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. (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.)
height on wall =5.8 sin 51
that is the lever arm for the horizontal force at the top
To find the lever arm for the horizontal force of the wall on the ladder, we need to understand the forces acting on the ladder and how they contribute to the equilibrium.
In this scenario, we have two main forces acting on the ladder: the weight of the painter (gravitational force) and the force exerted by the wall on the ladder.
Let's break down the forces acting on the ladder:
1. Gravitational Force on the Painter:
The gravitational force acting on the painter can be calculated as the product of the mass (m) and the acceleration due to gravity (g).
Gravitational Force on the painter (Fg) = m * g
In this case, the mass of the painter is given as 85.0 kg. The acceleration due to gravity is approximately 9.8 m/s^2.
Thus, Fg = 85.0 kg * 9.8 m/s^2 = 833 N
2. Weight of the Ladder:
The weight of the ladder acts at its center of mass. The weight (W) can be calculated in the same way as the gravitational force on the painter, using the mass of the ladder.
Weight of the Ladder (W) = m * g
The mass of the ladder is given as 37.0 kg.
Thus, W = 37.0 kg * 9.8 m/s^2 = 363.6 N
Now, we can determine the force exerted by the wall on the ladder. In equilibrium, the sum of all the vertical forces acting on the ladder must be zero. Therefore, the vertical component of the force exerted by the wall on the ladder must balance the weight of the painter and the ladder.
Vertical Component of the Wall Force = Fg + W
Since the wall is smooth and there is no vertical acceleration, the vertical component of the force exerted by the wall on the ladder is equal to the weight of the ladder and the painter combined.
Now, let's move on to calculating the horizontal force of the wall on the ladder.
Horizontal Component of the Wall Force = Frictional Force
The frictional force (Ff) can be calculated using the equation:
Ff = coefficient of friction * Normal force
The normal force (Fn) is the force exerted by the ground on the ladder perpendicular to the point of contact. In this case, it is equal to the weight of the ladder and the painter.
Fn = Fg + W
Now, we can calculate the frictional force:
Ff = 0.470 * (Fg + W)
Finally, to find the lever arm, we consider the torque equation:
Torque = Force * Lever Arm
Since the ladder is on the verge of slipping, the frictional force provides the torque to balance the torque due to the weight of the ladder and the painter. Therefore, we equate the torques:
Torque due to Weight = Torque due to Frictional Force
To find the lever arm, we rearrange the equation as follows:
Lever Arm = (Torque due to Weight) / Frictional Force
The torque due to weight is the product of the weight and its lever arm, while the frictional force is the horizontal force.
Lever Arm = (Weight * Lever Arm due to Weight) / Frictional Force
By substituting the values for the weight, lever arm due to weight, and frictional force, we can calculate the lever arm for the horizontal force of the wall on the ladder.