Static equilibrium of window cleaner

[brookbj]

A window cleaner of mass 95 Kg places a 22 KG ladder against a frictionless wall, at an angle 65 degrees with the horizontal. The ladder is 10 m long and rests on a wet floor with a coefficient of static friction equal to 0.40 What is the maximum length that the window cleaner can climb before the ladder slips.

I know I need to use Torque, in both X and Y directions, but how do I find the length that the painter can climb.

thanks

 

[tiny-tim]

I know I need to use Torque, in both X and Y directions, but how do I find the length that the painter can climb.

Hi brookbj!

(torque doesn't come in x and y directions … it's just a number … or, in 3D, it's a vector in the z direction )

Hint: take moments (torques) about the top of the ladder.

 

[baba89]

To find the maximum length that the window cleaner can climb before the ladder slips, we first need to calculate the maximum torque that the ladder can withstand before slipping. This can be done by considering the forces acting on the ladder: the weight of the window cleaner and the ladder itself, the normal force from the floor, and the friction force from the floor.

Since the ladder is in static equilibrium, the sum of the forces in the horizontal and vertical directions must be equal to zero. This means that the vertical component of the weight of the window cleaner and ladder must be equal to the normal force from the floor, and the horizontal component of the weight must be equal to the friction force from the floor.

Using trigonometry, we can calculate the vertical component of the weight as 95 kg * 9.8 m/s^2 * cos(65 degrees) = 428.38 N. The normal force from the floor must therefore also be 428.38 N.

To calculate the friction force, we use the coefficient of static friction and the normal force: 0.40 * 428.38 N = 171.35 N.

Now, to find the maximum torque that the ladder can withstand before slipping, we use the equation torque = force * distance. Since the ladder is 10 m long, the maximum torque is 171.35 N * 10 m = 1713.5 Nm.

To find the length that the window cleaner can climb, we use the fact that the ladder will start to slip when the torque applied by the window cleaner's weight exceeds the maximum torque calculated above. So, we set up the equation: weight of window cleaner * distance climbed = maximum torque.

Since we want to find the distance climbed, we rearrange the equation to: distance climbed = maximum torque / weight of window cleaner. Plugging in the values, we get: distance climbed = 1713.5 Nm / 95 kg = 18.04 m.

Therefore, the maximum length that the window cleaner can climb before the ladder slips is 18.04 m.