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 85.0 kg, length = 5.80 m, the ladder’s mass to be 37.0 kg, è = 51.0°, and the coefficient of friction between ground and ladder be 0.470. (c) If the ladder is uniform, what is the lever arm for the force of gravity acting on the ladder? (do not include the man)

To find the lever arm for the force of gravity acting on the ladder, we need to first understand what the lever arm is in this context.

The lever arm is the perpendicular distance between the line of action of the force and the axis of rotation. In this case, the ladder is acting as a lever with the axis of rotation being the point where the ladder touches the ground.

To calculate the lever arm, we can use the following formula:
Lever Arm = Length of the ladder * sin(è)

Given:
Length of the ladder (L) = 5.80 m
Angle between the ladder and the ground (è) = 51.0°

Now, let's calculate the lever arm.

Lever Arm = 5.80 m * sin(51.0°)
Lever Arm = 5.80 m * 0.7771
Lever Arm ≈ 4.50 m

Therefore, the lever arm for the force of gravity acting on the ladder (excluding the man) is approximately 4.50 meters.