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

Question

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

Answer 1

To find the maximum distance the painter can climb up the ladder, we need to consider the equilibrium condition of the ladder at the point where it is on the verge of slipping.

Let's break down the forces acting on the ladder and the painter:

1. Weight (Wp): The weight of the painter acting downwards, with a magnitude of (mass of painter x acceleration due to gravity) = (85.0 kg x 9.8 m/s^2) = 833 N.
2. Weight (Wl): The weight of the ladder acting downwards, with a magnitude of (mass of ladder x acceleration due to gravity) = (37.0 kg x 9.8 m/s^2) = 363 N.
3. Normal force (N): The force exerted by the ground on the ladder, perpendicular to the ground. Its magnitude is equal to the vertical component of the weight of the ladder and painter, which is N = Wp + Wl x cos(θ).
N = 833 N + 363 N x cos(51.0°) = 833 N + 363 N x 0.623 = 833 N + 226 N = 1059 N.

Now let's consider the maximum frictional force (Ff) that can be exerted between the ladder and the ground before slipping occurs:

4. Frictional force (Ff): The force that opposes the impending motion of slipping. Its magnitude is given by the coefficient of friction (μ) multiplied by the normal force (N). Ff = μ x N.
Ff = 0.470 x 1059 N = 498 N.

To find the maximum distance the painter can climb up the ladder, we need to find the point at which the horizontal component of the ladder's weight is equal to the maximum frictional force:

5. Horizontal component of ladder's weight (Wl,h): The component of the weight of the ladder acting parallel to the ground.
Wl,h = Wl x sin(θ) = 363 N x sin(51.0°) = 290 N.

At the point of maximum distance, this horizontal component of the ladder's weight is equal to the maximum frictional force:

6. Wl,h = Ff
290 N = 498 N

Since the horizontal component of the ladder's weight is less than the maximum frictional force, slipping will not occur. Therefore, the painter can climb up the ladder until he reaches its top end without slipping.

Hence, the maximum distance the painter can climb up the ladder is equal to the length of the ladder = 5.80 m.