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. (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)(d) Find the maximum distance the painter can climb up the ladder.

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Problem #7

When I followed that i got .26 meters and i think that is too small, right?

To answer part (c) of the question, we need to find the lever arm for the force of gravity acting on the ladder. The lever arm is the perpendicular distance between the line of action of the force and the axis of rotation. In this case, the axis of rotation is the point where the ladder touches the ground.

The force of gravity acts vertically downwards on the center of mass of the ladder, which is midway along its length. Since the ladder is uniform, the center of mass is located at the midpoint of the ladder.

To find the lever arm, we can use the concept of torque, which is the product of the force applied and the lever arm. The equation for torque is given by:

Torque = Force × Lever Arm

In this case, the force is the weight of the ladder and the lever arm is the distance between the midpoint of the ladder and the point of contact with the ground.

The weight of the ladder can be calculated using the equation:

Weight = mass × gravitational acceleration

where the gravitational acceleration is approximately 9.8 m/s².

Weight of the ladder = mass of the ladder × gravitational acceleration
= 21.0 kg × 9.8 m/s²

Now, to find the lever arm, we need to consider that the ladder is at an angle of 57.0° with the ground. We can use trigonometry to calculate the lever arm.

Lever arm = Length of the ladder × sin(θ)

Substituting the given values:

Lever arm = 2.80 m × sin(57.0°)

Calculating this value will give you the lever arm for the force of gravity acting on the ladder.

To answer part (d) of the question, we need to find the maximum distance the painter can climb up the ladder without it slipping. In this case, the maximum distance will be limited by the friction between the ladder and the ground.

The maximum distance can be calculated using the equation:

Maximum distance = Coefficient of friction × Normal force

where the normal force is the force exerted by the ground on the ladder in the vertical direction.

The normal force can be calculated using the equation:

Normal force = Weight of the ladder + Weight of the painter

The weight of the painter can be calculated using the same formula as before:

Weight of the painter = mass of the painter × gravitational acceleration
= 69.0 kg × 9.8 m/s²

Substituting the given values, you can calculate the normal force.

Once you have the normal force, you can use the given coefficient of friction to calculate the maximum distance.

Maximum distance = 0.290 × Normal force

Calculating this value will give you the maximum distance the painter can climb up the ladder.