A ladder of length 5m has mass of 25 Kg. the ladder is leaned against a frictionless vertical wall at an angle of 10 degrees with the vertical. A repairman with a mass of 82Kg needs to stand on an upper rung of the ladder that is 1 m from the end. What is the minimum coefficient of friction between the ladder and the floor such that the ladder doesn’t slip?
Thank you very much
To find the minimum coefficient of friction between the ladder and the floor to prevent slipping, we need to analyze the forces acting on the ladder.
Let's break down the problem step by step:
Step 1: Identify the forces acting on the ladder:
- The weight of the ladder (acting downwards) can be calculated as the product of the mass (25 kg) and the acceleration due to gravity (9.8 m/s^2).
- The normal force (acting perpendicular to the surface of contact) applies an equal and opposite force to support the ladder's weight.
Step 2: Split the weight of the ladder into components:
- Since the ladder is leaning against the wall, we need to resolve its weight into two components: a vertical component and a horizontal component.
- The vertical component is given by the weight of the ladder multiplied by the cosine of the angle (10 degrees).
- The horizontal component is given by the weight of the ladder multiplied by the sine of the angle (10 degrees).
Step 3: Consider the torque acting on the ladder:
- Torque is the amount of force applied around a pivot point and can cause an object to rotate.
- In this case, the pivot point is the contact between the ladder and the floor.
- The ladder will start to slip if the torque due to the horizontal component of the ladder's weight is greater than the product of the frictional force and the distance from the pivot point to the point where the repairman stands.
Step 4: Calculate the torque due to the horizontal component:
- The torque due to the horizontal component is given by the product of the horizontal component of the ladder's weight and its distance from the pivot point (1 m).
Step 5: Calculate the frictional force:
- The frictional force can be calculated by multiplying the coefficient of friction between the ladder and the floor by the normal force.
- We need to find the minimum coefficient of friction that prevents slipping, so the frictional force should be equal to the torque due to the horizontal component.
Step 6: Set up the equation and solve for the coefficient of friction:
- Equate the torque due to the horizontal component to the frictional force.
- Substitute the expression for the frictional force and solve for the coefficient of friction.
Following these steps, we can find the minimum coefficient of friction between the ladder and the floor such that the ladder doesn't slip.
Require that the moment about the upper support point be zero. The vertical force component at the lower end of the ladder is (25 + 82)*g = 1049 J, since the upper support point is frictionless.
The distance of the lower end of the ladder from the wall is
X = 5m * sin 10 = 0.868 m.
The repairman stands in the ladder at a horizontal distance of 1.0 m sin 10 = 0.1736 m from the wall.
Assume the Center of Mass of the ladder is midway between the ends. The CM is then 5 sin 10 = 0.868 m from the wall.
25*0.868*g + 82*0.1736*g = 107*g*Us(5 cos10
g cancels out. Solve for the required static friction coefficient Us