A board of mass m and length L is leaning against a wall. The wall is frictionless, and the coefficient of static friction between the board and the floor is µ. Find an expression for the minimum angle β at which the board can be leaned without slipping.
To find the minimum angle β at which the board can be leaned without slipping, we need to consider the forces acting on the board. Let's break down the problem into different force components:
1. Weight (W): The force exerted by gravity acting vertically downward. Its magnitude is given by W = mg, where m is the mass of the board and g is the acceleration due to gravity.
2. Normal force (N): The force exerted by the floor on the board perpendicular to its surface. This force acts vertically upward and is equal in magnitude to the weight of the board when in equilibrium.
3. Friction force (F_friction): The force exerted by the floor on the board parallel to its surface. This force opposes the tendency of the board to slip. Its maximum possible magnitude is given by F_friction = µN, where µ is the coefficient of static friction between the board and the floor.
4. Contact force (F_contact): The force exerted by the wall on the board. Since the wall is assumed to be frictionless, the contact force is perpendicular to the wall and has no horizontal component.
Considering the forces in the horizontal direction, we can use the condition for equilibrium:
F_friction = F_contact
µN = F_contact (1)
Considering the forces in the vertical direction, we have:
N + F_contact = W = mg (2)
To find the minimum angle β, we can consider two extreme cases:
Case 1: To find the minimum angle of inclination, we assume that slipping is just about to occur. In this case, the friction force F_friction reaches its maximum value µN.
Case 2: To find the maximum angle of inclination, we assume that slipping is not occurring. In this case, the friction force F_friction is less than the maximum value µN.
Considering Case 1, we substitute F_friction = µN in equation (2):
N + F_contact = mg
N = mg - F_contact
Substituting this in equation (1):
µ(mg - F_contact) = F_contact
µmg - µF_contact = F_contact
µmg = (µ + 1)F_contact
F_contact = µmg / (µ + 1)
Now, considering Case 2, where slipping is not occurring, the friction force F_friction is less than µN. Therefore, it is given by:
F_friction = µ'mg
where µ' is a reduced coefficient of static friction that is less than µ.
Comparing this to equation (1), we have:
F_friction = µ'mg = µN = µ(mg - F_contact)
µmg - µF_contact = µ'mg
µF_contact = (µ - µ')mg
F_contact = (µ - µ')mg / µ
For the board to be on the verge of slipping, we compare the maximum and minimum values of F_contact:
(µ + 1)F_contact = (µ - µ')mg / µ
µ(µ + 1)F_contact = (µ - µ')mg
Simplifying, we get:
µ + 1 = (µ - µ') / µ
µ + 1 = 1 - (µ' / µ')
µ + µ' = µ - µ'
2µ' = µ - µ'
µ' = µ / 3
Finally, to find the minimum angle β, we use the trigonometric relationship:
tan(β) = L / h
where h is the height of the board.
Since the board is assumed to be leaning against a wall, h is equal to L sin(β). Substituting this in:
tan(β) = L / (L sin(β))
tan(β) = 1 / sin(β)
Simplifying, we find:
sin(β) = 1 / sqrt(2)
β = arcsin(1 / sqrt(2))
Therefore, the expression for the minimum angle β at which the board can be leaned without slipping is β = arcsin(1 / sqrt(2)).