Final answer:
To determine the distance at which a vertical downward force can be applied to balance the system, we need to consider the equilibrium conditions.
The reaction force from the vertical wall is negligible, and the friction force between the end B of the bar and the block C can be calculated using the coefficient of friction and the normal force.
By setting the sum of the weight of block C and the friction force equal to the vertical downward force, we can solve for the distance λ, which is approximately 1.92 m.
Step-by-step explanation:
To determine at what distance λ the vertical downward force of 200 N can be applied on the bar to ensure the balance of the system, we need to consider the equilibrium conditions. First, we need to find the reaction force from the vertical wall. Since the bar rests against the smooth vertical wall, the reaction force from the wall will be perpendicular to the wall and equal in magnitude to the horizontal component of the weight of the bar.
The weight of the bar can be calculated using the equation:
Weight of bar = mass of bar * acceleration due to gravity
Since the weight of the bar is negligible, the reaction force from the wall will also be negligible.
Next, we need to find the friction force between the end B of the bar and the weight block C. The friction force can be calculated using the equation:
Friction force = coefficient of friction * normal force
The normal force is equal to the weight of the block C, which is 100 N.
The friction force between end B of the bar and block C is given by the equation:
Friction force = coefficient of friction * normal force
Using the given coefficient of static friction between end B and block C (0.4), the friction force can be calculated as:
Friction force = 0.4 * 100 N = 40 N
Since the bar is in equilibrium, the sum of all the vertical forces must equal zero. This means that the vertical downward force of 200 N can be balanced by the sum of the weight of block C and the friction force between end B and block C.
Let λ be the distance measured in the direction of the bar from end B. The weight of block C can be represented by the equation:
Weight of block C = λ * (weight of block C / total length of the bar)
Using the given length of the bar (3 m) and weight of block C (100 N), this equation can be simplified to:
Weight of block C = λ * (100 N / 3 m)
The sum of the weight of block C and the friction force must equal the vertical downward force of 200 N:
Weight of block C + Friction force = 200 N
Substituting the expressions for the weight of block C and the friction force into the equation:
λ * (100 N / 3 m) + 40 N = 200 N
Simplifying the equation:
λ * (100 N / 3 m) = 160 N
Dividing both sides of the equation by 100 N / 3 m:
λ = 1.92 m
Therefore, the vertical downward force of 200 N can be applied at a distance of 1.92 m measured from end B of the bar to ensure the balance of the system.