A uniform 12 kg ladder 2.8 m long rests against a vertical frictionless wall with it's lower end 1.1 m from the wall, if the ladder is to stay in place. What must be minimum coefficient of friction between the bottom of the ladder and the ground
To find the minimum coefficient of friction between the bottom of the ladder and the ground for it to stay in place, we need to consider the forces acting on the ladder.
First, let's determine the forces acting on the ladder:
1. Weight (W): The weight of the ladder acts downward from its center of gravity and can be calculated using the formula W = m * g, where m is the mass of the ladder and g is the acceleration due to gravity.
Given that the ladder has a mass of 12 kg, we can calculate the weight:
W = 12 kg * 9.8 m/s²
W = 117.6 N
2. Normal Force (N): The normal force acts perpendicular to the plane of contact, which in this case is the ground pushing back on the ladder. Since there is no vertical acceleration, the normal force is equal in magnitude and opposite in direction to the weight of the ladder. Therefore, the normal force is also 117.6 N.
Now let's analyze the forces in the horizontal direction:
1. Applied Force (F): Assuming the ladder is at rest, there must be a horizontal force counteracting any tendency to slide. This force is the frictional force between the ladder and the ground and can be calculated using the formula F = μ * N, where μ is the coefficient of friction and N is the normal force.
Since the ladder is on the verge of slipping and the frictional force is maximum, the applied force is equal to the static frictional force. Therefore:
F = μ * N
We know that the normal force (N) is 117.6 N, so the applied force can be written as:
F = μ * 117.6 N
2. Horizontal Component of Weight (W_h): The horizontal component of the weight acts in the opposite direction to the applied force and can be calculated using the formula W_h = W * sin(θ), where θ is the angle between the ladder and the ground.
The ladder forms a right triangle with the wall and the ground, and the angle θ can be determined using trigonometry. The opposite side is 1.1 m, and the hypotenuse (the length of the ladder) is 2.8 m. Therefore, sin(θ) = opposite/hypotenuse:
sin(θ) = 1.1 m / 2.8 m
sin(θ) ≈ 0.393
Now we can calculate the horizontal component of weight:
W_h = 117.6 N * sin(θ)
W_h ≈ 117.6 N * 0.393
W_h ≈ 46.2 N
For the ladder to remain at rest, the applied force (F) should be equal to the horizontal component of weight (W_h):
μ * 117.6 N = 46.2 N
Rearranging the equation to solve for the coefficient of friction (μ):
μ = (46.2 N) / (117.6 N)
μ ≈ 0.393
Therefore, the minimum coefficient of friction between the bottom of the ladder and the ground for it to stay in place is approximately 0.393.
To determine the minimum coefficient of friction between the bottom of the ladder and the ground, we can analyze the forces acting on the ladder.
1. Calculate the weight of the ladder:
The weight of the ladder can be calculated using the formula:
Weight = mass * acceleration due to gravity
Weight = 12 kg * 9.8 m/s^2
Weight = 117.6 N
2. Calculate the force exerted by the ladder:
The force exerted by the ladder can be calculated using the formula:
Force = weight * (distance from the bottom end to the center of mass / length of the ladder)
Force = 117.6 N * (1.4 m / 2.8 m)
Force = 58.8 N
3. Calculate the force normal to the ladder:
The force normal to the ladder is equal to the force exerted by the ladder (since there is no vertical acceleration).
Force normal = 58.8 N
4. Calculate the force parallel to the ladder:
The force parallel to the ladder is the reaction force caused by the friction between the bottom of the ladder and the ground. This force opposes the motion of the ladder. Therefore,
Force parallel = force normal * friction coefficient
5. Determine the minimum coefficient of friction:
To find the minimum coefficient of friction, we need to consider the equilibrium condition. Since the ladder is in equilibrium, the net force acting on the ladder in the horizontal direction is zero (no acceleration horizontally).
Net force = force parallel - force applied = 0
force parallel - 0 = 0
force parallel = 0
Since the force parallel is zero, it means that the reaction force caused by the friction between the bottom of the ladder and the ground is zero.
Therefore, the minimum coefficient of friction between the bottom of the ladder and the ground is 0.