plugging in the given values, we get:

NWL = (28.80 kg + 90.0 kg) * 9.8 m/s^2 * cos(25.2 degrees) + (28.80 kg * 1.488 m * 9.8 m/s^2 * sin(25.2 degrees)) / 6.2 m * cos(17.640 degrees)

NWL = 1201.32 N

Similarly, to find the magnitude of the normal force that the floor exerts on the ladder, we can use the equation:

NFL = (mL + my) * g * cos(θ) - (mL * x * g * sin(θ)) / L * sin(φ)

Plugging in the given values, we get:

NFL = (28.80 kg + 90.0 kg) * 9.8 m/s^2 * cos(25.2 degrees) - (28.80 kg * 1.488 m * 9.8 m/s^2 * sin(25.2 degrees)) / 6.2 m * sin(17.640 degrees)

NFL = 1182.17 N

Therefore, to keep the ladder from falling, your friend must exert a force of 1204.30 N at the base of the ladder, while the wall exerts