A painter climbs a ladder leaning against a smooth wall. At a certain height, the ladder is on the verge of slipping. Let the mass of the painter be 69.0 kg, length = 2.80 m, the ladder’s mass to be 21.0 kg, θ = 57.0°, and the coefficient of friction between ground and ladder be 0.290. (a) What is the lever arm for the horizontal force of the wall on the ladder with the axis of rotation taken at the base of the ladder? (The lever arm is the perpedicular distance between the base of the ladder and the force exerted by the wall.)
L•sinθ
To determine the lever arm for the horizontal force of the wall on the ladder, we need to consider the geometry of the ladder and the angle θ.
The lever arm is the perpendicular distance between the base of the ladder and the force exerted by the wall. In this case, the force exerted by the wall is the frictional force between the ladder and the ground.
To calculate the lever arm, we first need to determine the point of contact between the ladder and the ground. At this point, the ladder is on the verge of slipping, which means the frictional force is at its maximum value and is equal to the product of the coefficient of friction (μ) and the normal force.
The normal force is the force exerted by the ground perpendicular to the surface. In this case, the normal force is equal to the weight of the painter and the ladder, which can be calculated using the formula:
normal force = (mass of painter + mass of ladder) × gravitational acceleration
Given:
mass of painter (m1) = 69.0 kg
mass of ladder (m2) = 21.0 kg
length of the ladder (L) = 2.80 m
angle (θ) = 57.0°
coefficient of friction (μ) = 0.290
Gravitational acceleration is approximately 9.8 m/s^2.
Step 1: Calculate the normal force:
normal force = (69.0 kg + 21.0 kg) × 9.8 m/s^2
Step 2: Calculate the frictional force:
frictional force = coefficient of friction × normal force
Step 3: Find the point of contact on the ladder:
Using trigonometry, we can determine the vertical distance between the base of the ladder and the point of contact. This can be calculated as:
vertical distance = L × sin(θ)
Step 4: Calculate the lever arm:
The lever arm for the horizontal force of the wall on the ladder is the horizontal distance between the base of the ladder and the point of contact. This can be calculated as:
lever arm = L × cos(θ) - vertical distance
Substituting the values into the equations, we can find the lever arm.