A 4m ladder weighing 25 kg with its upper end against a smooth wall and lower end on rough ground.what should be the minimum coefficient of friction between the ground and the ladder it to be inclined at angle 60 with the horizontal without slipping?
To find the minimum coefficient of friction required to keep the ladder from slipping, we need to analyze the forces acting on the ladder.
Let's break down the forces on the ladder:
1. Weight (W): The weight of the ladder acts vertically downward from its center of mass. It can be calculated using the formula W = m * g, where m is the mass and g is the acceleration due to gravity. In this case, the weight is given as 25 kg, so W = 25 kg * 9.8 m/s^2 = 245 N.
2. Normal force (N): The normal force acts perpendicular to the surface and counterbalances the weight of the ladder. In this case, with the ladder leaning against the wall, the normal force acts at the point where the ladder makes contact with the ground. Therefore, N = W = 245 N.
3. Friction force (F): The friction force acts parallel to the surface of contact and opposes the impending motion between the ladder and the ground. Since the ladder is on rough ground, the friction force must be considered to prevent the ladder from slipping.
Now, let's consider the forces acting along the ladder:
1. Vertical component (Wv): The vertical component of the weight acts in the downward direction and can be determined by multiplying the weight by the sine of the angle at which the ladder is inclined. In this case, Wv = W * sin(60°) = 245 N * sin(60°) = 245 N * 0.866 = 212.97 N.
2. Horizontal component (Wh): The horizontal component of the weight acts perpendicular to the friction force and is equal to the force required to maintain equilibrium for the ladder. In this case, Wh = W * cos(60°) = 245 N * cos(60°) = 245 N * 0.5 = 122.5 N.
To determine the minimum coefficient of friction (μ) required, we need to consider the maximum friction force (Fmax) that can be exerted by the ground, which is equal to μ * N.
Since the ladder is in equilibrium, the horizontal component of the weight (Wh) should be equal to the maximum friction force (Fmax): Wh = Fmax.
Therefore, we can write the equation as follows:
Fmax = μ * N
Substituting the values we calculated earlier:
122.5 N = μ * 245 N
Simplifying the equation:
μ = 122.5 N / 245 N
μ = 0.5
Hence, the minimum coefficient of friction required between the ground and the ladder for it to be inclined at an angle of 60° without slipping is 0.5.