A ladder weighing 400N is 10m long. Its end ‘A’ is resting on smooth vertical wall and lower end ‘B’ is resting on rough horizontal floor having coefficient of friction is 0.4. The ladder makes an angle 45o with horizontal. The ladder is about to slip when a man weighing 600N standing at mid length of the ladder. Find reactions at supports A and B and limiting friction at the floor
To find the reactions at supports A and B and the limiting friction at the floor, we can use the principles of equilibrium.
First, let's draw a free-body diagram of the ladder. We have the weight of the ladder acting downward at its center, and the reaction forces at A and B. Additionally, we have the weight of the man acting downward at the midpoint of the ladder.
Considering the horizontal forces, the frictional force at B opposes the tendency of the ladder to slip. The ladder is about to slip, so the frictional force is at its maximum value, which is μN, where μ is the coefficient of friction and N is the normal force. The normal force at B is equal to the weight of the ladder plus the weight of the man.
Considering the vertical forces, at equilibrium, the sum of the vertical forces must be zero. The reaction force at A supports the weight of the ladder, the weight of the man, and the vertical component of the frictional force at B. The reaction force at B supports the vertical component of the ladder's weight.
Now, let's calculate the values.
Weight of the ladder = 400N
Weight of the man = 600N
Coefficient of friction (μ) = 0.4
Angle made by the ladder with the horizontal = 45 degrees
The vertical component of the ladder's weight can be calculated using the formula:
Vertical weight = Weight of the ladder * sin(θ)
Vertical weight = 400N * sin(45°)
The reaction force at B is equal to the vertical weight:
Reaction force at B = Vertical weight = 400N * sin(45°)
The normal force at B is:
Normal force at B = Weight of the ladder + Weight of the man
The frictional force at B is:
Frictional force at B = Coefficient of friction * Normal force at B
The reaction force at A can be calculated using the equation:
Reaction force at A = Weight of the ladder + Weight of the man + Vertical component of frictional force at B
Finally, you can substitute the known values into these equations to find the reactions at supports A and B and the limiting friction at the floor.