A 4 meter long ladder weighing 25kg rest 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 for it to be inclined at 60 degree with the horizontal without slipping?
To determine the minimum coefficient of friction needed for the ladder to not slip, we need to analyze the forces acting on the ladder. In this case, we have two forces: the weight of the ladder acting downwards and the friction force acting upwards.
Let's break down the forces acting on the ladder:
1. Weight of the ladder (acting downwards): The weight of the ladder can be calculated using the formula: weight = mass * gravitational acceleration. Given that the ladder weighs 25kg, and the gravitational acceleration is approximately 9.8 m/s^2, the weight of the ladder is 25kg * 9.8 m/s^2 = 245N.
2. Friction force (acting upwards): The friction force opposes the ladder's tendency to slide down due to gravity. The maximum value of the friction force can be given by the formula: friction force = coefficient of friction * normal force. In this case, the normal force is the force exerted by the ground perpendicular to the ladder, which counteracts the weight of the ladder.
To determine the normal force, we need to consider the equilibrium of forces acting in the vertical direction. At 60 degrees inclination, the vertical component of the weight (245N) is given by: vertical component = weight * sin(theta). Substituting theta = 60 degrees, the vertical component of the weight is 245N * sin(60°) = 212.13N.
Since the ladder is at rest, the normal force provided by the ground is equal to the vertical component of the weight: normal force = 212.13N.
Now, substituting the values into the formula for the friction force: friction force = coefficient of friction * normal force. We want to find the minimum coefficient of friction required for the ladder to not slip. Thus, to prevent the ladder from slipping, the friction force should be equal to the maximum available friction force.
At the point of impending slipping, the maximum available friction force can be found using the formula: maximum friction force = coefficient of static friction * normal force. The coefficient of static friction represents the roughness between the ladder and the ground, and it represents the maximum friction force when the ladder is just about to slip.
Since we're looking for the minimum coefficient of friction, we can set the maximum friction force equal to the friction force: coefficient of static friction * normal force = coefficient of friction * normal force.
Thus, the minimum coefficient of friction required for the ladder to be inclined at 60 degrees without slipping is equal to the coefficient of static friction. Therefore, the minimum coefficient of static friction between the ground and the ladder is 1.