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 1m from the end. What is the minimum coefficient of friction between the ladder and the floor such that the ladder doesn’t slip?
To find the minimum coefficient of friction between the ladder and the floor, we need to analyze the forces acting on the ladder.
Let's consider the forces acting on the ladder along the horizontal and vertical directions:
Horizontal forces:
1. The weight of the ladder acts vertically downward from its center of mass.
2. The normal force from the floor acts vertically upward from the contact point between the ladder and the floor.
3. The frictional force acts horizontally from the contact point between the ladder and the floor.
Vertical forces:
1. The weight of the ladder acts vertically downward from its center of mass.
2. The normal force from the wall acts vertically upward from the contact point between the ladder and the wall.
To start, let's analyze the horizontal forces. Because the ladder is not slipping, the frictional force balances the horizontal component of the weight of the ladder.
Horizontal forces equation:
Frictional force = Horizontal component of the ladder's weight
The horizontal component of the ladder's weight can be determined using trigonometry:
Horizontal component of weight = Weight of the ladder * sin(angle of inclination)
Substituting the given values:
Horizontal component of weight = 25 kg * sin(10 degrees)
Next, we need to consider the vertical forces acting on the ladder. The weight of the repairman also acts vertically downward from the point where he stands on the ladder.
Vertical forces equation:
Weight of the repairman + Vertical component of the ladder's weight = Normal force from the wall
The normal force from the wall can be determined using trigonometry:
Normal force from the wall = Weight of the ladder * cos(angle of inclination)
Substituting the given values:
Normal force from the wall = 25 kg * cos(10 degrees)
Lastly, to prevent the ladder from slipping, the frictional force should be equal to or greater than the vertical component of the ladder's weight plus the weight of the repairman.
Therefore, the minimum coefficient of friction can be obtained by dividing the frictional force by the normal force from the floor:
Minimum coefficient of friction = Frictional force / (Weight of the ladder * cos(angle of inclination))
Substituting the calculated values, we can find the minimum coefficient of friction.
To determine the minimum coefficient of friction between the ladder and the floor such that the ladder doesn't slip, we need to consider the forces acting on the ladder.
1. First, let's determine the forces acting on the ladder:
- Weight of the ladder (W_l): mass (m_l) x acceleration due to gravity (g)
- Weight of the repairman (W_r): mass (m_r) x acceleration due to gravity (g)
- Normal force exerted by the floor (N)
- Frictional force between the ladder and the floor (F_f)
2. Next, let's calculate the forces using the given information:
- Weight of the ladder: W_l = m_l x g = 25 kg x 9.8 m/s^2 = 245 N
- Weight of the repairman: W_r = m_r x g = 82 kg x 9.8 m/s^2 = 803.6 N
3. Now, let's analyze the forces acting on the ladder:
- Vertical forces:
- N - W_l - W_r x cos(10 degrees) = 0 (since the ladder is in equilibrium in the vertical direction)
- Horizontal forces:
- F_f - W_r x sin(10 degrees) = 0 (since the ladder is in equilibrium in the horizontal direction)
4. Rearranging the equations, we can solve for N and F_f:
- N = W_l + W_r x cos(10 degrees)
= 245 N + 803.6 N x cos(10 degrees)
≈ 1015.9 N
- F_f = W_r x sin(10 degrees)
= 803.6 N x sin(10 degrees)
≈ 139.4 N
5. Finally, we can calculate the coefficient of friction (μ) using the formula:
- μ = F_f / N
= 139.4 N / 1015.9 N
≈ 0.137
Therefore, the minimum coefficient of friction between the ladder and the floor such that the ladder doesn't slip is approximately 0.137.